Fundamentals of Mathematics

🧭 Overview

🧠 One-sentence thesis

The "Addition and Subtraction of Whole Numbers" chapter equips students to understand the Hindu-Arabic base-ten system, perform addition and subtraction operations, round numbers, and apply the commutative and associative properties of addition.

📌 Key points (3–5)

• Number system foundation: Students learn the Hindu-Arabic numeration system, the base-ten positional system, and how to identify and graph whole numbers.
• Core operations: The chapter covers both the conceptual understanding and mechanical skills for adding and subtracting whole numbers, including calculator use.
• Approximation technique: Rounding is introduced as a method of approximation to a specified position.
• Algebraic properties: The commutative and associative properties of addition are presented, along with zero as the additive identity.
• Common confusion: Distinguishing between numbers (abstract quantities) and numerals (written symbols) is explicitly addressed.

🔢 Number system foundations

🔢 Numbers vs numerals

The chapter begins by clarifying a fundamental distinction:

• Students will "know the difference between numbers and numerals."
• Numbers are abstract quantities; numerals are the written symbols we use to represent them.
• This distinction helps students understand that the same number can be written in different ways (e.g., Roman numerals vs Arabic numerals).

🌍 Hindu-Arabic numeration system

The Hindu-Arabic numeration system: the number system used in this text and in everyday life.

• Students will "know why our number system is called the Hindu-Arabic numeration system."
• This historical context grounds the student's understanding of the notation they use daily.

🏗️ Base-ten positional system

• Students will "understand the base ten positional number system."
• In a positional system, the location of a digit determines its value (e.g., the "2" in "20" means twenty, not two).
• The base-ten system uses ten digits (0–9) and each position represents a power of ten.

📍 Identifying and graphing whole numbers

• Students will "be able to identify and graph whole numbers."
• Graphing typically means placing numbers on a number line, which visualizes their relative size and order.

📖 Reading, writing, and rounding

📖 Reading and writing whole numbers

• Students will "be able to read and write a whole number."
• This skill ensures students can translate between spoken language (e.g., "three hundred forty-two") and written numerals (342).
• Example: A student should be able to write "one thousand five" as 1,005, not 1,5.

📏 Rounding as approximation

Rounding: a method of approximation.

• Students will "understand that rounding is a method of approximation" and "be able to round a whole number to a specified position."
• Rounding simplifies numbers for easier estimation or communication.
• "Specified position" means rounding to the nearest ten, hundred, thousand, etc.
• Example: Rounding 347 to the nearest hundred gives 300; rounding to the nearest ten gives 350.

➕ Addition of whole numbers

➕ The addition process

• Students will "understand the addition process."
• Understanding the process means grasping why addition works, not just memorizing steps.
• The text emphasizes conceptual understanding alongside mechanical skill.

➕ Performing addition

• Students will "be able to add whole numbers."
• This includes multi-digit addition with carrying (regrouping).
• Students will also "be able to use the calculator to add one whole number to another."
• Don't confuse: understanding the process vs using a calculator—both are valuable, but the first ensures the student knows what the calculator is doing.

➖ Subtraction of whole numbers

➖ The subtraction process

• Students will "understand the subtraction process."
• Like addition, the focus is on both conceptual understanding and technique.

➖ Performing subtraction

• Students will "be able to subtract whole numbers."
• This includes multi-digit subtraction with borrowing (regrouping).
• Students will also "be able to use a calculator to subtract one whole number from another whole number."
• Example: Subtracting 47 from 123 should be understood as "how much remains when 47 is removed from 123," not just a mechanical procedure.

🔄 Properties of addition

🔄 Commutative and associative properties

Property	What it means	Example (in words)
Commutative	Order doesn't matter	Adding A to B gives the same result as adding B to A
Associative	Grouping doesn't matter	Adding A, then B, then C gives the same result regardless of which pair you add first

• Students will "understand the commutative and associative properties of addition."
• These properties are foundational for algebra and mental arithmetic.
• Example: Commutative means 3 + 5 = 5 + 3; associative means (2 + 3) + 4 = 2 + (3 + 4).

🔄 Zero as the additive identity

The additive identity: the number that, when added to any number, leaves that number unchanged.

• Students will "understand why 0 is the additive identity."
• Adding zero to any whole number does not change the number.
• Example: 7 + 0 = 7 and 0 + 7 = 7.
• This property is important for understanding algebraic equations later.

🧭 Overview

🧠 One-sentence thesis

The Hindu-Arabic numeration system uses ten digits in a base-ten positional framework to represent whole numbers, where each position's value is ten times the preceding position moving right to left.

📌 Key points (3–5)

• Numbers vs numerals: A number is a mental concept; a numeral is the symbol used to represent it (though common usage treats them interchangeably).
• Hindu-Arabic system origin: Invented by Hindus around A.D. 500 with digits 0–9, then introduced to Europe by Leonardo Fibonacci in the 13th century and popularized by Arabs.
• Base-ten positional system: Each position has a value ten times the preceding position (ones, tens, hundreds, thousands, etc.), grouped into periods of three digits.
• Common confusion: The value of a digit depends on its position—the digit 5 means different amounts in 50 vs 500 vs 5,000.
• Whole numbers: Numbers formed using only digits 0–9, starting from 0 and continuing indefinitely (0, 1, 2, 3, ...).

🔢 Numbers versus numerals

🧠 Number as concept

A number is a concept. It exists only in the mind.

• Numbers are mental pictures of the size of collections.
• They cannot be written down directly—only represented.

✍️ Numeral as symbol

A numeral is a symbol that represents a number.

• Different systems use different symbols for the same number:
  • Hindu-Arabic: 4, 123, 1005
  • Roman: IV, CXXIII, MV
  • Egyptian: (various symbols)
• Written words like "four" or "one hundred twenty-three" also qualify as numerals because letters are symbols.
• Don't confuse: In everyday use, "number" and "numeral" are used interchangeably, and the excerpt follows this convention.

🌍 The Hindu-Arabic system

🕰️ Historical development

The Hindu-Arabic numeration system uses the digits 0 1 2 3 4 5 6 7 8 9.

• Origin: Hindus invented these numerals shortly before the third century.
• Spread to Europe: Leonardo Fibonacci of Pisa introduced the system to Europe around the 13th century (about a thousand years later).
• Name: Called "Hindu-Arabic" because it was popularized by the Arabs after its Hindu invention.

🔟 Why "base ten positional"

The Hindu mathematician who devised the system about A.D. 500 stated that "from place to place each is ten times the preceding."

• Digits: The ten symbols (0–9) are called digits.
• Positional: The same digit has different values depending on where it appears.
• Base ten: Each position is worth ten times the position to its right.

📍 Position values and periods

📊 How positions work

Each position has a specific value that increases by factors of ten from right to left:

Position (right to left)	Value
First	Ones
Second	Tens
Third	Hundreds
Fourth	Thousands
Fifth	Ten thousands
Sixth	Hundred thousands

📦 Periods (groups of three)

Commas are sometimes used to separate digits into groups of three. These groups of three are called periods.

• Each period contains three positions: ones, tens, and hundreds.
• Period names (right to left): units, thousands, millions, billions, trillions.
• Example: In 86,932,106,005, the digit 9 is in the hundreds position of the millions period, so its value is "9 hundred millions" or "9 hundred million."

🎯 Finding digit values

The value of a digit = its face value × its position value.

• Example: In 7,261, the digit 6 is in the tens position, so its value is 6 tens = 60.
• Example: In 102,001, the digit 2 is in the ones position of the thousands period, so its value is 2 one thousands = 2 thousand.
• Example: In 108, the digit 0 is in the tens position, so its value is zero tens = zero.

🔢 Whole numbers definition

🔢 What whole numbers are

Numbers that are formed using only the digits 0 1 2 3 4 5 6 7 8 9 are called whole numbers.

• The set: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …
• The three dots mean "and so on in this same pattern."
• They continue indefinitely—there is no largest whole number.

📈 Graphing whole numbers

📏 The number line

A number line is constructed by drawing a straight line and choosing any point on the line and labeling it 0.

• Origin: The point labeled 0 is called the origin.
• Construction: Mark off equal intervals to the right of 0, labeling each endpoint with the next whole number.
• The line extends indefinitely to the right.

🎯 How to graph

Graphing means to "visually display."

• To graph a whole number, draw a closed circle at the point labeled with that number.
• Example: To graph 3, 5, and 9, place closed circles at those three points on the number line.
• The number line can show breaks (indicated by a gap) to skip over unlisted numbers when space is limited.
• Example: A line showing 0, 106, 873, 874 would have breaks between 0 and 106, and between 106 and 873.

🔍 Reading graphs

• To specify which whole numbers are graphed, identify all points marked with closed circles.
• Example: If circles appear at 4, 5, 6, 113, and 978, those are the graphed whole numbers.

🧭 Overview

🧠 One-sentence thesis

Reading and writing whole numbers is straightforward in our positional number system because we use commas to separate periods and read each period individually with its name.

📌 Key points (3–5)

• How to read whole numbers: insert commas every three digits from right to left to mark periods, then read each period from left to right with its period name.
• How to write whole numbers: use commas in the verbal phrase as period separators, write each period in digits, then combine them with commas.
• Period structure: every period has the same three positions (ones, tens, hundreds) but different period names (ones, thousands, millions, billions, trillions).
• Common confusion: missing periods—when writing numbers from words, you must insert zeros for any period not mentioned (e.g., "ten million, five hundred twelve" needs zeros for the thousands period).
• Why it matters: our base-10 positional system makes reading and writing large numbers systematic and consistent.

📖 Reading whole numbers

📖 The two-step method

The excerpt provides a clear procedure:

1. Right to left: starting at the right, insert commas every three digits to separate periods.
2. Left to right: starting at the left, read each period individually and say the period name.

• This method works because our number system groups digits into periods of three.
• Example: 42958 becomes 42,958 → "Forty-two thousand, nine hundred fifty-eight."

🔢 Reading multi-period numbers

• Each comma marks a new period boundary.
• The period names from right to left are: ones (no name spoken), thousands, millions, billions, trillions.
• Example: 307,991,343 is read as "Three hundred seven million, nine hundred ninety-one thousand, three hundred forty-three."
• Don't confuse: you read the digits within each period as a group, then attach the period name; you don't read digit by digit.

🕳️ Handling zeros in periods

• If a period contains only zeros, you skip that period name entirely.
• Example: 36,000,000,000,001 becomes "Thirty-six trillion, one" (the billions, millions, and thousands periods are all zero, so they are omitted).

✍️ Writing whole numbers

✍️ The three-step method

To convert words to digits:

1. Identify periods: commas in the verbal phrase mark period boundaries.
2. Write each period: convert each period's words into digits (three digits per period).
3. Combine: use commas to join the periods into one number.

• Example: "Seven thousand, ninety-two" → the comma separates thousands from ones → 7,092.
• Example: "Fifty billion, one million, two hundred thousand, fourteen" → 50,001,200,014.

🕳️ Inserting missing periods

• If a period is not mentioned in the verbal phrase, you must insert zeros to hold that period's place.
• Example: "Ten million, five hundred twelve" has no thousands period mentioned → you write 10,000,512 (three zeros for the missing thousands).
• Don't confuse: the absence of a period name in words does not mean you skip those digit positions; you fill them with zeros.

📋 Period-by-period breakdown

Period name	Position in number (right to left)	Example digits
Ones	1st group (rightmost)	512
Thousands	2nd group	080
Millions	3rd group	003
Billions	4th group	020
Trillions	5th group	(and so on)

• Each period always occupies exactly three digit positions.
• When writing, if a period is mentioned, write its digits; if not mentioned, write 000.

🧩 Understanding the positional system

🧩 Why commas matter

• Commas are not arbitrary; they mark the boundaries of periods in our base-10 positional system.
• Every three digits from the right forms a new period with its own name.
• This structure is consistent: ones/tens/hundreds repeat in every period, but the period name changes.

🧩 Period names and values

• The excerpt lists period names from right to left: ones, thousands, millions, billions, trillions.
• Within each period, the three positions are always (from right to left): ones, tens, hundreds.
• Example: in 42,958, the "42" is in the thousands period, so it represents "42 thousands" or "forty-two thousand."

🧩 The role of zero

• Zero holds a place when a position has no value.
• Example: in 10,046, the zero in the hundreds position of the ones period means "zero hundreds," and the zero in the thousands position means "zero thousands."
• Don't confuse: zero is not "nothing"—it is a placeholder that keeps other digits in their correct positions.

🔍 Common patterns and pitfalls

🔍 Reading vs writing symmetry

• Reading: digits → commas → words (right-to-left grouping, then left-to-right reading).
• Writing: words → commas → digits (commas in words guide period placement).
• The comma is the key structural element in both directions.

🔍 Handling large numbers

• For very large numbers, the method remains the same: separate into periods, read or write each period.
• Example: 20,003,080,109,402 is "Twenty trillion, three billion, eighty million, one hundred nine thousand, four hundred two."
• The excerpt shows that even numbers with many periods follow the same pattern.

🔍 When periods are missing

• In reading: if a period is all zeros, omit it from the verbal phrase.
• In writing: if a period name is absent in the verbal phrase, insert 000 in that period's position.
• Example: "Eighty billion, thirty-five" → 80,000,000,035 (millions and thousands periods are missing, so insert six zeros).

🧭 Overview

🧠 One-sentence thesis

Rounding is a systematic method of approximation that replaces precise whole numbers with nearby values grouped by tens, hundreds, thousands, or higher place values, making numbers easier to work with when exact counts are unnecessary.

📌 Key points (3–5)

• What rounding does: approximates a precise number by thinking of it in groups of tens, hundreds, thousands, etc., rather than exact counts.
• The core method: mark the round-off digit (the place you're rounding to), then look at the digit immediately to its right—if it's less than 5, round down; if it's 5 or greater, round up.
• The halfway convention: when a number is exactly halfway between two values, always round to the higher number.
• Common confusion: rounding to a place value higher than the number itself (e.g., rounding 39 to the nearest hundred) results in 0, not an error.
• Why it matters: rounding is very useful in estimation and everyday communication (e.g., describing salary or population in simpler terms).

🔍 What rounding is and why we use it

🔍 Approximation vs. precision

Rounding: the process of approximation by mentally seeing a collection as occurring in groups of tens, hundreds, thousands, etc.

• Sometimes we only need the approximate number of objects in a collection, not the precise count.
• Example: a collection of 18 symbols might be described as "approximately 20."
• The excerpt emphasizes that rounding is a primary tool for approximation, not for exact counting.

📊 Rounding to different place values

• Rounding to the nearest ten: thinking of numbers in groups of tens.
• Rounding to the nearest hundred: thinking of numbers in groups of hundreds.
• This idea continues through thousands, ten thousands, hundred thousands, millions, etc.
• Example: a person earning $41,450 per year might say she makes "$41,000 per year"—she has rounded to the nearest thousand because 41,450 is closer to 41,000 than to 42,000.

🧮 The method of rounding whole numbers

🧮 Two-step procedure

The excerpt provides a systematic method:

1. Mark the position of the round-off digit (the place value you are rounding to).
2. Note the digit immediately to the right of the round-off digit:
   • If it is less than 5: replace it and all digits to its right with zeros; leave the round-off digit unchanged (round down).
   • If it is 5 or larger: replace it and all digits to its right with zeros; increase the round-off digit by 1 (round up).

🔢 The "less than 5" rule (round down)

• When the digit to the right of the round-off position is less than 5, you round down.
• Example: rounding 4,329 to the nearest hundred—the round-off digit is 3 (hundreds place), and the digit to its right is 2. Since 2 < 5, replace 2 and 9 with zeros, leaving the 3 unchanged → 4,300.
• Example: rounding 9,614,018,007 to the nearest ten million—the round-off digit is 1 (ten millions place), and the digit to its right is 4. Since 4 < 5, replace 4 and all digits to its right with zeros → 9,610,000,000.

🔢 The "5 or larger" rule (round up)

• When the digit to the right of the round-off position is 5 or greater, you round up.
• Example: rounding 67 to the nearest ten—the round-off digit is 6 (tens place), and the digit to its right is 7. Since 7 ≥ 5, replace 7 with 0 and increase 6 by 1 → 70.
• Example: rounding 3,426 to the nearest ten—the round-off digit is 2 (tens place), and the digit to its right is 6. Since 6 ≥ 5, replace 6 with 0 and add 1 to 2 (2 + 1 = 3) → 3,430.

⚖️ The halfway convention

By convention, when the number to be rounded is exactly halfway between two numbers, it is rounded to the higher number.

• Example: rounding 16,500 to the nearest thousand—16,500 is exactly halfway between 16,000 and 17,000, so it rounds up to 17,000.
• Don't confuse: this is a convention, not a mathematical necessity; it ensures consistency when the number is equidistant from both options.

🧩 Special cases and carrying

🧩 Rounding to a place value higher than the number

• If you round a number to a place value it doesn't reach, the result is 0.
• Example: rounding 148,422 to the nearest million—imagine a 0 in the millions place: 0,148,422. The digit to the right of the millions place is 1. Since 1 < 5, round down by replacing all digits with zeros → 0 (which is just 0).
• Example: rounding 39 to the nearest hundred—imagine 0 in the hundreds place: 0,039. The digit to the right is 0. Since 0 < 5, round down → 0.

🧩 Carrying when rounding up

• Sometimes rounding up requires carrying to the next higher place value.
• Example: rounding 397,000 to the nearest ten thousand—the round-off digit is 9 (ten thousands place), and the digit to its right is 7. Since 7 ≥ 5, replace 7 and all digits to its right with zeros and add 1 to 9. But 9 + 1 = 10, so write 0 in the ten thousands place and carry 1 to the hundred thousands place → 400,000.
• Don't confuse: this is the same carrying process used in addition; it applies when the round-off digit is 9 and you need to round up.

📐 Using rounding in context

📐 Number line visualization

• The excerpt uses the number line to show that rounding depends on which value the number is closer to.
• Example: 67 is more than halfway from 60 to 70 on the number line, so it rounds to 70.
• Example: 4,329 is less than halfway from 4,300 to 4,400, so it rounds to 4,300.
• The digit to the immediate right of the round-off digit is the indicator for this halfway judgment.

📐 Practical examples

Context	Original number	Rounded to	Result	Reason
Salary	$41,450	Nearest thousand	$41,000	Closer to 41,000 than 42,000
Diphtheria cases (1950)	5,796	Nearest hundred	5,800	Digit to right is 9 (≥5), round up
Food stamp recipients (1979)	19,309,000	Nearest ten thousand	19,310,000	Digit to right is 9 (≥5), round up
School enrollment (1980)	1,105,000	Nearest million	1,000,000	Digit to right is 1 (<5), round down

• The excerpt notes that for some situations, you should "round to the position you think is most reasonable"—e.g., rounding an average salary of $16,096 might be done to the nearest thousand ($16,000) or nearest hundred ($16,100), depending on context.

🧭 Overview

🧠 One-sentence thesis

Addition combines two or more numbers (addends) into a total (sum), and the process requires aligning digits by place value and carrying when column sums exceed 9.

📌 Key points (3–5)

• What addition is: the process of combining collections of objects (real or abstract) to form a total.
• Key terminology: numbers being added are called addends or terms; the result is the sum; use + for addition and = for equals.
• The basic process: write numbers vertically with matching place values in columns, then add each column from right to left.
• Carrying mechanism: when a column sum exceeds 9, convert to the next higher place value and carry it left.
• Common confusion: improper alignment of place values leads to incorrect sums—always align ones with ones, tens with tens, etc.

🔢 What addition means

🔢 The combining process

Addition: The process of combining two or more objects (real or intuitive) to form a third, the total.

• Addition takes separate collections and merges them into one larger collection.
• Example: combining a collection of four objects with a collection of three objects yields a collection of seven objects.

🏷️ Terminology

• Addends or terms: the numbers being added.
• Sum: the total result.
• Plus symbol (+): indicates addition.
• Equal symbol (=): represents "equal."
• Example: 4 + 3 = 7 means "four added to three equals seven."

📏 Number line visualization

• Start at 0, move right by the first addend, then move right again by the second addend.
• Example: To find 4 + 3, start at 0, move right 4 units (now at 4), then move right 3 more units (now at 7).
• This visual method confirms 4 + 3 = 7.

➕ The basic addition process

➕ Step-by-step procedure

The excerpt illustrates adding 25 and 43:

1. Write numbers vertically: place corresponding positions (ones, tens, hundreds) in the same column.
2. Add each column: start at the right (ones position) and move left, placing each sum at the bottom.

Example:

• 25 + 43
• Ones: 5 + 3 = 8
• Tens: 2 + 4 = 6
• Sum: 68

⚠️ Alignment is critical

• Don't confuse: misaligned columns produce wrong answers.
• Always ensure ones align with ones, tens with tens, etc.
• The excerpt warns: "Confusion and incorrect sums can occur when the numbers are not aligned in columns properly."

📝 Simple examples (no carrying)

• 276 + 103: ones (6+3=9), tens (7+0=7), hundreds (2+1=3) → sum is 379.
• 1459 + 130: ones (9+0=9), tens (5+3=8), hundreds (4+1=5), thousands (1+0=1) → sum is 1589.
• These examples work when each column sum stays below 10.

🔄 Carrying in addition

🔄 When and why to carry

• Carrying happens when a column sum exceeds 9.
• Convert the excess to the next higher place value.
• Example: adding 18 + 34 in the ones column gives 8 + 4 = 12 ones, which converts to 1 ten and 2 ones.
• Write the 2 in the ones place and carry the 1 to the tens column.

🔄 Carrying mechanism step-by-step

Adding 1875 + 358:

• Ones: 5 + 8 = 13 → write 3, carry 1 ten.
• Tens: 1 (carried) + 7 + 5 = 13 → write 3, carry 1 hundred.
• Hundreds: 1 (carried) + 8 + 3 = 12 → write 2, carry 1 thousand.
• Thousands: 1 (carried) + 1 = 2.
• Sum: 2233.

🔄 Multiple addends

• You can add more than two numbers at once.
• Example: 2648 + 1359 + 861:
  • Ones: 8 + 9 + 1 = 18 → write 8, carry 1.
  • Tens: 1 + 4 + 5 + 6 = 16 → write 6, carry 1.
  • Hundreds: 1 + 6 + 3 + 8 = 18 → write 8, carry 1.
  • Thousands: 1 + 2 + 1 = 4.
  • Sum: 4868.

🔄 Carrying numbers other than 1

• The excerpt notes that numbers other than 1 can be carried.
• Example: when adding six numbers, a column might sum to 19, so you write 9 and carry 1; another column sums to 26, so write 6 and carry 2; another sums to 33, so write 3 and carry 3.

🧮 Using calculators

🧮 Calculator procedure

The excerpt describes calculators that do not require an ENTER key:

Step	Action	Display
Type first number	34	34
Press +		34
Type second number	21	21
Press =		55 (the sum)

🧮 Running subtotals for multiple addends

Adding 106 + 85 + 322 + 406:

• Type 106, press +, type 85, press = → running subtotal 191.
• Type 322, press + → running subtotal 513.
• Type 406, press = → final sum 919.
• The calculator keeps a running subtotal as you add each number.

🌍 Real-world application

🌍 Enrollment example

• The excerpt gives a scenario: Riemann College had enrollments of 10,406 (1984), 9,289 (1985), 10,108 (1986), and 11,412 (1987).
• Question: What was the total enrollment for 1985, 1986, and 1987?
• Add 9,289 + 10,108 + 11,412:
  • Ones: 9 + 8 + 2 = 19 → write 9, carry 1.
  • Tens: 1 + 8 + 0 + 1 = 10 → write 0, carry 1.
  • Hundreds: 1 + 2 + 1 + 4 = 8.
  • Thousands: 9 + 0 + 1 = 10 → write 0, carry 1.
  • Ten-thousands: 1 + 1 = 3 (corrected in excerpt to 30,809 total).
• Total enrollment: 30,809 students.

🧭 Overview

🧠 One-sentence thesis

Subtraction determines the remainder when part of a total is removed, and mastering the borrowing process allows you to subtract any whole numbers correctly.

📌 Key points (3–5)

• What subtraction measures: the process of determining what remains when part of a total is removed.
• Key terminology: minuend (original number), subtrahend (number to be removed), and difference (result).
• Subtraction as the opposite of addition: any subtraction can be checked by adding the difference and subtrahend to get the minuend.
• Common confusion: borrowing vs. not borrowing—when a digit in the minuend is smaller than the corresponding digit in the subtrahend, you must borrow from the next column to the left.
• Special borrowing cases: borrowing from zero (single or group) requires converting zeros to nines and the rightmost zero to ten.

🔢 Core concepts and terminology

🔢 What subtraction is

Subtraction is the process of determining the remainder when part of the total is removed.

• Visualized on a number line: starting at 11 and moving 4 units left lands you at 7, so 7 units remain.
• Example: If you have 11 items and remove 4, you have 7 left.

➖ The minus symbol and its parts

The minus symbol (−) indicates subtraction.

• Minuend: the number immediately in front of or above the minus symbol; it represents the original number of units.
• Subtrahend: the number immediately following or below the minus symbol; it represents the number of units to be removed.
• Difference: the result of the subtraction.
• Example: In 11 − 4 = 7, the minuend is 11, the subtrahend is 4, and the difference is 7.

🔄 Subtraction as the opposite of addition

• Any subtraction statement can be rewritten as an addition statement.
• Example: 8 − 5 = 3 since 3 + 5 = 8.
• Example: 9 − 3 = 6 since 6 + 3 = 9.
• This relationship helps check your work: add the difference and subtrahend to see if you get the minuend.

🧮 The basic subtraction process

🧮 Step-by-step process for simple subtraction

To subtract two whole numbers:

1. Write the numbers vertically, placing corresponding positions (ones, tens, hundreds) in the same column.
2. Subtract the digits in each column, starting at the right (ones position) and moving left, placing the difference at the bottom.

📝 Examples without borrowing

Problem	Ones	Tens	Hundreds	Difference
48 − 35	8 − 5 = 3	4 − 3 = 1	—	13
275 − 142	5 − 2 = 3	7 − 4 = 3	2 − 1 = 1	133
977 − 235	7 − 5 = 2	7 − 3 = 4	9 − 2 = 7	742

• Example: To find the difference between 977 and 235, write the larger number on top, line up the columns, and subtract each column from right to left.
• Example: If Flags County had 1,159 cable installations and Keys County had 809, the difference is 1,159 − 809 = 350 more installations in Flags County.

🔄 Subtraction involving borrowing

🔄 When and why to borrow

• When to borrow: when a digit in the minuend (top number) is less than the digit in the same position in the subtrahend (bottom number).
• Example: In 84 − 27, you cannot subtract 7 from 4 in the ones place.
• How borrowing works: convert (borrow) 1 unit from the next higher place value, then add it to the current place.

🔢 Single borrowing example

For 84 − 27:

1. Borrow 1 ten from the 8 tens, leaving 7 tens.
2. Convert the 1 ten to 10 ones.
3. Add 10 ones to 4 ones to get 14 ones.
4. Now subtract: 14 − 7 = 7 (ones) and 7 − 2 = 5 (tens), giving 57.

• The new name for 84 is "7 tens + 14 ones."
• Don't confuse: borrowing does not change the value of the number, only its representation.

🔁 Borrowing more than once

Sometimes you must borrow in multiple columns.

Example: For 641 − 358:

1. Borrow 1 ten from 4 tens (leaving 3 tens), convert to 10 ones, add to 1 one to get 11 ones; subtract 11 − 8 = 3.
2. Borrow 1 hundred from 6 hundreds (leaving 5 hundreds), convert to 10 tens, add to 3 tens to get 13 tens; subtract 13 − 5 = 8.
3. Subtract 5 − 3 = 2 in the hundreds place.
4. Result: 283.

Example: For 534 − 5:

1. Borrow in the ones and tens places as needed.
2. After borrowing, subtract column by column.
3. Result: 449.

🔟 Borrowing from zero

🔟 Borrowing from a single zero

To borrow from a single zero:

1. Decrease the digit immediately to the left of the zero by one.
2. Draw a line through the zero and make it a 10.
3. Proceed to subtract as usual.

Example: For 503 − 37:

1. The digit to the left of 0 is 5; decrease it by 1 to get 4.
2. Change the 0 to 10.
3. Borrow 1 ten from 10 tens (leaving 9 tens), convert to 10 ones, add to 3 ones to get 13 ones.
4. Subtract: 13 − 7 = 6 (ones), 9 − 3 = 6 (tens), 4 − 0 = 4 (hundreds).
5. Result: 466.

🔟🔟 Borrowing from a group of zeros

To borrow from a group of zeros:

1. Decrease the digit immediately to the left of the group of zeros by one.
2. Draw a line through each zero in the group and make it a 9, except the rightmost zero—make it 10.
3. Proceed to subtract as usual.

Example: For 5,000 − 37:

1. The digit to the left of the group of zeros is 5; decrease it by 1 to get 4.
2. Change the first three zeros to 9 and the rightmost zero to 10.
3. Now you have 4,999 with an extra 1 in the ones place (represented as 10).
4. Subtract: 10 − 7 = 3 (ones), 9 − 3 = 6 (tens), 9 − 0 = 9 (hundreds), 9 − 0 = 9 (thousands), 4 − 0 = 4 (ten-thousands).
5. Result: 4,963.

Example: For 40,000 − 125:

1. Decrease 4 by 1 to get 3.
2. Make each 0 a 9 except the rightmost, which becomes 10.
3. Subtract column by column.
4. Result: 39,875.

Example: For 8,000,006 − 41,107:

1. Decrease 8 by 1 to get 7.
2. Make each zero in the group a 9 except the rightmost, which becomes 10.
3. Borrow from the 10 in the ones place to perform the subtraction.
4. Result: 7,958,899.

🧰 Practical applications and tools

🧰 Word problems and real-world scenarios

• "Subtract X from Y" means Y − X (start at Y, remove X).
• "Find the difference between A and B" means subtract the smaller from the larger.
• "How much bigger is A than B?" means A − B.

Example: If the sun shines 74% of the time in July and 59% in November, the difference is 74 − 59 = 15% more in July.

Example: If 11,330,000 people were arrested for drunk driving and 83,000 for prostitution, the difference is 11,330,000 − 83,000 = 11,247,000 more arrests for drunk driving.

🖩 Using calculators

To find the difference between two whole numbers with a calculator:

1. Type the minuend (larger number).
2. Press the minus (−) button.
3. Type the subtrahend (smaller number).
4. Press equals (=).
5. Read the difference on the display.

Example: For 1,006 − 284, type 1006, press −, type 284, press =, and the display reads 722.

• Note: If you type the smaller number first, you will get a different result (negative numbers, covered in a later chapter).

✅ Checking your work

• Add the difference and the subtrahend; if you get the minuend, your subtraction is correct.
• Example: If 84 − 27 = 57, check by computing 57 + 27 = 84. ✓

🔍 Common confusions and tips

🔍 Borrowing vs. not borrowing

• No borrowing needed: when every digit in the minuend is greater than or equal to the corresponding digit in the subtrahend.
• Borrowing needed: when any digit in the minuend is less than the corresponding digit in the subtrahend.
• Don't confuse: borrowing changes the representation of the number (e.g., 84 becomes 7 tens + 14 ones) but not its value.

🔍 Single zero vs. group of zeros

Situation	Method
Single zero	Decrease the digit to the left by 1; change the 0 to 10
Group of zeros	Decrease the digit to the left of the group by 1; change each 0 to 9 except the rightmost, which becomes 10

• Don't confuse: a single zero requires only one adjustment, but a group of zeros requires changing multiple digits.

🔍 Order matters in subtraction

• Subtraction is not commutative: 84 − 27 ≠ 27 − 84.
• Always place the larger number (minuend) on top when writing vertically.
• "From" indicates the starting point: "subtract 63 from 92" means 92 − 63, not 63 − 92.

🧭 Overview

🧠 One-sentence thesis

Addition has three fundamental properties—commutative, associative, and identity—that describe how the order and grouping of numbers do not change sums, and how zero preserves any number's value.

📌 Key points (3–5)

• Commutative property: two whole numbers can be added in any order without changing the sum.
• Associative property: when adding three numbers, grouping the first two or last two first yields the same result.
• Additive identity: zero added to any whole number leaves that number unchanged.
• Common confusion: commutative involves order (two numbers), while associative involves grouping (three numbers with parentheses).
• Why it matters: these properties simplify calculations and show that addition is flexible in how we approach it.

🔄 The Commutative Property

🔄 What it means

Commutative Property of Addition: If two whole numbers are added in any order, the sum will not change.

• This property is about order: you can swap the positions of two numbers being added.
• The sum remains identical regardless of which number comes first.

🧮 How it works

• Example from the excerpt: 8 + 5 = 13 and 5 + 8 = 13
• The numbers 8 and 5 can be added in either sequence, and both produce 13.
• Another example: 12 + 41 = 53 and 41 + 12 = 53

⚠️ Don't confuse with associative

• Commutative deals with two numbers and their order.
• It does not involve parentheses or grouping—just swapping positions.

🧩 The Associative Property

🧩 What it means

Associative Property of Addition: If three whole numbers are to be added, the sum will be the same if the first two are added first, then that sum is added to the third, or, the second two are added first, and that sum is added to the first.

• This property is about grouping: when adding three numbers, which pair you combine first doesn't matter.
• Parentheses show which pair to add first.

🧮 How it works

• The excerpt uses parentheses to indicate grouping.
• Example: (17 + 32) + 25 = 49 + 25 = 74
• Same numbers, different grouping: 17 + (32 + 25) = 17 + 57 = 74
• Both approaches yield 74.

📝 Parentheses notation

• Using parentheses is a common mathematical practice to show which pair of numbers to combine first.
• The parentheses tell you what to calculate before moving to the next step.

⚠️ Don't confuse with commutative

• Associative deals with three numbers and grouping (parentheses).
• It does not change the order of numbers, only which two you add first.

🆔 The Additive Identity

🆔 What it means

0 Is the Additive Identity: The whole number 0 is called the additive identity, since when it is added to any whole number, the sum is identical to that whole number.

• Zero has a special role: adding it to any number leaves that number unchanged.
• The term "identity" is used because the original number retains its identity after the addition.

🧮 How it works

• Example from the excerpt: 29 + 0 = 29 and 0 + 29 = 29
• Zero added to 29 does not change the identity of 29.
• This works with any whole number: 8 + 0 = 8, 0 + 5 = 5

🔤 Using variables

• If x represents any whole number, then x + 0 = x and 0 + x = x.
• Example: if x = 17, then x + 0 = 17 + 0 = 17.

💡 Why "identity" is appropriate

• The excerpt explains: "its partner in addition remains identically the same after that addition."
• Zero is unique among whole numbers in preserving the other number exactly as it was.

📊 Summary comparison

Property	Numbers involved	What it describes	Example
Commutative	Two	Order doesn't matter	8 + 5 = 5 + 8 = 13
Associative	Three	Grouping doesn't matter	(17 + 32) + 25 = 17 + (32 + 25) = 74
Identity	Any number + zero	Zero preserves the number	29 + 0 = 29

🧭 Overview

🧠 One-sentence thesis

Zero is the additive identity because adding zero to any whole number leaves that number unchanged, and this property—along with commutative and associative properties—governs how addition behaves.

📌 Key points (3–5)

• Additive identity: Zero added to any whole number does not change the value of that number.
• Why "identity" fits: The term is appropriate because the partner in addition remains identically the same after adding zero.
• Commutative property: A first number + a second number = a second number + a first number (order does not matter).
• Associative property: (First + second) + third = first + (second + third) (grouping does not matter).
• Common confusion: Identity vs commutative—identity is about zero preserving a number; commutative is about swapping order of any two numbers.

🔢 The additive identity: zero

🔢 What the additive identity means

The additive identity property: 0 + any number = that particular number.

• Zero does not change the value when added.
• The excerpt shows: 29 + 0 = 29 and 0 + 29 = 29.
• "Zero added to 29 does not change the identity of 29."

🪞 Why the term "identity" is appropriate

• The excerpt asks: "Why is the term identity so appropriate?"
• Answer given: "…because its partner in addition remains identically the same after that addition."
• In other words: the original number keeps its identity—it stays exactly what it was.

Example:

• 8 + 0 = 8
• 0 + 5 = 5
• x + 0 = x (for any whole number x)

Don't confuse: Identity is specifically about zero; it is not about rearranging numbers or grouping them.

🔄 Commutative property of addition

🔄 What commutative means

Commutative property: A first number + a second number = a second number + a first number.

• Order of addition does not affect the sum.
• The excerpt illustrates with 15 and 8: "15 + 8 = 8 + 15 = 23."

Example:

• 15 + 8 = 8 + 15 = 23
• Any two whole numbers can be swapped without changing the result.

Don't confuse: Commutative is about swapping two numbers; associative is about regrouping three or more numbers.

🧮 Associative property of addition

🧮 What associative means

Associative property: (A first number + a second number) + third number = a first number + (a second number + a third number).

• Grouping (which pair you add first) does not affect the sum.
• The excerpt shows multiple examples:
  • (11 + 27) + 9 = 11 + (27 + 9)
  • (80 + 52) + 6 = 80 + (52 + 6) = 138
  • (114 + 226) + 108 = 114 + (226 + 108)
  • (731 + 256) + 171 = 731 + (256 + 171) = 1,158

Example:

• (6 + 5) + 11 = 6 + (5 + 11)
• Left side: 11 + 11 = 22; right side: 6 + 16 = 22.

Don't confuse: Associative is about changing parentheses/grouping; commutative is about changing order of the same two numbers.

📋 Summary comparison

Property	What it says	Example from excerpt
Identity	0 + any number = that number	29 + 0 = 29; 0 + 29 = 29
Commutative	First + second = second + first	15 + 8 = 8 + 15 = 23
Associative	(First + second) + third = first + (second + third)	(80 + 52) + 6 = 80 + (52 + 6) = 138

📋 How to distinguish them

• Identity: involves zero; the other number stays the same.
• Commutative: involves two numbers; swap their order.
• Associative: involves three or more numbers; change which pair you add first.

🧭 Overview

🧠 One-sentence thesis

Multiplication is a shorthand for repeated addition, and mastering the process—from single-digit to multi-digit multipliers and numbers ending in zeros—enables efficient calculation of products both by hand and with calculators.

📌 Key points (3–5)

• What multiplication represents: repeated addition of the same number (the multiplicand) a certain number of times (the multiplier).
• Key terminology: the numbers being multiplied are called factors (specifically, the multiplicand and multiplier), and the result is the product.
• How to multiply multi-digit numbers: use partial products for each digit of the multiplier, align them correctly by place value, then add them together.
• Common confusion: when zeros appear at the end of factors, align rightmost nonzero digits in the same column and attach all zeros to the final product—don't multiply through every zero.
• Carrying in multiplication: just like addition, when a product exceeds 9 in any place value, carry the extra to the next column.

🔢 Core concepts

🔢 What multiplication means

Multiplication is a description of repeated addition.

• Instead of writing 5 + 5 + 5, we write 3 × 5 (three times five).
• The number being repeated (5) is the multiplicand.
• The number that counts how many times (3) is the multiplier.
• Example: 7 + 7 + 7 + 7 + 7 + 7 becomes 6 × 7 (multiplier is 6, multiplicand is 7).

🏷️ Factors and products

In a multiplication, the numbers being multiplied are called factors. The result is called the product.

• In 3 × 5 = 15, both 3 and 5 are factors; 15 is the product.
• Factors can also be called multiplier and multiplicand, depending on their role.

✖️ Multiplication symbols

• Multiple notations exist: × (times sign), · (dot), and parentheses.
• All of these mean the same thing: 3 × 5, 3 · 5, 3(5), (3)5, (3)(5).

🧮 Single-digit multiplication process

🧮 How carrying works

• Multiply each digit of the multiplicand by the single-digit multiplier, starting from the rightmost (ones) place.
• If a product is 10 or greater, write the ones digit and carry the tens digit to the next column.
• Example: 7 × 38
  • 7 × 8 = 56 → write 6, carry 5
  • 7 × 3 = 21 → add the carried 5 → 21 + 5 = 26
  • Product: 266

📝 Step-by-step pattern

1. Multiply the ones digit.
2. Write the ones part of the result; carry any tens.
3. Multiply the next digit to the left.
4. Add any carried amount to that result.
5. Repeat until all digits are processed.

🔨 Multi-digit multiplication process

🔨 Partial products method

When the multiplier has two or more digits, multiply in parts:

Step	Action	Alignment
Part 1	Multiply by the ones digit → first partial product	Rightmost digit in ones column
Part 2	Multiply by the tens digit → second partial product	Rightmost digit in tens column
Part 3	Continue for hundreds, thousands, etc.	Each partial product shifts one column left
Part 4	Add all partial products → total product	Standard addition with carrying

🎯 Example walkthrough

Multiply 326 by 48:

• Part 1: 326 × 8 = 2,608 (first partial product)
• Part 2: 326 × 4 = 1,304, written as 13040 (aligned in tens column)
• Part 4: 2,608 + 13,040 = 15,648 (total product)

⚠️ Don't confuse: zero in the multiplier

• If a digit in the multiplier is 0 (e.g., 206), that partial product is 0 and doesn't change the sum.
• You can skip writing it and move to the next digit.
• Example: 1,508 × 206 → skip the tens place (0 × 1,508), go straight to the hundreds place (2 × 1,508).

🎯 Shortcuts with trailing zeros

🎯 Aligning nonzero digits

When factors end in zeros, use this shortcut:

1. Align the rightmost nonzero digits in the same column.
2. Draw a mental vertical line separating zeros from nonzeros.
3. Multiply only the nonzero parts.
4. Attach the total count of zeros from both factors to the result.

📊 Example comparison

Multiply 49,000 by 1,200:

• Nonzero parts: 49 × 12 = 588
• Total zeros: 3 (from 49,000) + 2 (from 1,200) = 5 zeros
• Product: 58,800,000

Don't confuse: This is not "ignoring" zeros—it's efficiently placing them at the end after multiplying the significant digits.

🖩 Using calculators

🖩 Standard multiplication

• Enter the first number.
• Press the × key.
• Enter the second number.
• Press = to see the product.
• Example: 75,891 × 263 = 19,959,333

🔢 Very large numbers

• Calculators may not display all digits for extremely large products.
• You may need to count and attach zeros manually (e.g., 4,510,000,000,000 × 1,700 → calculator shows 7667, you add 12 zeros → 7,667,000,000,000,000).
• Some calculators display results in scientific notation (e.g., 2.2546563 12) for numbers that exceed the display limit.

✅ Verification tip

• Use a calculator to check hand-calculated products, especially for multi-digit problems.

🧭 Overview

🧠 One-sentence thesis

Prime factorization provides a systematic way to break down composite numbers into their unique prime building blocks, which then enables finding the greatest common factor shared by multiple numbers.

📌 Key points (3–5)

• Prime vs composite: Prime numbers have exactly two factors (1 and themselves), while composite numbers have more than two factors.
• Fundamental Principle of Arithmetic: Every natural number (except 1) can be factored into primes in exactly one way, regardless of factor order.
• Prime factorization method: Repeatedly divide by the smallest prime that works, moving to the next prime when needed, until the quotient is smaller than the divisor.
• Common confusion: A prime number has no prime factorization—prime factorization applies only to composite numbers.
• GCF application: The greatest common factor is found by taking common prime bases with their smallest exponents from the factorizations.

🔢 Understanding prime and composite numbers

🔢 What makes a number prime

Prime number: a whole number that has exactly two factors, 1 and itself.

• The number must be divisible only by 1 and by itself.
• Example: 47 is prime because division trials show it is only divisible by 1 and 47.
• Example: 3 is prime (factors: 1, 3).

🧱 What makes a number composite

Composite number: a whole number that has more than two factors.

• If a number can be divided evenly by any number other than 1 and itself, it is composite.
• Example: 39 is composite because 3 divides into it (39 ÷ 3 = 13).
• Some composite numbers listed: 4, 6, 8, 9, 10, 12, 15.

⚠️ Don't confuse

• Prime numbers cannot be broken down further; they are the "building blocks."
• Composite numbers can be expressed as products of smaller whole numbers.
• The number 1 is neither prime nor composite.

🏗️ Prime factorization

🏗️ The Fundamental Principle

Fundamental Principle of Arithmetic: Except for the order of the factors, every natural number other than 1 can be factored in one and only one way as a product of prime numbers.

• This uniqueness is central—no matter how you factor a composite number, you end up with the same set of primes.
• Example: 10 = 5 · 2 (both 2 and 5 are prime, so this is the prime factorization).

🛠️ The factorization method

Step-by-step process:

1. Divide the number repeatedly by the smallest prime that divides it evenly (without remainder).
2. When that prime no longer works, move to the next larger prime.
3. Continue until the quotient is smaller than the divisor.
4. The prime factorization is the product of all these prime divisors.

Using divisibility tests helps identify which primes to try:

• Even last digit → divisible by 2.
• Sum of digits divisible by 3 → divisible by 3.
• Last digit 0 or 5 → divisible by 5.

📐 Examples of factorization

Example: 60

• 60 is even, so divide by 2: 60 = 2 · 30
• 30 is even, divide by 2 again: 60 = 2 · 2 · 15
• 15 is not even but 1+5=6 (divisible by 3): 15 = 3 · 5
• Result: 60 = 2 · 2 · 3 · 5 = 2² · 3 · 5

Example: 441

• Not divisible by 2 (last digit is odd).
• Sum of digits: 4+4+1=9, divisible by 3.
• Divide by 3: 441 = 3 · 147
• 147: sum 1+4+7=12, divisible by 3: 147 = 3 · 49
• 49 is not divisible by 3 or 5, but is divisible by 7: 49 = 7 · 7
• Result: 441 = 3 · 3 · 7 · 7 = 3² · 7²

Example: 31

• Not divisible by 2, 3, 5, or 7 (through trial division).
• When quotient becomes smaller than divisor, stop.
• Conclusion: 31 is prime (has no prime factorization).

⚠️ Remember

• Prime factorization applies only to composite numbers.
• Use exponents to write repeated factors compactly (e.g., 2 · 2 · 2 = 2³).

🔗 Greatest Common Factor (GCF)

🔗 What the GCF means

Common factor: a number that appears as a factor in two or more numbers.

Greatest Common Factor (GCF): the largest common factor of two or more whole numbers.

• Example: For 30 and 42, both have 2 and 3 as factors; the GCF combines these common factors.
• The GCF is useful for working with fractions (mentioned for later study).

🧮 Method for finding the GCF

Four-step process:

1. Write the prime factorization of each number using exponents.
2. List each base (prime) that is common to all the numbers.
3. For each common base, attach the smallest exponent that appears on it in any of the factorizations.
4. The GCF is the product of these numbers from step 3.

📊 GCF examples

Numbers	Prime factorizations	Common bases	Smallest exponents	GCF
12, 18	12 = 2² · 3<br>18 = 2 · 3²	2, 3	2¹, 3¹	2 · 3 = 6
18, 60, 72	18 = 2 · 3²<br>60 = 2² · 3 · 5<br>72 = 2³ · 3²	2, 3	2¹ (from 18)<br>3¹ (from 60)	2 · 3 = 6
700, 1880, 6160	700 = 2² · 5² · 7<br>1880 = 2³ · 5 · 47<br>6160 = 2⁴ · 5 · 7 · 11	2, 5	2² (from 700)<br>5¹ (from 1880 or 6160)	2² · 5 = 20

🎯 Why smallest exponents matter

• The GCF must divide all numbers evenly.
• Taking the smallest exponent ensures the factor appears at least that many times in every number.
• Example: If one number has 2¹ and another has 2³, the GCF can only include 2¹ because that's the most guaranteed to divide both.

⚠️ Key distinction

• Don't confuse "common factors" (any shared factors) with "greatest common factor" (the largest one).
• The GCF is the product of common primes with their minimum exponents, not the sum or maximum.

🧭 Overview

🧠 One-sentence thesis

Fractions can be added or subtracted only when they share a common denominator, and this principle applies whether working with simple fractions, mixed numbers, or complex fractions.

📌 Key points (3–5)

• Like denominators: When fractions have the same denominator, add or subtract only the numerators and keep the denominator unchanged.
• Unlike denominators: Convert each fraction to an equivalent fraction with the least common denominator (LCD) before adding or subtracting.
• Mixed numbers: Convert mixed numbers to improper fractions first, then perform the operation.
• Common confusion: Do NOT add or subtract denominators—this leads to absurd results (e.g., two halves would equal one half).
• Complex fractions: Simplify the numerator and denominator separately before dividing.

➕ Adding and subtracting fractions with like denominators

➕ The basic rule for like denominators

Method of Adding Fractions Having Like Denominators: To add two or more fractions that have the same denominators, add the numerators and place the resulting sum over the common denominator. Reduce, if necessary.

• The denominators stay the same; only the numerators are combined.
• Example: 3/7 + 2/7 = (3+2)/7 = 5/7
• After adding, always check if the result can be reduced.

➖ Subtraction with like denominators

Subtraction of Fractions with Like Denominators: To subtract two fractions that have like denominators, subtract the numerators and place the resulting difference over the common denominator. Reduce, if possible.

• Example: 3/5 − 1/5 = (3−1)/5 = 2/5
• The process mirrors addition: operate on numerators only.

⚠️ Why we never add or subtract denominators

• Don't confuse: Adding denominators produces nonsense.
• Example showing the error: 1/2 + 1/2 = (1+1)/(2+2) = 2/4 = 1/2, which claims two halves equal one half—preposterous!
• Subtracting denominators can lead to division by zero: 7/15 − 4/15 = (7−4)/(15−15) = 3/0, which is undefined.

🔄 Adding and subtracting fractions with unlike denominators

🔄 The fundamental rule

A Basic Rule: Fractions can only be added or subtracted conveniently if they have like denominators.

• Think of coins: to combine a quarter (25/100) and a dime (10/100), you need the same denomination (same denominator).
• Same denomination → same denominator.

🔢 Finding the least common denominator (LCD)

• The LCD is the least common multiple (LCM) of the original denominators.
• Example: For 1/6 + 3/4, find the LCD of 6 and 4.
  • 6 = 2 · 3
  • 4 = 2²
  • LCD = 2² · 3 = 12

🔁 Converting to equivalent fractions

Step-by-step process:

1. Find the LCD of all denominators.
2. Divide the LCD by each original denominator to get a quotient.
3. Multiply both the numerator and denominator of each fraction by that quotient.
4. Add or subtract the new numerators.
5. Place the result over the common denominator and reduce if needed.

• Example: 1/2 + 2/3
  • LCD = 6
  • 1/2 = (1·3)/(2·3) = 3/6
  • 2/3 = (2·2)/(3·2) = 4/6
  • 3/6 + 4/6 = 7/6 or 1 1/6

🔢 Working with mixed numbers

🔢 The conversion method

Method: To add or subtract mixed numbers, convert each mixed number to an improper fraction, then add or subtract the resulting improper fractions.

• A mixed number like 3 1/8 becomes (3·8+1)/8 = 25/8.
• After converting all mixed numbers, follow the rules for adding/subtracting fractions.

📐 Step-by-step example

Adding 8 3/5 + 5 1/4:

1. Convert: 8 3/5 = 43/5 and 5 1/4 = 21/4
2. Find LCD of 5 and 4: LCD = 20
3. Convert: 43/5 = 172/20 and 21/4 = 105/20
4. Add: 172/20 + 105/20 = 277/20
5. Convert back: 277/20 = 13 17/20

➖ Subtracting mixed numbers

• Same process: convert to improper fractions first.
• Example: 3 1/8 − 5/6 = 25/8 − 5/6 = 75/24 − 20/24 = 55/24 = 2 7/24

📊 Comparing fractions

📊 Understanding order

Ordered number system: Our number system is ordered because the numbers can be placed in order from smaller to larger.

• On a number line, numbers to the right are larger.
• > means "greater than"
• < means "less than"

⚖️ How to compare fractions

Comparing Fractions: If two fractions have the same denominators, the fraction with the larger numerator is the larger fraction.

Process:

1. Convert all fractions to equivalent fractions with the LCD.
2. Compare the numerators.
3. The fraction with the larger numerator is larger.

• Example: Compare 8/9 and 14/15
  • LCD = 45
  • 8/9 = 40/45 and 14/15 = 42/45
  • Since 40 < 42, we have 8/9 < 14/15

🔍 Comparing mixed numbers

• Same whole number parts: Compare only the fractional parts.
• Different whole number parts: Compare only the whole numbers.
• Example: 6 3/4 < 8 6/7 because 6 < 8 (whole number parts differ).

🧩 Complex fractions

🧩 Simple vs. complex fractions

Simple fraction: Any fraction in which the numerator is any whole number and the denominator is any nonzero whole number.

Complex fraction: Any fraction in which the numerator and/or the denominator is a fraction; it is a fraction of fractions.

• Simple: 1/2, 4/3, 763/1000
• Complex: (3/4)/(5/6), (1/3)/2, 6/(9/10)

🔄 Converting complex to simple fractions

Method:

1. Simplify the numerator completely.
2. Simplify the denominator completely.
3. Divide the numerator by the denominator.

• Example: (3/8)/(15/16) = 3/8 ÷ 15/16 = 3/8 · 16/15 = 2/5
• Example with operations: (5 + 3/4)/(46) = (23/4)/46 = 23/4 ÷ 46/1 = 23/4 · 1/46 = 1/8

🎯 More complex examples

When the numerator or denominator contains operations:

• Example: (1/4 + 3/8)/(1/2 + 13/24)
  1. Simplify numerator: 1/4 + 3/8 = 2/8 + 3/8 = 5/8
  2. Simplify denominator: 1/2 + 13/24 = 12/24 + 13/24 = 25/24
  3. Divide: 5/8 ÷ 25/24 = 5/8 · 24/25 = 3/5

🔢 Order of operations with fractions

🔢 The standard order

Order of Operations:

1. Perform operations inside grouping symbols ( ), [ ], { }, working innermost to outermost.
2. Perform exponential and root operations.
3. Perform all multiplications and divisions from left to right.
4. Perform all additions and subtractions from left to right.

🎯 Applying the order

• Example: 1/4 + 5/8 · 2/15
  1. Multiply first: 5/8 · 2/15 = 1/12
  2. Then add: 1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3

🧮 Complex expressions

Example: 8 − (15/426)(2 − 1 4/15)(3 1/5 + 2 1/8)

1. Simplify each parenthesis separately.
2. Perform multiplications.
3. Perform final subtraction.
4. Always work step-by-step, following the order strictly.

📐 With exponents and roots

• Example: (3/4)² · 8/9 − 5/12
  1. Square first: (3/4)² = 9/16
  2. Multiply: 9/16 · 8/9 = 1/2
  3. Subtract: 1/2 − 5/12 = 6/12 − 5/12 = 1/12

🧭 Overview

🧠 One-sentence thesis

When fractions share the same denominator, you add or subtract only the numerators and keep the common denominator unchanged.

📌 Key points (3–5)

• Core rule for addition: add the numerators together and place the sum over the common denominator; reduce if needed.
• Core rule for subtraction: subtract the numerators and place the difference over the common denominator; reduce if possible.
• Common confusion: never add or subtract the denominators—doing so produces incorrect or undefined results.
• Visual understanding: diagrams show that combining parts with the same "size" (denominator) means counting how many of those parts you have total.
• Reduction reminder: always simplify the resulting fraction to lowest terms when possible.

➕ Adding Fractions with Like Denominators

🎨 Visual foundation

• The excerpt uses a diagram showing shaded regions: 2 one-fifths plus 1 one-fifth equals 3 one-fifths.
• This illustrates that when denominators match, you're counting pieces of the same size.
• Example: 2/5 + 1/5 = 3/5 (two pieces plus one piece equals three pieces, all fifths).

📐 The addition method

Method of Adding Fractions Having Like Denominators: To add two or more fractions that have the same denominators, add the numerators and place the resulting sum over the common denominator. Reduce, if necessary.

Step-by-step:

• Check that denominators are identical.
• Add only the numerators.
• Write the sum over the common denominator.
• Simplify the result if possible.

🔢 Worked examples

Problem	Work	Result
3/7 + 2/7	(3+2)/7	5/7
1/8 + 3/8	(1+3)/8 = 4/8	1/2 (reduced)
4/9 + 5/9	(4+5)/9 = 9/9	1
7/8 + 5/8	(7+5)/8 = 12/8	3/2

⚠️ Why you never add denominators

• The excerpt demonstrates the error with 1/2 + 1/2.
• Wrong approach: (1+1)/(2+2) = 2/4 = 1/2.
• Why it's absurd: this claims two halves equals one half—clearly false.
• The excerpt calls this "preposterous" because it violates basic logic.
• Another example given: 3/4 + 3/4 wrongly becomes 6/8 = 3/4, meaning two three-fourths equals one three-fourth.

➖ Subtracting Fractions with Like Denominators

📐 The subtraction method

Subtraction of Fractions with Like Denominators: To subtract two fractions that have like denominators, subtract the numerators and place the resulting difference over the common denominator. Reduce, if possible.

Step-by-step:

• Verify denominators are the same.
• Subtract the second numerator from the first.
• Place the difference over the common denominator.
• Simplify if needed.

🔢 Worked examples

Problem	Work	Result
3/5 - 1/5	(3-1)/5	2/5
8/6 - 2/6	(8-2)/6 = 6/6	1
16/9 - 2/9	(16-2)/9	14/9

⚠️ Why you never subtract denominators

• The excerpt shows what happens with 7/15 - 4/15 if you wrongly subtract denominators.
• Wrong approach: (7-4)/(15-15) = 3/0.
• Why it fails: division by zero is undefined—the operation becomes meaningless.
• The rule is clear: denominators stay constant in subtraction.

🎯 Practice applications

✅ Addition practice results

• 1/10 + 3/10 = 4/10 = 2/5 (reduced)
• 1/4 + 1/4 = 2/4 = 1/2 (reduced)
• 7/11 + 4/11 = 11/11 = 1
• 3/5 + 1/5 = 4/5

✅ Subtraction practice results

• 10/13 - 8/13 = 2/13
• 5/12 - 1/12 = 4/12 = 1/3 (reduced)
• 1/2 - 1/2 = 0/2 = 0
• 26/10 - 14/10 = 12/10 = 6/5 (reduced)

🧭 Overview

🧠 One-sentence thesis

Decimal fractions extend the positional number system to the right of the ones place, allowing us to represent fractional values as powers of ten, and we read and write them by naming the position of the rightmost digit.

📌 Key points (3–5)

• Positional values to the right: each position to the right of the ones place has one-tenth the value of the position to its left, creating fractional values (tenths, hundredths, thousandths, etc.).
• Decimal point as separator: the decimal point marks where the whole number part ends and the fractional part begins.
• Reading method: read the whole number, say "and" for the decimal point, then read the fractional part as a whole number followed by the name of the rightmost digit's position.
• Common confusion: inserting zeros to the right of the rightmost digit does not change the value (e.g., 0.7 = 0.70), but zeros must be inserted correctly when writing to place digits in the proper position.
• Converting back to fractions: decimals can be converted to fractions by reading them in words and writing the corresponding fraction, then reducing if necessary.

🔢 How positions work to the right of ones

🔢 Each position is one-tenth of the previous

• The excerpt explains that each position has ten times the value of the position to its right.
• Flipping this: each position has one-tenth the value of the position to its left.
• The word "of" translates to multiplication.

📍 Calculating fractional position values

• First position to the right (tenths): one-tenth of 1 = one-tenth times 1 = one-tenth.
• Second position (hundredths): one-tenth of one-tenth = one-tenth times one-tenth = one over ten squared = one over one hundred.
• Third position (thousandths): one-tenth of one-hundredth = one-tenth times one-hundredth = one over ten cubed = one over one thousand.
• This pattern continues indefinitely.
• All these fractional values are powers of 10 in the denominator.

📐 What decimal fractions are

📐 Definition and notation

Decimal fraction: a fraction in which the denominator is a power of 10.

• The decimal point separates the whole number part from the fractional part.
• The word "decimal" comes from the Latin prefix "deci" meaning ten, reflecting the base-ten system.
• Numbers written with a decimal point are called decimal fractions or simply decimals.

🔍 Position names have the suffix "th"

• Tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, ten millionths, etc.
• Example from the excerpt: 42.6 means 42 and six-tenths, which equals 42 plus six over ten.
• Example: 9.8014 has 8 in tenths, 0 in hundredths, 1 in thousandths, 4 in ten thousandths, so it equals 9 plus 8014 over 10,000.

🧮 Special cases and notation tips

• Leading zero: a zero is often inserted in front of the decimal point (in the units position) when the value is less than one, to prevent overlooking the decimal point.
  • Example: 0.93 (not just .93).
• Trailing zeros: zeros can be inserted to the right of the rightmost digit without changing the value.
  • Example: 0.7 = 0.70 because seven-tenths equals seventy-hundredths (both reduce to seven over ten).
• Don't confuse: trailing zeros are harmless, but when writing a decimal from words, you must insert leading zeros after the decimal point to place the rightmost digit in the correct position.

📖 Reading decimal fractions

📖 Step-by-step reading method

1. Read the whole number part as usual (if the whole number is less than 1, skip steps 1 and 2).
2. Read the decimal point as the word "and."
3. Read the number to the right of the decimal point as if it were a whole number.
4. Say the name of the position of the last digit.

🗣️ Examples from the excerpt

Decimal	How to read it
6.8	Six and eight tenths
14.116	Fourteen and one hundred sixteen thousandths
0.0019	Nineteen ten-thousandths
81 (or 81.0)	Eighty-one (whole numbers are also decimal numbers)

• The excerpt notes that some people say "six point eight," which conveys the message but is not technically correct; "six and eight tenths" is the proper phrasing.

✍️ Writing decimal fractions

✍️ Step-by-step writing method

1. Write the whole number part.
2. Write a decimal point for the word "and."
3. Write the decimal part so that the rightmost digit appears in the position indicated in the word name.
4. If necessary, insert zeros to the right of the decimal point so the rightmost digit lands in the correct position.

🧪 Examples from the excerpt

• Thirty-one and twelve hundredths: the hundredths position is indicated, so write 31.12.
• Two and three hundred-thousandths: the hundred thousandths position is indicated, so insert zeros: 2.00003.
• Six thousand twenty-seven and one hundred four millionths: the millionths position is indicated, so write 6,027.000104.
• Seventeen hundredths: no whole number part (less than one), so write 0.17.

⚠️ Common pitfall: placing the rightmost digit

• The position name tells you where the last digit must go.
• Example: "three hundred-thousandths" means the 3 must be in the hundred thousandths place, so you write 2.00003 (not 2.3).
• Don't confuse: "three hundred thousandths" (300 over 1000) vs. "three hundred-thousandths" (3 over 100,000).

🔄 Converting decimals back to fractions

🔄 Converting ordinary decimals

• The excerpt explains: essentially, say the decimal in words, then write what you say as a fraction.
• Reduce the fraction if possible.

Decimal	Read as	Fraction form	Reduced
0.6	Six tenths	six over ten	three over five
0.903	Nine hundred three thousandths	903 over 1000	(already reduced)
18.61	Eighteen and sixty-one hundredths	18 and 61 over 100	(already reduced)
508.0005	Five hundred eight and five ten-thousandths	508 and 5 over 10,000	508 and 1 over 2,000

🔄 Converting complex decimals

Complex decimals: numbers such as 0.11 and two-thirds, where a fraction appears in a decimal position.

• The fractional part refers to a fraction of the position value.
• Example: 0.11 and two-thirds means "eleven and two-thirds hundredths."
  • Write as: (11 and two-thirds) over 100.
  • Convert the mixed number: (11 times 3 plus 2) over 3 = 35 over 3.
  • So: (35 over 3) over 100 = (35 over 3) divided by (100 over 1) = (35 over 3) times (1 over 100) = 35 over 300 = 7 over 60.
• Example: 4.006 and one-fourth equals 4 plus 0.006 and one-fourth.
  • The one-fourth refers to one-fourth of a thousandth.

🧩 Why this works

• The method relies on the fact that the position name directly tells you the denominator.
• Reading the decimal aloud gives you the numerator and denominator in words.
• Don't confuse: a complex decimal like 0.11 and two-thirds is not the same as 0.11 plus two-thirds; the fraction is part of the last decimal place value.

🧭 Overview

🧠 One-sentence thesis

Rounding decimals follows a systematic method that determines whether to leave a digit unchanged or increase it by one based on the digit immediately to its right, with different handling for digits left versus right of the decimal point.

📌 Key points (3–5)

• Core method: mark the round-off position, check if the next digit is less than 5 (leave unchanged) or 5 or greater (add 1).
• Position matters: digits right of the decimal point are simply eliminated after rounding; digits left of the decimal point require replacing intervening digits with zeros and removing the decimal portion.
• Terminology: "round to three decimal places" means the round-off digit is in the third position after the decimal point (thousandths place).
• Common confusion: don't mix up the handling—right-of-decimal rounding eliminates trailing digits, but left-of-decimal rounding replaces with zeros and removes the entire decimal part.

📐 The three-step rounding method

📍 Step 1: Mark the round-off digit

• Identify which position you're rounding to (e.g., nearest hundredth, nearest ten, three decimal places).
• Mark that digit with an arrow or check.
• Example: rounding 32.116 to the nearest hundredth means marking the second digit after the decimal (the 1 in the hundredths place).

🔍 Step 2: Check the digit to the immediate right

Two cases determine what happens next:

Condition	Action
Digit is less than 5	Leave the round-off digit unchanged
Digit is 5 or greater	Add 1 to the round-off digit

• Example: in 32.116, the digit right of the hundredths place is 6, and 6 > 5, so add 1 to the round-off digit (1 becomes 2).
• Example: in 633.14216 rounded to nearest hundred, the digit right of the hundreds place is 3, and 3 < 5, so the 6 stays unchanged.

✂️ Step 3: Handle remaining digits based on position

If the round-off digit is to the right of the decimal point:

• Eliminate all digits to its right.
• Example: 32.116 rounded to hundredths becomes 32.12 (the 6 is dropped).

If the round-off digit is to the left of the decimal point:

• Replace all digits between it and the decimal point with zeros.
• Eliminate the decimal point and all decimal digits.
• Example: 633.14216 rounded to nearest hundred becomes 600 (the 33.14216 portion is replaced with 00 and the decimal part removed).

🗣️ Understanding rounding terminology

📏 "Decimal places" phrasing

"Round to three decimal places" means the round-off digit is the third decimal digit (the digit in the thousandths position).

• This is shorthand for specifying position by counting places after the decimal point.
• Example: 67.129558 rounded to 3 decimal places means rounding to the thousandths position → 67.130.
• Don't confuse: "two decimal places" = hundredths position, not the second digit overall in the number.

🎯 Position names

Common positions mentioned in the excerpt:

• Tenth (first decimal place)
• Hundredth (second decimal place)
• Thousandth (third decimal place)
• Ten thousandth (fourth decimal place)
• One (the units position, no decimals)
• Ten, hundred (positions left of the decimal point)

💡 Worked examples from the excerpt

💡 Example: Right-of-decimal rounding

• Problem: Round 32.116 to the nearest hundredth.
• Process: Mark the hundredths digit (1). Next digit is 6, which is ≥ 5, so add 1 → 1 becomes 2. Eliminate digits to the right.
• Result: 32.12

💡 Example: Left-of-decimal rounding

• Problem: Round 633.14216 to the nearest hundred.
• Process: Mark the hundreds digit (6). Next digit is 3, which is < 5, so leave unchanged. Replace digits between it and decimal with zeros, remove decimal part.
• Result: 600

💡 Example: Cascading changes

• Problem: Round 60.98 to the nearest one.
• Process: Mark the ones digit (0). Next digit is 9, which is ≥ 5, so add 1 → 0 becomes 1 (and this affects the tens place). Eliminate decimal part.
• Result: 61

💡 Example: Trailing zeros

• Problem: Round 67.129558 to 3 decimal places.
• Process: Mark the thousandths digit (9). Next digit is 5, which is ≥ 5, so add 1 → 9 becomes 10, carrying over.
• Result: 67.130 (the zero is kept to show three decimal places)

🧭 Overview

🧠 One-sentence thesis

Decimal operations follow whole-number rules but require careful alignment of decimal points and tracking of decimal places to maintain correct positional value.

📌 Key points (3–5)

• Rounding decimals: identify the round-off digit, look at the digit to its right, and round up if that digit is 5 or greater.
• Adding and subtracting decimals: align decimal points vertically so corresponding positions match, then compute as with whole numbers.
• Multiplying decimals: multiply as whole numbers, then count total decimal places in both factors to place the decimal point in the product.
• Common confusion: when adding/subtracting, students may forget to align decimal points—misalignment causes positional errors (tenths with hundredths, etc.).
• Powers of 10 shortcut: multiplying by 10, 100, 1000, etc. shifts the decimal point without full computation.

🔢 Rounding decimals

🎯 How rounding works

The excerpt explains a systematic procedure:

1. Identify the round-off digit (the position you're rounding to).
2. Look at the digit immediately to its right.
3. If that digit is 5 or greater, increase the round-off digit by 1; otherwise leave it unchanged.
4. Drop all digits to the right of the round-off position (or replace with zeros if they're to the left of the decimal point).

Example: 633.14216 rounded to the nearest hundred becomes 600 (the round-off digit is 6 in the hundreds place; the digit to its right is 3, which is less than 5, so 6 stays; all digits between the round-off position and the decimal point become zeros).

📏 Rounding to decimal places

• "Round to three decimal places" means the round-off digit is in the thousandths position (the third digit after the decimal point).
• Example: 67.129558 rounded to 3 decimal places is 67.130 (the round-off digit is 9 in the thousandths place; the next digit is 5, so 9 rounds up to 10, carrying over).

⚠️ Edge cases

• When rounding 99.9999 to two decimal places, the result is 100.00—rounding can change the whole-number part.
• Don't confuse: "nearest tenth" vs "one decimal place" mean the same thing; "nearest hundredth" = "two decimal places."

➕ Adding and subtracting decimals

🧮 The alignment method

To add or subtract decimals: (1) Align the numbers vertically so that the decimal points line up under each other and the corresponding decimal positions are in the same column. (2) Add or subtract the numbers as if they were whole numbers. (3) Place a decimal point in the resulting sum or difference directly under the other decimal points.

• The key is vertical alignment of decimal points—this ensures tenths align with tenths, hundredths with hundredths, etc.
• You can write trailing zeros (e.g., 47.016 becomes 47.0160) to help columns line up without changing the value.

🔍 Why alignment matters

The excerpt shows the logic using fractions:

• 4.37 + 3.22 = 4 and 37/100 + 3 and 22/100 = (437/100) + (322/100) = 759/100 = 7.59.
• When decimal points are aligned, each column represents the same fractional unit (ones, tenths, hundredths), so addition works like whole numbers.

Example: To compute 1.314 − 0.58, write a 0 in the thousandths position of 0.58 (making it 0.580) so columns align properly.

💰 Practical application

Example from the excerpt: Wendy has $643.12 in her account and writes a check for $16.92. Subtract 16.92 from 643.12 to find the new balance: $626.20.

🖩 Calculator use

• Calculators are useful for sums and differences, but eight-digit display calculators cannot handle numbers with more than eight digits.
• For numbers like 3,891.001786 (more than eight digits), you must add by hand.

✖️ Multiplying decimals

🧩 The decimal-place rule

The excerpt derives the method from fraction multiplication:

• (3.2)(1.46) = (32/10) × (146/100) = (32 × 146)/(10 × 100) = 4672/1000 = 4.672.
• Notice: 3.2 has 1 decimal place, 1.46 has 2 decimal places, and the product 4.672 has 3 decimal places (1 + 2 = 3).

The sum of the number of decimal places in the factors equals the number of decimal places in the product.

📐 The method

1. Multiply the numbers as if they were whole numbers (ignore decimal points).
2. Count the total number of decimal places in both factors.
3. Place the decimal point in the product so it has that many decimal places.

Example: To multiply 3.2 × 1.46, compute 32 × 146 = 4672, then insert the decimal point 3 places from the right: 4.672.

⚡ Multiplying by powers of 10

The excerpt mentions a shortcut for multiplying by 10, 100, 1000, etc., which shifts the decimal point to the right without full multiplication (though the excerpt does not provide detailed examples of this shortcut in the provided text).

🔤 Multiplication in terms of "of"

The excerpt lists "Multiplication in Terms of 'Of'" as a section heading, indicating that the word "of" in word problems signals multiplication (e.g., "find 0.5 of 20" means 0.5 × 20), but the excerpt does not provide further detail.

🧪 Practice and review

📋 Exercise structure

The excerpt includes:

• Rounding exercises to various decimal places (tenth, hundredth, thousandth, ten thousandth).
• Addition and subtraction problems with instructions to align decimal points and optionally use a calculator.
• Multiplication setup (the excerpt cuts off before showing full multiplication exercises).

🔁 Review problems

The excerpt references prior sections (indicated by "[link]"):

• Place value (e.g., "What is the value of 2 in 421,916,017?" Answer: Ten million).
• Exponents (e.g., "Find the value of 4 to the 4th power." Answer: 256).
• Fraction conversions (e.g., "Convert 3.16 to a mixed number fraction." Answer: 3 and 4/25).

These review problems reinforce connections between decimals, fractions, and whole-number operations.

🧭 Overview

🧠 One-sentence thesis

Multiplying decimals works by treating the numbers as whole numbers during calculation, then placing the decimal point in the product so that the total decimal places equals the sum of decimal places in both factors.

📌 Key points (3–5)

• Core method: Multiply as if the numbers were whole numbers, then count total decimal places from both factors to position the decimal point in the product.
• Decimal place rule: The number of decimal places in the product equals the sum of decimal places in the two factors being multiplied.
• Powers of 10 shortcut: Multiplying by 10, 100, 1,000, etc., simply moves the decimal point to the right by the number of zeros in the power of 10.
• Common confusion: Calculator limitations—an eight-digit display may not show products with more than eight digits; hand calculation or rounding may be needed.
• "Of" means multiply: The word "of" in decimal problems translates to the multiplication operation.

🔍 Why the method works

🔍 The logic from fractions

The excerpt demonstrates the reasoning using an example:

• 3.2 × 1.46 can be rewritten as fractions: (32/10) × (146/100)
• This equals (32 × 146)/(10 × 100) = 4672/1000 = 4.672
• Notice: 3.2 has 1 decimal place, 1.46 has 2 decimal places, and the product 4.672 has 3 decimal places (1 + 2 = 3).

The sum of the number of decimal places in the factors equals the number of decimal places in the product.

🧮 From fractions to the general rule

• The denominator in fraction form determines where the decimal point goes.
• Multiplying denominators (powers of 10) is equivalent to adding the number of decimal places.
• This observation leads directly to the multiplication method for decimals.

🛠️ The multiplication method

🛠️ Three-step procedure

Method of Multiplying Decimals:

1. Multiply the numbers as if they were whole numbers.
2. Find the sum of the number of decimal places in the factors.
3. The number of decimal places in the product is the sum found in step 2.

📝 Worked examples

Example: 6.5 × 4.3

• Multiply as whole numbers: 65 × 43 = 2795
• Count decimal places: 6.5 has 1, 4.3 has 1 → total = 2
• Place the decimal: 27.95

Example: 23.4 × 1.96

• Multiply: 234 × 196 = 45864
• Decimal places: 1 + 2 = 3
• Result: 45.864

Example: 0.251 × 0.00113 (rounded to three decimal places)

• Multiply: 251 × 113 = 28363
• Decimal places: 3 + 5 = 8
• Before rounding: 0.00028363
• After rounding to three places: 0.000

⚠️ Don't confuse

• The method counts decimal places, not the position of the decimal point itself.
• Even if a factor is less than 1 (like 0.251), you still count all digits after the decimal point.

🖩 Using calculators

🖩 Display limitations

• An eight-digit display calculator cannot handle products that result in more than eight digits.
• The excerpt notes that inexpensive calculators ($50–$75 at the time) may have larger displays.

🖩 When calculators fall short

Example: 0.0026 × 0.11976

• Expected decimal places: 4 + 5 = 9
• An eight-digit display cannot show all nine decimal places.
• Solution: Use hand calculation for exact value, or round to fewer decimal places.
• If rounding to three decimal places: calculator shows 0.0003114 → rounds to 0.000

✅ When calculators work well

Example: 2.58 × 8.61

• Calculator steps: Type 2.58 → Press × → Type 8.61 → Press =
• Display: 22.2138
• Since this fits in the display and matches expected decimal places (2 + 2 = 4), the result is reliable.

⚡ Multiplying by powers of 10

⚡ The pattern

Multiplication	Zeros in power of 10	Decimal moves right
10 × 8.315274 = 83.15274	1	1 place
100 × 8.315274 = 831.5274	2	2 places
1,000 × 8.315274 = 8,315.274	3	3 places
10,000 × 8.315274 = 83,152.74	4	4 places

⚡ The shortcut rule

Multiplying a Decimal by a Power of 10: Move the decimal point to the right as many places as there are zeros in the power of 10. Add zeros if necessary.

📝 Examples with the shortcut

Example: 100 × 34.876

• 100 has 2 zeros → move decimal 2 places right
• Result: 3,487.6

Example: 10,000 × 56.82

• 10,000 has 4 zeros → move decimal 4 places right
• Original: 56.82 → need to add two zeros → 568,200
• Since there's no fractional part, drop the decimal point: 568200

Example: (10,000,000,000)(52.7)

• 10 billion has 10 zeros → move decimal 10 places right
• Result: 527,000,000,000

🔤 The word "of" in multiplication

🔤 Translation rule

• The word "of" translates to the multiplication operation (×).
• This is a recall from earlier arithmetic: "of" means "multiply."

📝 Examples using "of"

Example: Find 4.1 of 3.8

• Translate: 4.1 × 3.8
• Calculate: 15.58
• Answer: 4.1 of 3.8 is 15.58

Example: Find 0.95 of the sum of 2.6 and 0.8

• First find the sum: 2.6 + 0.8 = 3.4
• Then multiply: 0.95 × 3.4 = 3.230
• Answer: 0.95 of (2.6 + 0.8) is 3.230

⚠️ Order of operations reminder

• When "of" appears with other operations (like "sum of"), handle the operations inside parentheses or described by other words first.
• Then apply the multiplication indicated by "of."

🧭 Overview

🧠 One-sentence thesis

Division of decimals works by converting the divisor to a whole number through moving decimal points, then dividing as with whole numbers, with special shortcuts for powers of 10.

📌 Key points (3–5)

• Dividing by whole numbers: place the decimal point in the quotient directly above the dividend's decimal point, then divide normally.
• Dividing by decimals: move both divisor and dividend decimal points the same number of places to the right until the divisor becomes a whole number.
• Dividing by powers of 10: move the decimal point left by the number of zeros in the power of 10 (shortcut method).
• Common confusion: "moving" the decimal point actually means multiplying by a power of 10; the divisor and dividend must both be adjusted by the same factor.
• Nonterminating divisions: some divisions never produce a remainder of zero and repeat endlessly, shown with three dots or a bar over repeating digits.

🔢 Dividing by whole numbers

🔢 The basic method

Method of Dividing a Decimal by a Nonzero Whole Number: Write a decimal point above the division line directly over the decimal point of the dividend, then proceed to divide as if both numbers were whole numbers.

• The key insight: the decimal point in the quotient aligns vertically with the decimal point in the dividend.
• After placing the decimal point, the division proceeds exactly like whole-number division.
• Example: 114.1 ÷ 7 = 16.3 (the decimal point in 16.3 sits directly above the decimal point in 114.1).

🔍 Handling leading zeros

• If the first nonzero digit in the quotient appears to the right of the decimal point but not in the tenths position, place zeros in each position between the decimal point and that first nonzero digit.
• Example: 0.02068 ÷ 4 = 0.00517 (zeros fill the tenths and hundredths positions).
• Don't confuse: these zeros are part of the quotient's value, not just placeholders; they indicate the magnitude of the result.

🔄 Dividing by decimals

🔄 Converting the divisor to a whole number

The excerpt explains the logic using fractions:

• Dividing 4.32 by 1.8 is the same as dividing 432/100 by 18/10.
• To make the divisor (18/10) a whole number, multiply it by 10 to get 18.
• Because we multiplied the denominator by 10, we must also multiply the numerator (432/100) by 10 to get 432/10 = 43.2.
• Result: 4.32 ÷ 1.8 becomes 43.2 ÷ 18.

📏 The decimal-point rule

Method of Dividing a Decimal by a Decimal Number: Convert the divisor to a whole number by moving the decimal point to the position immediately to the right of the divisor's last digit; move the decimal point of the dividend to the right the same number of digits; set the decimal point in the quotient directly above the newly located decimal point in the dividend; divide as usual.

Step	What to do	Why
1	Move divisor's decimal point to the far right	Makes divisor a whole number
2	Move dividend's decimal point the same number of places	Keeps the ratio unchanged (multiplies both by the same power of 10)
3	Place quotient's decimal point above the new dividend position	Aligns the result correctly
4	Divide normally	Now it's whole-number division

• Example: 32.66 ÷ 7.1 → move both decimal points 1 place right → 326.6 ÷ 71 = 4.6.
• Example: 12 ÷ 0.00032 → move both 5 places right → 1,200,000 ÷ 32 = 37,500 (add five zeros to 12).

⚠️ What "moving" really means

• The excerpt emphasizes: "The word 'move' actually indicates the process of multiplication by a power of 10."
• Don't confuse: you are not just sliding symbols; you are multiplying both dividend and divisor by the same power of 10 to preserve the quotient's value.

⚡ Shortcuts for powers of 10

⚡ The pattern

Dividing a Decimal Fraction by a Power of 10: Move the decimal point of the decimal fraction to the left as many places as there are zeros in the power of 10; add zeros if necessary.

• Example: 8,162.41 ÷ 10 = 816.241 (one zero → move left one place).
• Example: 8,162.41 ÷ 100 = 81.6241 (two zeros → move left two places).
• Example: 3.28 ÷ 10,000 = 0.000328 (four zeros → move left four places, adding three zeros).

🔍 Why this works

• The excerpt shows the long division for 8,162.41 ÷ 10 and 8,162.41 ÷ 100.
• Observation: the number of zeros in the divisor matches the number of places the decimal point moves left in the quotient.
• This is a direct consequence of dividing by 10, 100, 1,000, etc., which scales the number down by factors of ten.

🔁 Nonterminating divisions

🔁 What makes a division nonterminating

Terminating division: a division in which the quotient terminates after several divisions (the remainder is zero). Also called exact divisions.

Nonterminating division: a division that, regardless of how far we carry it out, always has a remainder.

• Example of terminating: 9.8 ÷ 3.5 eventually produces a remainder of zero.
• Example of nonterminating: 4 ÷ 3 = 1.333... (the pattern of 3s repeats endlessly).

🔁 Repeating decimals

Repeating decimal: a decimal quotient in which a pattern repeats itself endlessly.

• Notation with three dots: 4 ÷ 3 = 1.333...
• Notation with a bar: write a bar above the repeating sequence of digits (the excerpt mentions this but does not show the symbol clearly in plain text).
• Don't confuse: the three dots or bar indicate an infinite pattern, not an approximation or rounding.

🧮 Using calculators

🧮 Recognizing approximations

• Calculators have limited digits (e.g., eight-digit displays).
• If the display fills with digits, the result may be an approximation because the true answer has more digits than the calculator can show.
• Example: 0.8215199 ÷ 4.113 fills an eight-digit display with 0.1997373; checking by multiplication shows the answer should have 10 decimal places, so the display is approximate.

✅ How to check

• Multiply the suspected quotient by the divisor.
• Count the total decimal places in the multiplication.
• If the product requires more decimal places than the quotient has, the quotient is approximate.
• Example: 4.113 (3 places) × 0.1997373 (7 places) should produce 10 decimal places, but the quotient only shows 8, confirming approximation.
• Recommendation: round approximate results to a specified number of decimal places (e.g., five or four).

🧭 Overview

🧠 One-sentence thesis

The greatest common factor (GCF) is the largest number that divides two or more whole numbers without a remainder, and it can be systematically found using prime factorizations and selecting the smallest exponents on common prime bases.

📌 Key points (3–5)

• What the GCF is: the largest whole number that divides evenly into all given numbers.
• How to find it: use prime factorizations, identify common bases, attach the smallest exponent appearing on each common base, then multiply.
• Why smallest exponents matter: the GCF must divide all numbers, so it can only include prime factors to the extent they appear in every number.
• Common confusion: don't pick the largest exponent—the GCF uses the smallest exponent on each common prime base.
• Where it's useful: working with fractions (mentioned for future use).

🔍 Understanding common factors

🔍 What a common factor is

Common factor: a number that appears as a factor in two or more numbers.

• If a prime appears in the factorization of multiple numbers, it is a common factor of those numbers.
• Example: For 30 = 2 · 3 · 5 and 42 = 2 · 3 · 7, both 2 and 3 are common factors because they appear in both factorizations.

🏆 What makes the GCF "greatest"

• Among all common factors, the GCF is the largest one.
• It is the biggest number that divides all the given numbers without leaving a remainder.
• Example: The GCF of 30 and 42 is 6, because 6 is the largest number that divides both evenly.

🛠️ The method for finding the GCF

📝 Step-by-step procedure

The excerpt provides a four-step method:

1. Write prime factorizations using exponents on repeated factors.
2. List common bases that appear in all the numbers.
3. Attach the smallest exponent that appears on each common base across all factorizations.
4. Multiply the results from step 3 to get the GCF.

⚠️ Why the smallest exponent

• The GCF must divide every number in the set.
• A prime factor can only be included as many times as it appears in the number where it appears least often.
• Don't confuse: using the largest exponent would give a number too big to divide all the originals.

📊 Worked examples from the excerpt

📊 Example: 12 and 18

Step	Work
Prime factorizations	12 = 2² · 3; 18 = 2 · 3²
Common bases	2 and 3
Smallest exponents	2¹ (from either) and 3¹ (from 12)
GCF	2 · 3 = 6

📊 Example: 18, 60, and 72

Step	Work
Prime factorizations	18 = 2 · 3²; 60 = 2² · 3 · 5; 72 = 2³ · 3²
Common bases	2 and 3 (note: 5 is not common to all three)
Smallest exponents	2¹ (from 18) and 3¹ (from 60)
GCF	2 · 3 = 6

📊 Example: 700, 1,880, and 6,160

Step	Work
Prime factorizations	700 = 2² · 5² · 7; 1,880 = 2³ · 5 · 47; 6,160 = 2⁴ · 5 · 7 · 11
Common bases	2 and 5 (7, 47, and 11 are not common to all)
Smallest exponents	2² (from 700) and 5¹ (from 1,880 or 6,160)
GCF	2² · 5 = 4 · 5 = 20

🔑 Key insight from examples

• Only primes appearing in all factorizations are included.
• The smallest exponent ensures the GCF divides every number.
• Example: In the three-number case above, 7 appears in 700 and 6,160 but not in 1,880, so it is excluded from the GCF.

🧭 Overview

🧠 One-sentence thesis

Fractions can be converted to decimals by dividing the numerator by the denominator, and operations involving both decimals and fractions require converting all numbers to the same form before calculating.

📌 Key points (3–5)

• Conversion method: A fraction bar represents division, so 3/4 means "3 divided by 4."
• Handling non-terminating decimals: When division repeats indefinitely, round to the specified number of decimal places.
• Mixed numbers: Convert the fractional part to a decimal, then add it to the whole number part.
• Common confusion: Complex decimals (like 0.16 and 1/4) must be interpreted correctly—the fraction applies to the last decimal place value.
• Combined operations: When mixing decimals and fractions, convert everything to one form (all decimals or all fractions) before performing calculations.

🔄 Converting fractions to decimals

🔄 The basic conversion rule

A fraction bar can also be a division symbol, so 3/4 means both "3 objects out of 4" and "3 divided by 4."

• To convert any fraction to a decimal, divide the numerator (top number) by the denominator (bottom number).
• Example: 3/4 becomes 0.75 because 3 divided by 4 equals 0.75.
• Example: 1/5 becomes 0.2 because 1 divided by 5 equals 0.2.

🔁 Non-terminating decimals

• Some divisions produce repeating patterns that never end.
• When the excerpt instructs to "round to two decimal places," calculate until the pattern is clear, then round.
• Example: 5/6 equals 0.833... (the 3 repeats), which rounds to 0.83 when rounding to two decimal places.

🧮 Mixed numbers

• A mixed number like 5 and 1/8 means 5 plus 1/8.
• Convert only the fractional part to a decimal, then add it to the whole number.
• Example: 5 and 1/8 = 5 + 0.125 = 5.125 (because 1 divided by 8 equals 0.125).

🧩 Complex decimals

🧩 What complex decimals are

• A complex decimal contains both a decimal and a fraction, such as 0.16 and 1/4.
• This is read as "sixteen and one-fourth hundredths"—the fraction applies to the place value of the last decimal digit.
• Don't confuse: 0.16 and 1/4 is not the same as 0.16 plus 0.25.

🔢 Converting complex decimals

The excerpt shows a multi-step process:

1. Identify the place value of the last decimal digit (e.g., hundredths).
2. Write the decimal part with the fraction over that place value: 16 and 1/4 over 100.
3. Convert the mixed numerator to an improper fraction: (16 times 4 plus 1) divided by 4 equals 65/4.
4. Simplify: 65/4 divided by 100 equals 65 over 400, which simplifies to 13/80.
5. Convert the resulting fraction to a decimal: 13 divided by 80 equals 0.1625.

Example: 0.16 and 1/4 = 0.1625.

➕ Combining operations with decimals and fractions

➕ The conversion-first rule

• When a problem mixes decimals and fractions, convert all numbers to the same form before calculating.
• You can convert everything to decimals or everything to fractions—either works.
• The excerpt examples convert to decimals.

🔢 Order of operations still applies

• Multiply and divide before adding and subtracting.
• Example: 1.85 + 3/8 times 4.1
  • First convert 3/8 to 0.375.
  • Then multiply: 0.375 times 4.1 equals 1.5375.
  • Finally add: 1.85 plus 1.5375 equals 3.3875.

🧮 Working with parentheses

• Perform operations inside parentheses first.
• Example: 5/13 times (4/5 minus 0.28)
  • Convert 0.28 to a fraction: 28/100 simplifies to 7/25.
  • Subtract inside parentheses: 4/5 minus 7/25 equals 20/25 minus 7/25 equals 13/25.
  • Multiply: 5/13 times 13/25 equals 1/5.

📊 Multi-step problems

The excerpt shows a complex example:

Step	Operation	Result
Start	1 and 1/3 + 1/16 - 0.1211	—
Convert	1 and 1/3 = 4/3; express as eighths	1/8 times 3/4
Simplify	3/32 + 1/16	3/32 + 2/32
Add fractions	5/32	0.15625
Subtract	0.15625 - 0.1211	0.03515
Final	Convert back to fraction if needed	703/20,000

Don't confuse: Even when the final answer is a decimal, you may need to convert it back to fraction form depending on the problem requirements.

🧭 Overview

🧠 One-sentence thesis

This document outlines the learning objectives for a chapter on "Ratios and Rates" from a mathematics textbook, specifying what students should be able to do after completing the chapter.

📌 Key points (3–5)

• What the chapter covers: Ratios, rates, proportions, percents, and their applications
• Core skills: Distinguishing between types of numbers, solving proportions, converting between fractions/decimals/percents
• Problem-solving focus: Using the five-step method to solve proportion problems
• Practical applications: Understanding how these concepts apply to real-world situations

📚 Chapter structure

📚 Main topics covered

The chapter is organized into several sections:

• Ratios and Rates (distinguishing denominate vs. pure numbers)
• Proportions (describing and solving them, including rate problems)
• Applications of Proportions (five-step problem-solving method)
• Percent (relationships and conversions)
• Fractions of One Percent (understanding and converting)
• Applications of Percents (base, percent, and percentage)

🎯 Learning outcomes

After completing the chapter, students should be able to:

• Distinguish between different types of numbers and comparisons
• Set up and solve proportions
• Apply systematic methods to solve word problems
• Convert flexibly between fractions, decimals, and percents
• Identify and work with the components of percent problems

🔧 Key skills

🔧 Fundamental distinctions

Students learn to differentiate between:

• Denominate vs. pure numbers: Numbers with units vs. numbers alone
• Ratios vs. rates: Comparing like quantities vs. unlike quantities

🧮 Computational skills

• Proportion solving: Finding missing values in proportional relationships
• Format conversions: Moving between fractions, decimals, and percents
• Percent calculations: Finding percentage, percent, or base

🛠️ Problem-solving method

The five-step method provides a systematic approach to applications, helping students organize their thinking and work through complex problems methodically.

🧭 Overview

🧠 One-sentence thesis

The excerpt provides learning objectives and foundational concepts for comparing quantities through ratios and rates, emphasizing the distinction between denominate and pure numbers and the two methods of comparison—subtraction and division.

📌 Key points (3–5)

• Denominate vs pure numbers: denominate numbers include units (e.g., 8 miles), while pure numbers stand alone (e.g., 8).
• Like vs unlike denominate numbers: like denominate numbers share the same unit (e.g., 8 miles and 3 miles); unlike denominate numbers have different units (e.g., 8 miles and 5 gallons).
• Two ways to compare: subtraction shows how much more one number is; division shows how many times larger or smaller.
• Common confusion: subtraction only works for like denominate numbers or pure numbers; unlike denominate numbers cannot be compared by subtraction.
• Units in division: when dividing like denominate numbers, units cancel out; when dividing unlike denominate numbers, units remain and form a rate.

🔢 Types of numbers

🔢 Denominate numbers

Denominate numbers: numbers together with some specified unit.

• These represent amounts of quantities, not just abstract values.
• Example: 8 miles, 5 gallons, 12 seconds.

🔢 Pure numbers

Pure numbers: numbers that exist purely as numbers and do not represent amounts of quantities.

• They stand alone without units.
• Example: 8, 254, 0, 21 5/8, 2/5, 0.07.

🔍 Like vs unlike denominate numbers

• Like denominate numbers: units are the same.
  • Example: 8 miles and 3 miles.
• Unlike denominate numbers: units are different.
  • Example: 8 miles and 5 gallons.
• Don't confuse: only like denominate numbers (or pure numbers) can be compared by subtraction.

➖ Comparing by subtraction

➖ What subtraction tells you

Comparison of two numbers by subtraction indicates how much more one number is than another.

• Subtraction measures the difference between two quantities.
• It answers: "How much more is one than the other?"

✅ When subtraction works

• Subtraction is valid only if both numbers are:
  • Like denominate numbers, or
  • Both pure numbers.
• Example: 8 miles − 3 miles = 5 miles means "8 miles is 5 miles more than 3 miles."
• Example: 12 − 5 = 7 means "12 is 7 more than 5."

❌ When subtraction fails

• Comparing unlike denominate numbers by subtraction makes no sense.
• Example: 8 miles − 5 gallons = ? (meaningless).
• Don't confuse: you cannot subtract quantities with different units.

➗ Comparing by division

➗ What division tells you

Comparison by division indicates how many times larger or smaller one number is than another.

• Division measures the ratio or multiple between two quantities.
• It answers: "How many times as large is one compared to the other?"

🔄 Dividing like denominate numbers

• When dividing like quantities, units cancel out.
• Example: (8 miles) ÷ (2 miles) = 4 means "8 miles is 4 times as large as 2 miles."
• The result is a pure number because the units divide out.
• Example of use: "I jog 8 miles to your 2 miles. I jog 4 times as many miles as you jog."

🔄 Dividing unlike denominate numbers

• When dividing unlike quantities, units remain and form a rate.
• Example: (30 miles) ÷ (2 gallons) = 15 miles per 1 gallon.
• The result is a rate with compound units (miles per gallon).
• Example of use: "A car goes 30 miles on 2 gallons of gasoline, which is the same as getting 15 miles per gallon."

🔍 Key distinction

Comparison type	Units	Result
Like denominate numbers	Cancel out	Pure number (ratio)
Unlike denominate numbers	Remain	Rate (compound unit)

• Don't confuse: division of like quantities gives a dimensionless ratio; division of unlike quantities gives a rate with units.

🧭 Overview

🧠 One-sentence thesis

Ratios and rates are both comparisons by division, but ratios compare like quantities (or pure numbers) while rates compare unlike quantities, and understanding this distinction is essential for correctly interpreting and simplifying these comparisons.

📌 Key points (3–5)

• Two ways to compare numbers: subtraction shows "how much more," while division shows "how many times larger or smaller."
• Denominate vs pure numbers: denominate numbers include units (like "8 miles"), while pure numbers stand alone (like "8").
• Ratio vs rate distinction: ratios compare like quantities and units cancel out; rates compare unlike quantities and units remain.
• Common confusion: when dividing like quantities (ratio), drop the units; when dividing unlike quantities (rate), keep both units to preserve meaning.
• Simplification: both ratios and rates can be expressed as fractions and reduced to simpler equivalent forms.

🔢 Types of Numbers

🔢 Denominate numbers

Denominate numbers: numbers together with some specified unit.

• These are measurements or quantities with attached units.
• Example: 8 miles, 5 gallons, 12 people.

Like vs unlike denominate numbers:

• Like: same units (8 miles and 3 miles).
• Unlike: different units (8 miles and 5 gallons).

🔢 Pure numbers

Pure numbers: numbers that exist purely as numbers and do not represent amounts of quantities.

• These have no units attached.
• Example: 8, 254, 0.07, fractions like two-fifths.

⚖️ Two Methods of Comparison

➖ Comparison by subtraction

Comparison of two numbers by subtraction indicates how much more one number is than another.

• Rule: You can only subtract if both numbers are pure OR both are like denominate numbers.
• Example: 8 miles minus 3 miles equals 5 miles → "8 miles is 5 miles more than 3 miles."
• Don't confuse: Subtracting unlike quantities (8 miles minus 5 gallons) makes no sense.

➗ Comparison by division

Comparison by division indicates how many times larger or smaller one number is than another.

• Works for pure numbers, like denominate numbers, and unlike denominate numbers.
• Example: 36 divided by 4 equals 9 → "36 is 9 times as large as 4."
• This comparison can be written as a fraction: 36 over 4 equals 9.

🎯 Ratios vs Rates

🎯 What is a ratio?

A ratio: a comparison, by division, of two pure numbers or two like denominate numbers.

• When comparing like quantities, units cancel out (divide out).
• Example: 8 miles over 2 miles equals 4 (no units remain) → "8 miles is 4 times as many miles as 2 miles."
• Example: 36 over 4 equals 9 (pure numbers).

How to write ratios:

• As a fraction: a over b
• In words: "a to b"
• Can be reduced like any fraction.

🎯 What is a rate?

A rate: a comparison, by division, of two unlike denominate numbers.

• When comparing unlike quantities, keep both units.
• Example: 30 miles over 2 gallons equals 15 miles over 1 gallon → "15 miles per gallon."
• Example: 40 dollars over 5 tickets (dollars per ticket).

Why units matter:

• The units tell you what the rate means.
• Example: 16 bananas over 2 bags equals 8 bananas per bag.

🔍 Key distinction table

Comparison type	What it compares	What happens to units	Example
Ratio	Pure numbers OR like denominate numbers	Units cancel out	8 miles ÷ 2 miles = 4
Rate	Unlike denominate numbers	Both units remain	30 miles ÷ 2 gallons = 15 miles/gallon

🧮 Working with Ratios and Rates

🧮 Expressing as fractions

• Both ratios and rates can be written as fractions.
• Example: "3 to 2" becomes 3 over 2.
• Example: "5 books to 4 people" becomes 5 books over 4 people.

🧮 Reducing ratios and rates

• Since they are fractions, they can be simplified.
• Example (ratio): 30 over 2 reduces to 15 over 1 → "30 to 2" is equivalent to "15 to 1."
• Example (rate): 4 televisions over 12 people reduces to 1 television over 3 people → "for every 1 television, there are 3 people."

Meaning preservation:

• Reducing does not change the relationship.
• The simplified form is equivalent to the original.

🧮 Converting between forms

• Fraction form: a over b
• Verbal form: "a to b"
• Example: 9 over 5 becomes "9 to 5."
• Example: 25 miles over 2 gallons becomes "25 miles to 2 gallons."

📝 Practice Scenarios

📝 Subtraction comparisons

• 10 diskettes minus 2 diskettes equals 8 diskettes → "10 diskettes is 8 diskettes more than 2 diskettes."
• 16 bananas minus 2 bags → comparison makes no sense (unlike units).

📝 Division comparisons

• 10 diskettes over 2 diskettes equals 5 → "10 diskettes is 5 times as many diskettes as 2 diskettes."
• 16 bananas over 2 bags equals 8 bananas per bag → a rate showing distribution.

📝 Simplification examples

• 2 mechanics over 4 wrenches reduces to 1 mechanic over 2 wrenches → "1 mechanic to 2 wrenches."
• 15 video tapes over 18 video tapes reduces to 5 over 6 → "5 to 6" (units cancel because they're alike).

🧭 Overview

🧠 One-sentence thesis

Proportions are statements that two ratios or rates are equal, and they can be solved by using cross products to find an unknown value through a missing-factor approach.

📌 Key points (3–5)

• What a proportion is: a statement that two ratios or rates are equal.
• How to solve for an unknown: use cross products to create a multiplication statement, then divide the product by the known factor.
• Common confusion: when setting up proportions involving rates, the same type of unit must appear in the same position (numerator or denominator) on both sides; mixing positions produces incorrect comparisons.
• Reading proportions: "3 is to 5 as 12 is to 20" corresponds to the fraction form 3/5 = 12/20.
• Why it matters: proportions are used to solve many practical problems by comparing quantities in a consistent way.

📐 What proportions are and how to read them

📐 Definition and structure

A proportion is a statement that two ratios or rates are equal.

• A ratio is a comparison by division of two pure numbers or two like denominate numbers.
• A rate is a comparison by division of two unlike denominate numbers.
• Proportions can be written in fraction form or read aloud as sentences.

🗣️ Reading and writing proportions

Fraction form examples:

Written form	Read as
3/5 = 12/20	3 is to 5 as 12 is to 20
(10 items)/(5 dollars) = (2 items)/(1 dollar)	10 items is to 5 dollars as 2 items is to 1 dollar
50/1 = 300/6	50 milligrams of vitamin C is to 1 tablet as 300 milligrams of vitamin C is to 6 tablets

Sentence-to-fraction examples:

• "15 is to 4 as 75 is to 20" → 15/4 = 75/20
• "2 plates are to 1 tray as 20 plates are to 10 trays" → (2 plates)/(1 tray) = (20 plates)/(10 trays)

🔍 Finding the missing factor in a proportion

🔍 The cross-product method

• Many practical problems involve three known numbers and one unknown (represented by a letter such as x).
• Two fractions are equivalent if and only if their cross products are equal.
• Example: in the proportion x/4 = 20/16, the cross product gives 16·x = 4·20.

Key insight:

Regardless of where the unknown letter appears, the cross product always produces:

(number) · (letter) = (number) · (number)

This is a missing factor statement.

➗ Solving for the unknown

The missing factor can be determined by dividing the product by the known factor:
x = (product) ÷ (known factor)

Step-by-step examples:

Proportion	Cross product	Solve for x	Result
x/4 = 20/16	16·x = 4·20 = 80	x = 80 ÷ 16	x = 5
5/x = 20/16	5·16 = 20·x → 80 = 20·x	x = 80 ÷ 20	x = 4
16/3 = 64/x	16·x = 64·3 = 192	x = 192 ÷ 16	x = 12
9/8 = x/40	9·40 = 8·x → 360 = 8·x	x = 360 ÷ 8	x = 45

✅ Checking your answer

After finding x, substitute it back into the original proportion to verify that the two ratios are equal.

Example: if x/4 = 20/16 and you find x = 5, check that 5/4 = 20/16 (both simplify to the same value).

⚠️ Proportions involving rates: the form matters

⚠️ Correct setup for rates

When a rate involves two types of units (unit type 1 and unit type 2), you can write:

• (unit type 1)/(unit type 2) = (unit type 1)/(unit type 2)
  or
• (unit type 2)/(unit type 1) = (unit type 2)/(unit type 1)

Both forms produce a cross product of the type:

(unit type 1) · (unit type 2) = (unit type 1) · (unit type 2)

which is a valid comparison.

🚫 Incorrect setup: mixing unit positions

Do NOT write:

(unit type 1)/(unit type 2) = (unit type 2)/(unit type 1)

This produces a cross product of the form:

(unit type 1) · (unit type 1) = (unit type 2) · (unit type 2)

which is an incorrect comparison.

Example of an incorrect proportion:

(2 hooks)/(3 poles) = (6 poles)/(4 hooks)

• Numerically wrong: 2·4 ≠ 3·6
• Logically wrong: the cross product produces "hooks are to hooks as poles are to poles," which makes no sense.

Examples of correctly expressed proportions:

• (3 joggers)/(100 feet) = (6 joggers)/(200 feet)
• (40 miles)/(2 gallons) = (80 miles)/(4 gallons)
• (18 grams cobalt)/(10 grams silver) = (36 grams cobalt)/(20 grams silver)

🧭 How to avoid confusion

• Always place the same type of unit in the same position (numerator or denominator) on both sides.
• Before solving, check that your proportion makes sense when read aloud.
• Example: "3 joggers are to 100 feet as 6 joggers are to 200 feet" is a sensible comparison; "3 joggers are to 100 feet as 200 feet are to 6 joggers" is not.

🛠️ The five-step method for solving proportion problems

🛠️ Overview of the method

The excerpt introduces a systematic approach to solving practical problems using proportions:

1. Identify the unknown quantity and represent it with a letter (there will be only one unknown).
2. Identify the three specified numbers given in the problem.
3. Determine which comparisons are to be made and set up the proportion correctly.

(The excerpt ends before completing the five steps; the remaining steps are not provided.)

🎯 Why the first step is most important

The first and most important part of solving a proportion problem is to determine, by careful reading, what the unknown quantity is and to represent it with some letter.

• Careful reading ensures you understand what you are solving for.
• There will be only one unknown in a problem.
• Example: "5 hats are to 4 coats as x hats are to 24 coats" → the unknown is the number of hats, represented by x.

📝 Example applications

Sentence-to-proportion examples:

Sentence	Proportion	Solution
5 hats are to 4 coats as x hats are to 24 coats	5/4 = x/24	x = 30
1 spacecraft is to 7 astronauts as 5 spacecraft are to x astronauts	1/7 = 5/x	x = 35
18 calculators are to 90 calculators as x students are to 150 students	18/90 = x/150	x = 30

Don't confuse: the unknown can appear in any position (numerator or denominator, left or right side); the cross-product method works the same way regardless.

🧭 Overview

🧠 One-sentence thesis

Proportion problems can be systematically solved using a five-step method that begins by identifying the unknown quantity and ends with interpreting the numerical answer in context.

📌 Key points (3–5)

• The five-step method: a structured approach to solving proportion problems—identify the unknown, identify three known numbers, set up the proportion, solve it, and interpret the result.
• Step 1 is critical: many problems fail because students skip determining what the unknown quantity is and representing it with a letter.
• How to set up proportions: match corresponding comparisons (e.g., inches to miles on one side, inches to unknown miles on the other).
• Common confusion: forgetting to interpret the final number—the numeric answer must be translated back into a sentence with appropriate units.
• Cross-multiplication solves the proportion: multiply diagonally across the equals sign, then divide to isolate the unknown.

🛠️ The Five-Step Method

🛠️ What the method is

The five-step method for solving proportion problems:

1. By careful reading, determine what the unknown quantity is and represent it with some letter.
2. Identify the three specified numbers.
3. Determine which comparisons are to be made and set up the proportion.
4. Solve the proportion (using cross-multiplication).
5. Interpret and write a conclusion in a sentence with the appropriate units of measure.

• The excerpt emphasizes that Step 1 is extremely important: "Many problems go unsolved because time is not taken to establish what quantity is to be found."
• There will be only one unknown in a problem.
• Always begin by determining the unknown quantity and representing it with a letter.

🔍 Why Step 1 matters

• The excerpt states: "When solving an applied problem, always begin by determining the unknown quantity and representing it with a letter."
• Without a clear unknown, you cannot set up the correct proportion.
• Example: In a map problem, the unknown might be "number of miles represented by 8 inches," so you write "Let x = number of miles."

📐 Setting Up the Proportion (Steps 2 and 3)

📐 Identifying the three specified numbers (Step 2)

• After defining the unknown, list the three numbers given in the problem.
• Example (map problem): 2 inches, 25 miles, 8 inches.
• Example (acid solution): 7 parts water, 2 parts acid, 20 parts acid.

🔗 Matching comparisons (Step 3)

• Determine which quantities correspond to each other and write two ratios.
• The key is to match like to like: inches to miles on one side, inches to miles on the other.
• Example (map):
  • 2 inches to 25 miles → 2 inches / 25 miles
  • 8 inches to x miles → 8 inches / x miles
  • Proportion: 2/25 = 8/x
• The excerpt notes: "Proportions involving ratios and rates are more readily solved by suspending the units while doing the computations."
  • This means you can drop the units temporarily and work with just the numbers: 2/25 = 8/x.

🧩 Common patterns

Problem type	First ratio	Second ratio
Map scale	known inches / known miles	given inches / unknown miles
Mixture	known parts A / known parts B	given parts A / unknown parts B
Shadow	known height / known shadow	unknown height / given shadow
Ratio in population	known men / known women	given men / unknown women

🧮 Solving the Proportion (Step 4)

🧮 Cross-multiplication

• Once the proportion is set up (e.g., 2/25 = 8/x), perform cross-multiplication:
  • Multiply diagonally: 2 · x = 8 · 25
  • This gives: 2x = 200
• Then divide to isolate the unknown:
  • x = 200 / 2
  • x = 100
• Example (acid solution): 7/2 = n/20
  • Cross-multiply: 7 · 20 = 2 · n → 140 = 2n
  • Divide: n = 140 / 2 = 70

🔢 Working with mixed numbers

• Example (shadow problem): A 5-foot girl casts a 3 and 1/3 foot shadow; a person casts a 3-foot shadow.
  • Proportion: 5 / (3 and 1/3) = h / 3
  • Cross-multiply: 5 · 3 = (3 and 1/3) · h → 15 = (10/3) · h
  • Divide 15 by 10/3: h = 15 · (3/10) = 9/2 = 4 and 1/2 feet.

📝 Interpreting the Result (Step 5)

📝 Writing a conclusion

• Step 5 requires you to interpret and write a conclusion in a sentence with the appropriate units of measure.
• Don't confuse: the numeric answer (e.g., x = 100) is not the final answer—you must state what it represents.
• Example (map): "If 2 inches represents 25 miles, then 8 inches represents 100 miles."
• Example (acid solution): "7 parts water to 2 parts acid indicates 70 parts water to 20 parts acid."
• Example (shadow): "A person who casts a 3-foot shadow at this particular time of the day is 4 and 1/2 feet tall."

🎯 Why interpretation matters

• The excerpt emphasizes that Step 1 (defining the unknown) and Step 5 (interpreting the result) bookend the process.
• Without Step 5, you have a number but no meaning.
• Always include the correct units (miles, parts, feet, etc.) in your final sentence.

🧪 Worked Examples

🧪 Map scale

• Problem: On a map, 2 inches represents 25 miles. How many miles are represented by 8 inches?
• Step 1: Let x = number of miles represented by 8 inches.
• Step 2: 2 inches, 25 miles, 8 inches.
• Step 3: 2/25 = 8/x
• Step 4: 2x = 200 → x = 100
• Step 5: 8 inches represents 100 miles.

🧪 Mixture ratio

• Problem: An acid solution is 7 parts water to 2 parts acid. How many parts of water are in a solution with 20 parts acid?
• Step 1: Let n = number of parts of water.
• Step 2: 7 parts water, 2 parts acid, 20 parts acid.
• Step 3: 7/2 = n/20
• Step 4: 140 = 2n → n = 70
• Step 5: 70 parts water to 20 parts acid.

🧪 Shadow problem

• Problem: A 5-foot girl casts a 3 and 1/3 foot shadow. How tall is a person who casts a 3-foot shadow?
• Step 1: Let h = height of the person.
• Step 2: 5 feet (height), 3 and 1/3 feet (shadow), 3 feet (shadow).
• Step 3: 5 / (3 and 1/3) = h / 3
• Step 4: 15 = (10/3)h → h = 9/2 = 4 and 1/2
• Step 5: The person is 4 and 1/2 feet tall.

🧪 Population ratio

• Problem: The ratio of men to women in a town is 3 to 5. How many women are there if there are 19,200 men?
• Step 1: Let x = number of women.
• Step 2: 3 (men), 5 (women), 19,200 (men).
• Step 3: 3/5 = 19,200/x
• Step 4: 3x = 96,000 → x = 32,000
• Step 5: There are 32,000 women in town.

🧪 Win-loss rate

• Problem: A baseball team's win-to-loss rate is 9 to 2. How many games did they lose if they won 63?
• Step 1: Let n = number of games lost.
• Step 2: 9 (wins), 2 (losses), 63 (wins).
• Step 3: 9/2 = 63/n
• Step 4: 9n = 126 → n = 14
• Step 5: The team had 14 losses.

🔑 Key Reminders

🔑 One unknown only

• The excerpt states: "There will be only one unknown in a problem."
• If you find yourself with two unknowns, re-read the problem to identify which quantity is actually being asked for.

🔑 Units can be suspended during computation

• The excerpt advises: "Proportions involving ratios and rates are more readily solved by suspending the units while doing the computations."
• This means you can work with just the numbers (e.g., 2/25 = 8/x) and reattach units in Step 5.

🔑 Don't skip Step 5

• A common mistake is to stop after finding the numeric value.
• Always write a full sentence with units to interpret what the number means in the context of the problem.

🧭 Overview

🧠 One-sentence thesis

This excerpt does not contain substantive content on division of fractions; instead, it presents unrelated practice problems on ratios, proportions, and a section on percent conversions.

📌 Key points (3–5)

• The excerpt titled "6 Division of Fractions" does not actually explain division of fractions.
• Most content consists of practice problems on ratios, proportions, and unit conversions (e.g., paper stacks, fertilizer coverage, photograph enlargement).
• A substantive section on percents is included, explaining the relationship between ratios, fractions, decimals, and percents.
• Common confusion: percent means "per hundred," not simply "a part of something"—it is always a ratio comparing to 100.
• The excerpt provides conversion techniques between fractions, decimals, and percents but does not address fraction division.

📊 What the excerpt actually contains

📊 Practice problems (non-division content)

• The excerpt opens with miscellaneous word problems:
  • Ratio problems (e.g., algebra to geometry class ratios).
  • Odds and probability problems.
  • Proportion problems (e.g., paper stacks, fertilizer coverage, recipe scaling, photograph enlargement).
• These problems do not explain or demonstrate division of fractions as a concept.

📊 Review exercises

• The excerpt includes "Exercises for Review" referencing prior sections (e.g., finding products, determining missing numerators, subtracting decimals, solving proportions).
• None of these exercises focus on dividing fractions.

🔢 Percent concepts (the main substantive content)

🔢 What percent means

Percent means "for each hundred," or "for every hundred." The symbol % is used to represent the word percent.

• The word comes from Latin: "per" (for each) + "centum" (hundred).
• A percent is a ratio in which one number is compared to 100.
• Example: The ratio 26 to 100 can be written as 26% (read as "twenty-six percent").
• Example: 25 cents is 25/100 of a dollar, so 25 cents is 25% of one dollar.

🔢 Writing percents as fractions

• Any percent can be written as a fraction with denominator 100.
• Example: 38% = 38/100.
• Example: 210% = 210/100 or the mixed number 2 and 10/100, which simplifies to 2.1.
• Don't confuse: percents greater than 100% are valid—they represent ratios greater than 1.

🔄 Conversion techniques

🔄 Overview of conversions

The excerpt provides a table summarizing how to convert among fractions, decimals, and percents:

From	To Decimal	To Percent	To Fraction
Fraction	Divide numerator by denominator	Convert to decimal first, then move decimal point 2 places right and affix %	Already a fraction
Decimal	Already a decimal	Move decimal point 2 places right and affix %	Read the decimal and reduce the resulting fraction
Percent	Move decimal point 2 places left and drop %	Already a percent	Drop % and write the number "over" 100; reduce if possible

🔄 Decimal to percent

• Move the decimal point 2 places to the right and affix the % symbol.
• Example: 0.75 = 75/100 = 75%.
• Example: 5.64 = 564%.

🔄 Percent to decimal

• Move the decimal point 2 places to the left and drop the % symbol.
• Example: 12% = 12/100 = 0.12.
• Example: 461% = 4.61.

🔄 Fraction to percent

• First convert the fraction to a decimal (divide numerator by denominator).
• Then convert the decimal to a percent (move decimal point 2 places right and affix %).
• Example: 3/5 = 0.6 = 60%.
• Example: 11/8 = 1.375 = 137.5%.
• Don't confuse: fractions greater than 1 yield percents greater than 100%.

🔄 Percent to fraction

• Drop the % symbol and write the number over 100, then reduce if possible.
• Example: 42% = 42/100 = 21/50.
• Alternative: convert percent to decimal first, then write as a fraction (42% = 0.42 = 42/100 = 21/50).
• Example: 80% = 80/100 = 4/5.
• Example: 12.5% = 12.5/100 = 1/8.

🧩 Practice exercises included

🧩 Decimal-to-percent conversions

• Convert 0.25 → 25%.
• Convert 1.42 → 258% (note: this appears to be a typo in the excerpt; 1.42 should be 142%, not 258%).
• Convert 533.01 → 53,301%.

🧩 Percent-to-decimal conversions

• Convert 15% → 0.15.
• Convert 0.78% → 0.0078.
• Convert 0.001% → 0.000001 (implied by the pattern).

🧩 Fraction-to-percent conversions

• Convert 1/5 → 20%.
• Convert 5/8 → 62.5%.
• Convert 41/25 → 164%.

🧩 Percent-to-fraction conversions

• Convert 80% → 4/5.
• Convert 25% → 1/4.
• Convert 512.5% → 41/8 or 5 and 1/8.

⚠️ Note on the title mismatch

• The excerpt is titled "6 Division of Fractions" but contains no explanation or examples of dividing fractions.
• The substantive content is entirely about percents and their conversions.
• This appears to be a mislabeling or an excerpt taken from the wrong section of the source material.

🧭 Overview

🧠 One-sentence thesis

Fractions of one percent (like ½% or ⅗%) represent portions smaller than 1%, and converting them to fractions or decimals requires understanding that they mean "that fraction of 1%," which translates to multiplying by 1/100.

📌 Key points (3–5)

• What fractions of 1% mean: percents like ½% or ⅗% represent fractions of 1%, not full percentages—they haven't reached 1% yet.
• Core conversion method: "of" means "times," so ½% = ½ × 1/100 = 1/200.
• Nonterminating fractions require rounding: when converting fractions like ⅔% to decimals, round the fraction first, then shift the decimal point.
• Common confusion: don't treat ½% as ½ directly; it's ½ of 1%, which is much smaller (1/200, not 1/2).
• Decimal place planning: to get a final decimal with n places, round the fraction to (n - 2) places before removing the percent sign.

🔢 Understanding fractions of one percent

🔢 What "fraction of 1%" means

Fractions of one percent: percents where 1% has not been attained, such as ½%, ⅗%, ⅝%, or 7/11%.

• These represent portions of the 1% unit.
• The excerpt emphasizes: ½% = ½ of 1%, not ½ as a standalone number.
• Example: ⅗% means "three-fifths of one percent," which is much smaller than 3/5.

🧮 Why "of" means "times"

• "Percent" means "for each hundred," so 1% = 1/100.
• "Of" in mathematics translates to multiplication.
• Therefore: ½% = ½ of 1% = ½ × 1/100 = 1/200.
• Example: ⅝% = ⅝ × 1/100 = 5/800.

Don't confuse: ½% is not the same as ½ (which would be 50%). It's ½ × 1/100 = 0.005, a much smaller value.

🔄 Converting to fractions

🔄 The multiplication method

The excerpt shows a consistent pattern:

Percent	Interpretation	Calculation	Result
½%	½ of 1%	½ × 1/100	1/200
⅗%	⅗ of 1%	⅗ × 1/100	3/500
⅝%	⅝ of 1%	⅝ × 1/100	5/800
7/11%	7/11 of 1%	7/11 × 1/100	7/1100

📝 Step-by-step process

1. Recognize the fraction before the percent sign.
2. Multiply that fraction by 1/100.
3. Simplify if possible.

Example from excerpt: ⅔% = ⅔ × 1/100 = (2×1)/(3×100) = 2/300 = 1/150.

🔄 Converting to decimals (nonterminating cases)

⚠️ The nonterminating problem

• Some fractions (like ⅔ = 0.6666...) have decimal representations that never end.
• The excerpt states: "it cannot be expressed exactly as a decimal."
• Solution: round to a specified number of decimal places.

📐 The two-step conversion method

To convert a nonterminating fraction of 1% to a decimal:

1. Convert the fraction to a rounded decimal.
2. Move the decimal point two digits to the left and remove the percent sign.

Key insight: If you want n final decimal places, round the fraction to (n - 2) places first, because removing the percent sign accounts for two places.

🧮 Worked examples from the excerpt

Example 1: Convert ⅔% to a three-place decimal.

• Want 3 final places → round ⅔ to (3 - 2) = 1 place.
• ⅔ = 0.6666... → rounds to 0.7.
• So ⅔% = 0.7%.
• Move decimal two places left: 0.007.

Example 2: Convert 5 4/11% to a four-place decimal.

• Want 4 final places → round 4/11 to (4 - 2) = 2 places.
• 4/11 = 0.3636... → rounds to 0.36.
• So 5 4/11% = 5.36%.
• Move decimal two places left: 0.0536.

Example 3: Convert 28 5/9% to ten-thousandths (four places).

• Want 4 final places → round 5/9 to 2 places.
• 5/9 = 0.5555... → rounds to 0.56.
• So 28 5/9% = 28.56%.
• Move decimal two places left: 0.2856.

🎯 Why the (n - 2) rule works

• Removing the percent sign = moving decimal two places left.
• This automatically adds two decimal places.
• If you want 3 total places and the shift gives you 2, you need 1 more from rounding the fraction.
• Example: 0.7% → 0.007 (the shift from % to decimal gave 2 places; rounding to 1 place gave the third).

🧭 Overview

🧠 One-sentence thesis

When fractions share the same denominator, you add or subtract only the numerators and keep the common denominator unchanged.

📌 Key points (3–5)

• The core rule: for like denominators, add or subtract numerators only; place the result over the common denominator.
• Never add or subtract denominators: doing so produces nonsensical results (e.g., two halves would equal one half).
• Always reduce: after adding or subtracting, simplify the resulting fraction to lowest terms if possible.
• Common confusion: mistakenly adding denominators leads to absurd conclusions; denominators represent the "size of the pieces" and stay constant.

➕ Adding fractions with like denominators

🧮 The addition method

Method of Adding Fractions Having Like Denominators: To add two or more fractions that have the same denominators, add the numerators and place the resulting sum over the common denominator. Reduce, if necessary.

• The denominator tells you the size of each piece; it does not change when you combine pieces.
• Only the numerators (the count of pieces) are added together.

📐 How it works step-by-step

1. Check that denominators are identical.
2. Add the numerators.
3. Write the sum over the common denominator.
4. Reduce the result if possible.

Example: 3/7 + 2/7 = (3+2)/7 = 5/7
Example: 1/8 + 3/8 = (1+3)/8 = 4/8 = 1/2 (reduced)

⚠️ Why you never add denominators

• The excerpt shows what happens if you mistakenly add denominators:
  • 1/2 + 1/2 = (1+1)/(2+2) = 2/4 = 1/2
  • This claims "two halves equals one half," which is absurd.
• Don't confuse: the denominator is the shared unit (like "fifths" or "eighths"); adding units makes no sense.

➖ Subtracting fractions with like denominators

🧮 The subtraction method

Subtraction of Fractions with Like Denominators: To subtract two fractions that have like denominators, subtract the numerators and place the resulting difference over the common denominator. Reduce, if possible.

• The logic mirrors addition: only the numerators change; the denominator stays the same.

📐 How it works step-by-step

1. Verify denominators are the same.
2. Subtract the second numerator from the first.
3. Place the difference over the common denominator.
4. Reduce if needed.

Example: 3/5 − 1/5 = (3−1)/5 = 2/5
Example: 8/6 − 2/6 = (8−2)/6 = 6/6 = 1

⚠️ Why you never subtract denominators

• If you mistakenly subtract denominators:
  • 7/15 − 4/15 = (7−4)/(15−15) = 3/0
  • Division by zero is undefined—the operation breaks down completely.
• Don't confuse: subtracting the "size of pieces" (denominators) destroys the meaning of the fraction.

🔍 Key examples and practice patterns

Operation	Example	Step-by-step	Result
Addition	4/9 + 5/9	(4+5)/9	9/9 = 1
Addition (reduce)	7/8 + 5/8	(7+5)/8 = 12/8	3/2
Subtraction	16/9 − 2/9	(16−2)/9	14/9
Subtraction (result zero)	1/2 − 1/2	(1−1)/2	0

• Notice that results can be whole numbers (9/9 = 1), improper fractions (14/9), or zero.
• Always check if the final fraction can be reduced.

🧭 Overview

🧠 One-sentence thesis

The excerpt does not contain substantive content on addition and subtraction of fractions with like denominators; it consists of exercise problems, learning objectives for estimation techniques, and an introduction to estimation by rounding.

📌 Key points (3–5)

• The excerpt includes scattered exercise problems on ratios, decimals, percents, and fractions, but no instructional content on adding or subtracting fractions with like denominators.
• A section on "Estimation by Rounding" introduces the concept of estimating computation results by rounding numbers to one or two nonzero digits.
• Rounding for estimation is done by convenience rather than strict rounding rules, so different people may produce different estimated results.
• The excerpt emphasizes that estimation helps determine an expected value before or after a computation to check reasonableness.
• Common confusion: estimation rounding vs. standard rounding—estimation prioritizes convenience (e.g., rounding 26 to 20 instead of 30 when dividing 80, because 80 ÷ 20 is easier).

🔍 What the excerpt actually covers

🔍 Exercise problems only

• The excerpt opens with a series of exercise problems covering ratios, decimal-to-percent conversions, percent-to-fraction conversions, and word problems.
• These problems do not explain the title topic (addition and subtraction of fractions with like denominators).
• No worked examples or instructional text on fraction operations appear in the excerpt.

🔍 Learning objectives for estimation

• The excerpt lists objectives for a chapter on "Techniques of Estimation," including:
  • Understanding the reason for estimation.
  • Estimating results of addition, multiplication, subtraction, or division using rounding.
  • Understanding clustering and the distributive property.
  • Estimating sums of fractions by rounding fractions.
• These objectives are not developed in the excerpt; only the introduction to "Estimation by Rounding" begins.

🧮 Estimation by Rounding (the only instructional content)

🧮 Why estimate

Estimation: the process of determining an expected value of a computation.

• Before starting a computation, estimation gives an idea of what value to expect.
• After completing a computation, estimation helps verify if the result is reasonable.
• Common words used in estimation: about, near, and between.

🧮 How the rounding technique works

Estimation by rounding: the rounding technique estimates the result of a computation by rounding the numbers involved to one or two nonzero digits.

• The excerpt emphasizes rounding to one or two nonzero digits to simplify mental arithmetic.
• Example: to estimate 2,357 + 6,106, round 2,357 to 2,400 (two nonzero digits) and 6,106 to 6,100 (two nonzero digits), then add 2,400 + 6,100.

⚠️ Rounding by convenience, not strict rules

• Key difference from standard rounding: estimation rounding uses convenience as the guide, not the hard-and-fast rounding rules taught elsewhere.
• Example: when estimating 80 ÷ 26, you might round 26 to 20 instead of 30, because 80 is more conveniently divided by 20 (80 ÷ 20 = 4) than by 30.
• Don't confuse: this is not "wrong" rounding—it is deliberate simplification to make mental math easier.

🔄 Results may vary

• Because different people find different numbers convenient, estimation results can differ from person to person.
• The excerpt explicitly states: "Results may vary."
• This variability is acceptable in estimation, since the goal is an approximate sense of the answer, not precision.

📝 Summary of missing content

📝 No instruction on the title topic

• The excerpt does not explain how to add or subtract fractions with like denominators.
• No definitions, worked examples, or procedures for fraction operations appear.
• The title topic is not addressed in the provided text.

🧭 Overview

🧠 One-sentence thesis

Rounding fractions to simple benchmarks (¼, ½, ¾, 0, and 1) provides a quick way to estimate sums and differences, helping students check whether their exact answers are reasonable.

📌 Key points (3–5)

• What the technique does: rounds fractions to common benchmarks (¼, ½, ¾, 0, 1) before adding or subtracting.
• Why it matters: gives a quick sense of the expected result and helps verify that exact calculations are reasonable.
• How to apply it: look at each fraction and decide which benchmark it is closest to, then perform simple arithmetic with those benchmarks.
• Common confusion: estimates may vary between people because rounding choices can differ—there is no single "correct" estimate.
• Works for mixed numbers too: separate the whole-number parts and fractional parts, estimate each, then combine.

🎯 Core concept

🎯 What estimation by rounding fractions means

Estimation by rounding fractions: the process of approximating the result of a computation involving fractions by rounding each fraction to a simple benchmark value.

• The excerpt emphasizes that fractions are "commonly rounded to ¼, ½, ¾, 0, and 1."
• This is not about getting the exact answer; it is about getting close enough to know if your exact answer makes sense.
• The technique trades precision for speed and simplicity.

🔄 Why results may vary

• Different people may round the same fraction differently depending on what seems closest or most convenient.
• The excerpt repeatedly notes "results may vary" and "remember that rounding may cause estimates to vary."
• Example: Someone might round 5/12 to ½, while another person might round it to 0; both are acceptable estimates depending on perspective.

🧮 How to estimate with simple fractions

🧮 Rounding individual fractions

• Look at the numerator and denominator and decide which benchmark (0, ¼, ½, ¾, or 1) the fraction is nearest.
• Example from the excerpt: "3/5 is about ½, and 5/12 is about ½."
• Then add or subtract the rounded benchmarks instead of the original fractions.

✅ Comparing estimate to exact value

• After estimating, the excerpt recommends finding the exact value to see how close the estimate was.
• Example: The estimate for 3/5 + 5/12 is ½ + ½ = 1. The exact value is 61/60, "a little more than 1."
• This comparison helps build intuition about whether an estimate is reasonable.

🔢 Working with mixed numbers

🔢 Separate whole and fractional parts

• When estimating sums of mixed numbers, handle the whole-number parts and fractional parts separately.
• Example from the excerpt: "5 3/8 + 4 9/10 + 11 1/5"
  • Whole parts: 5 + 4 + 11 = 20
  • Fractional parts: 3/8 ≈ ¼, 9/10 ≈ 1, 1/5 ≈ ¼
  • Estimate of fractional sum: ¼ + 1 + ¼ = 1½
  • Total estimate: 20 + 1½ = 21½
• The exact answer is 21 19/40, "a little less than 21½."

🎲 Choosing benchmarks for mixed-number fractions

• The excerpt shows flexibility: 3/8 can be rounded to ¼, 9/10 to 1 (because it is very close), and 1/5 to ¼.
• Don't confuse: you are not required to round every fraction to the same benchmark; each fraction is rounded independently based on its own value.

📊 Practice patterns from the excerpt

📊 Example patterns

Original sum	Rounded estimate	Exact value	Comparison
5/8 + 5/12	½ + ½ = 1	25/24 = 1 1/24	Very close
7/9 + 3/5	1 + ½ = 1½	1 17/45	Close
8 4/15 + 3 7/10	8¼ + 3¾ = 12	11 29/30	Slightly off
16 1/20 + 4 7/8	16 + 5 = 21	20 37/40	Reasonably close

🧪 When estimates are less accurate

• The excerpt shows that sometimes the estimate can be a full unit away from the exact answer (e.g., estimate 21, exact 20 37/40).
• This is acceptable because the goal is a rough sense of magnitude, not precision.
• The technique is still useful for catching major errors (e.g., if you calculated 30 instead of 21, you would know something went wrong).

⚠️ Key reminders

⚠️ No single correct estimate

• Because rounding is flexible, two students might produce different estimates for the same problem and both be correct.
• Example: For 1/20, one person might round to 0, another to ¼; both are defensible.

⚠️ Purpose is verification, not replacement

• The excerpt consistently asks students to "find the exact value and compare this result to the estimated value."
• Estimation does not replace exact calculation; it complements it by providing a reasonableness check.

🧭 Overview

🧠 One-sentence thesis

The clustering technique provides a quicker way to estimate sums of more than two numbers when several numbers are close to one particular value, allowing you to group and multiply instead of adding each number individually.

📌 Key points (3–5)

• When to use clustering: when adding more than two numbers and several of them are close to (cluster around) one particular number.
• How clustering works: group numbers that are near the same value, multiply that value by how many numbers cluster around it, then add the group totals.
• Clustering vs rounding: rounding could also be used, but clustering is quicker when several numbers are close to one value.
• Common confusion: clustering requires noticing which numbers are "close to" a common value—don't force clustering if numbers are spread out; the technique works best when natural groupings exist.
• Results may vary: estimates depend on which cluster values you choose, but they should be close to the actual sum.

🔍 What clustering means

🔍 The concept of clustering

Cluster: when numbers are seen to be close to one particular number.

• Clustering is not about exact equality; it's about numbers being "near" or "close to" a common value.
• The excerpt emphasizes that this technique applies when you have more than two numbers to add.
• Example: In the sum 32 + 68 + 29 + 73, notice that 68 and 73 are both close to 70, and 32 and 29 are both close to 30.

🎯 Why clustering is useful

• The excerpt states that clustering "provides a quicker estimate" than rounding when several numbers cluster around one value.
• Instead of rounding each number separately and then adding, you identify groups and use multiplication.
• Don't confuse: clustering is a choice—you can still use rounding, but clustering is faster when the pattern exists.

🧮 How to apply the clustering technique

🧮 Step-by-step process

1. Identify clusters: look at all the numbers and notice which ones are close to the same value.
2. Count how many cluster: determine how many numbers are near each cluster value.
3. Multiply: for each cluster, multiply the cluster value by the count of numbers in that group.
4. Add the group totals: sum the results from each cluster to get the estimate.

📐 Example breakdown

The excerpt gives: 32 + 68 + 29 + 73

• First cluster: 68 and 73 both cluster around 70, so their sum is about 2 times 70 equals 140.
• Second cluster: 32 and 29 both cluster around 30, so their sum is about 2 times 30 equals 60.
• Estimated sum: (2 times 30) plus (2 times 70) equals 60 plus 140 equals 200.
• Actual sum: 32 + 68 + 29 + 73 equals 202 (the estimate is very close).

🔢 Handling different cluster sizes

• The excerpt shows examples with different numbers of items per cluster:
  • Two numbers clustering (e.g., 88 and 91 near 90).
  • Three numbers clustering (e.g., 17, 21, and 18 near 20).
  • Mixed clusters in one problem (e.g., 61 and 57 near 60; 48, 49, and 52 near 50).
• Example: 17 + 21 + 48 + 18 → three numbers (17, 21, 18) cluster near 20, so 3 times 20 equals 60; 48 is about 50; total estimate is 60 plus 50 equals 110 (actual sum is 104).

📊 Patterns and variations

📊 Cluster values at different scales

The excerpt demonstrates clustering at various magnitudes:

Example sum	Cluster groups	Estimate calculation	Actual sum
27 + 48 + 31 + 52	(27, 31 near 30) and (48, 52 near 50)	(2 times 30) plus (2 times 50) equals 160	158
88 + 21 + 19 + 91	(88, 91 near 90) and (21, 19 near 20)	(2 times 90) plus (2 times 20) equals 220	219
706 + 321 + 293 + 684	(706, 684 near 700) and (321, 293 near 300)	(2 times 700) plus (2 times 300) equals 2,000	2,004

• The technique scales from small numbers (around 20–50) to large numbers (around 300–700).
• The principle remains the same regardless of magnitude.

🎲 Results may vary

• The excerpt repeatedly notes "Results may vary."
• Different people might choose slightly different cluster values (e.g., clustering 28 around 30 vs 25).
• The goal is a reasonable estimate, not a single "correct" answer.
• Don't confuse: variation in estimates is expected and acceptable as long as the reasoning is sound.

⚠️ When clustering applies

⚠️ Requirements for clustering

• More than two numbers: the excerpt specifies this technique is for adding more than two numbers.
• Visible clustering: several numbers must be "seen to cluster" around particular values.
• If numbers are widely spread with no natural groupings, clustering may not be the best choice.

⚠️ Clustering vs other estimation methods

• The excerpt mentions that "the rounding technique could also be used."
• Clustering is preferred when the clustering pattern exists because it is quicker.
• Example: if you have 27 + 48 + 31 + 52, noticing the pairs (27, 31) and (48, 52) cluster makes the calculation faster than rounding each individually.

🧭 Overview

🧠 One-sentence thesis

The distributive property allows us to obtain exact multiplication results by distributing a factor to each addend or subtrahend in parentheses, which is especially useful for mental arithmetic when one factor ends in 0 or 5.

📌 Key points (3–5)

• What the distributive property does: distributes a factor to each addend (or subtrahend) inside parentheses, turning one multiplication into simpler pieces.
• Two ways to compute: the same expression can be solved using order of operations or the distributive property, yielding the same result.
• When it works best: the method is most efficient for mental arithmetic when one factor ends in 0 or 5.
• Common confusion: the distributive property is not just "doing operations in order"—it actively splits one multiplication into multiple smaller multiplications that are added or subtracted.
• Flexibility in breaking numbers: a number can be split as a sum (e.g., 23 = 20 + 3) or a difference (e.g., 23 = 30 − 7), depending on which is easier to compute.

🧮 What the distributive property is

🧮 Definition and core mechanism

Distributive property: a characteristic of numbers that involves both addition and multiplication; it distributes the factor to each addend in the parentheses.

• The property works for both sums and differences.
• Instead of computing inside parentheses first, you multiply the outside factor by each term inside, then add or subtract the results.
• Example: 3(2 + 5) can be computed as 3 · 2 + 3 · 5 = 6 + 15 = 21.

🔄 Why it works (repeated addition view)

• Multiplication describes repeated addition.
• 3(2 + 5) means "2 + 5 appears 3 times": (2 + 5) + (2 + 5) + (2 + 5).
• By the commutative property of addition, this becomes (2 + 2 + 2) + (5 + 5 + 5) = 3 · 2 + 3 · 5.
• The 3 has been distributed to both the 2 and the 5.

⚖️ Two methods, same result

Method	Steps for 3(2 + 5)	Result
Order of operations	Add inside parentheses first: 2 + 5 = 7, then multiply: 3 · 7	21
Distributive property	Distribute: 3 · 2 + 3 · 5 = 6 + 15	21

• Both methods are valid; the distributive property is particularly useful for mental arithmetic.

🎯 Using the distributive property for mental arithmetic

🎯 When it works best

• The method is most efficient when one factor ends in 0 or 5.
• These numbers are easier to multiply mentally (e.g., 25 · 10, 15 · 30).
• Example: 25 · 23 is easier to compute as 25(20 + 3) = 25 · 20 + 25 · 3 = 500 + 75 = 575.

✂️ Breaking numbers apart

• You can write a number as a sum or a difference, whichever is simpler.
• Example for 25 · 23:
  • As a sum: 23 = 20 + 3, so 25 · 23 = 25 · 20 + 25 · 3 = 500 + 75 = 575.
  • As a difference: 23 = 30 − 7, so 25 · 23 = 25 · 30 − 25 · 7 = 750 − 175 = 575.
• Choose the split that makes mental calculation easier.

🧩 Step-by-step process

1. Identify the factor ending in 0 or 5.
2. Break the other factor into a sum or difference (usually tens + ones or a nearby round number minus a small number).
3. Distribute the first factor to each part.
4. Compute each simpler multiplication.
5. Add or subtract the results.

Example: 15 · 37

• Write 37 as 30 + 7.
• Distribute: 15 · 37 = 15 · 30 + 15 · 7.
• Compute: 15 · 30 = 450 and 15 · 7 = 105.
• Add: 450 + 105 = 555.

Alternatively, write 37 as 40 − 3:

• 15 · 37 = 15 · 40 − 15 · 3 = 600 − 45 = 555.

📝 Practice patterns

📝 Common factor patterns

• Factors ending in 5: 15, 25, 35, 45, 55, 65, 75, 85, 95.
• Factors ending in 0: 10, 20, 30, 40, 50, 60, 70, 80, 90.
• These are easy to multiply by multiples of 10 or small single-digit numbers.

📝 Example computations from the excerpt

• 25 · 12 = 25(10 + 2) = 250 + 50 = 300
• 35 · 14 = 35(10 + 4) = 350 + 140 = 490
• 80 · 58 = 80(50 + 8) = 4,000 + 640 = 4,640
• 65 · 62 = 65(60 + 2) = 3,900 + 130 = 4,030

🔁 Subtraction form examples

• 15 · 16 = 15(20 − 4) = 300 − 60 = 240
• 25 · 38 = 25(40 − 2) = 1,000 − 50 = 950
• 40 · 89 = 40(90 − 1) = 3,600 − 40 = 3,560

Don't confuse: The distributive property applies to both addition and subtraction inside parentheses; choose the form that makes mental calculation simpler.

🧭 Overview

🧠 One-sentence thesis

Rounding fractions to common benchmarks (¼, ½, ¾, 0, and 1) provides a quick way to estimate sums and differences, though the technique naturally produces results that may vary from exact values.

📌 Key points (3–5)

• What the technique does: rounds fractions to ¼, ½, ¾, 0, or 1 before computing, making mental arithmetic easier.
• When to use it: estimating sums or differences of fractions or mixed numbers.
• Key characteristic: estimates may vary because rounding is judgment-based; the goal is closeness, not precision.
• Common pattern: whole number parts are added separately, then fractional parts are rounded and combined.
• How accuracy works: the exact answer is usually close to the estimate, sometimes a little more or a little less.

🎯 The rounding technique

🎯 What fractions are rounded to

Estimation by rounding fractions commonly rounds fractions to ¼, ½, ¾, 0, and 1.

• These five values are easy to work with mentally.
• The choice depends on which benchmark the original fraction is closest to.
• Example: 3/8 is close to ¼ (since 3/8 = 0.375 and ¼ = 0.25); 5/12 is close to ½.

🔍 Why results may vary

• The excerpt repeatedly states "results may vary."
• Different people may round the same fraction differently depending on judgment.
• The technique prioritizes speed and simplicity over exact agreement.

🧮 How to apply the method

🧮 For simple fraction sums

• Round each fraction to a benchmark.
• Add the rounded values.
• Example from the excerpt: estimate 3/5 + 5/12.
  • 3/5 is about ½.
  • 5/12 is about ½.
  • Estimate: ½ + ½ = 1.
  • Exact value: 61/60, a little more than 1.

🧮 For mixed numbers

• Separate whole number parts from fractional parts.
• Add the whole numbers.
• Round each fractional part and add those.
• Combine the two sums.
• Example from the excerpt: estimate 5 3/8 + 4 9/10 + 11 1/5.
  • Whole parts: 5 + 4 + 11 = 20.
  • Fractional parts: 3/8 ≈ ¼, 9/10 ≈ 1, 1/5 ≈ ¼.
  • Sum of fractions: ¼ + 1 + ¼ = 1½.
  • Total estimate: 20 + 1½ = 21½.
  • Exact value: 21 19/40, a little less than 21½.

📊 Rounding decisions in practice

Original fraction	Rounded to	Reasoning (from examples)
3/8	¼	Close to one-quarter
5/12	½	Close to one-half
9/10	1	Very close to one whole
1/5	¼	Close to one-quarter
1/20	0	Very small, close to zero

✅ Comparing estimate to exact value

✅ The pattern in the examples

• After making an estimate, the excerpt always shows the exact computation.
• The exact answer is described as "a little more than" or "a little less than" the estimate.
• This confirms the estimate was reasonable.

✅ Purpose of the comparison

• Estimation is not meant to replace exact calculation.
• It provides a quick check: if your exact answer is far from the estimate, you may have made an error.
• Example: if you estimate 1 but calculate an exact answer of 5, something went wrong.

⚠️ Don't confuse

• Estimation vs exact calculation: estimation sacrifices precision for speed; exact calculation gives the true value.
• "Results may vary" does not mean "anything goes": estimates should be close to the exact value, just not identical.

🧭 Overview

🧠 One-sentence thesis

The excerpt provides practice exercises for estimating and computing exact values when combining addition, subtraction, multiplication, and division operations with fractions and mixed numbers, emphasizing both approximation techniques and precise calculation.

📌 Key points (3–5)

• What the exercises cover: estimation followed by exact computation for combined operations involving fractions and mixed numbers.
• Estimation strategies: rounding fractions to nearby whole numbers or simple fractions (like 1/2) before performing operations.
• Exact computation: after estimating, finding the precise answer by performing the operations with the original fractions.
• Common confusion: distinguishing between the estimated value (used for quick mental checks) and the exact value (the true mathematical result).
• Why it matters: estimation helps verify that exact answers are reasonable and builds number sense with fractions.

🧮 Estimation techniques

🧮 Rounding fractions for quick estimates

The excerpt demonstrates rounding fractions to simpler values before computing:

• Fractions close to 1 (like 15/16) are rounded to 1.
• Fractions close to 1/2 (like 5/8, 11/20, 17/30) are rounded to 1/2.
• Fractions close to 0 (like 1/25) are rounded to 0.

Example: To estimate 15/16 + 5/8, round to 1 + 1/2 = 1 1/2, then compute the exact value (1 9/16).

🔢 Estimating with mixed numbers

For mixed numbers, round the fractional part:

• 8 9/16 rounds to 8 1/2 (since 9/16 is close to 1/2).
• 14 1/12 rounds to 14 (since 1/12 is close to 0).
• 1 17/36 rounds to 1 1/2.

Example: To estimate 8 9/16 + 14 1/12, round to 8 1/2 + 14 = 22 1/2, then find the exact value (22 31/48).

Don't confuse: The estimate is not the answer—it's a tool to check whether your exact calculation is in the right ballpark.

🎯 Exact computation after estimation

🎯 Finding precise values

After making an estimate, the excerpt shows how to compute the exact result:

• Add or subtract the whole number parts.
• Add or subtract the fractional parts, finding common denominators as needed.
• Simplify the result.

Problem	Estimate	Exact Value
15/16 + 5/8	1 + 1/2 = 1 1/2	1 9/16
1/25 + 11/20 + 17/30	0 + 1/2 + 1/2 = 1	1 47/300
8 9/16 + 14 1/12	8 1/2 + 14 = 22 1/2	22 31/48
5 4/9 + 1 17/36 + 6 5/12	5 1/2 + 1 1/2 + 6 1/2 = 13 1/2	13 1/3

✅ Comparing estimate to exact value

The parenthetical exact values show how close the estimates are:

• The estimate helps catch large errors in computation.
• Small differences between estimate and exact value are expected due to rounding.

Example: Estimating 5 4/9 + 1 17/36 + 6 5/12 as 13 1/2 is very close to the exact answer of 13 1/3, confirming the calculation is reasonable.

🧭 Overview

🧠 One-sentence thesis

Measurement is comparison to a standard, and both the United States and metric systems allow conversions between units through systematic methods—unit fractions for the U.S. system and decimal-point shifts for the metric system.

📌 Key points (3–5)

• What measurement means: comparison to a standard unit that should be accessible, invariant, and reproducible.
• Two major systems: the United States system (feet, pounds, gallons) and the metric system (meters, grams, liters with prefixes).
• U.S. conversions use unit fractions: place the target unit in the numerator and the original unit in the denominator, then multiply.
• Metric conversions use decimal shifts: moving from larger to smaller units shifts the decimal right; smaller to larger shifts it left.
• Common confusion: in the U.S. system, choosing the wrong unit fraction (numerator/denominator reversed) will give an incorrect answer; in the metric system, shifting the decimal the wrong direction inverts the conversion.

📏 What measurement is

📏 Definition and purpose

Measurement: comparison to some standard.

• Measurement is not an absolute quantity; it is the result of comparing two quantities.
• The quantity used for comparison is the standard unit of measure.

🏛️ Properties of a good standard

The excerpt lists three desirable properties:

Property	Meaning
Accessibility	People should have access to the standard to make comparisons
Invariance	The standard should not change over time
Reproducibility	The standard should be easy to reproduce so measurements are convenient and accessible

• Historical example: in the past, 1 inch was defined as the distance from the king's thumb tip to knuckle, or the length of 16 barley grains end to end—these were not invariant or reproducible.
• Today, the Bureau of Standards in Washington D.C. maintains standard units that rarely change.

🇺🇸 The United States system

🇺🇸 Common units

The excerpt provides a table of U.S. units:

Category	Conversions
Length	1 ft = 12 in.; 1 yd = 3 ft; 1 mi = 5,280 ft
Weight	1 lb = 16 oz; 1 T = 2,000 lb
Liquid Volume	1 tbsp = 3 tsp; 1 fl oz = 2 tbsp; 1 c = 8 fl oz; 1 pt = 2 c; 1 qt = 2 pt; 1 gal = 4 qt
Time	1 min = 60 sec; 1 hr = 60 min; 1 da = 24 hr; 1 wk = 7 da

• These units do not follow a single pattern; each category has its own conversion factors.

🔄 Converting with unit fractions

Unit fraction: a fraction with a value of 1, formed by using two equal measurements.

• One measurement goes in the numerator, the other in the denominator.
• Placement rule: put the unit you are converting to in the numerator; put the unit you are converting from in the denominator.
• Because a unit fraction equals 1, multiplying by it does not change the value—only the units.

Example from the excerpt: Convert 11 yards to feet.

• Equal measurements: 1 yd = 3 ft.
• Unit fractions: (1 yd)/(3 ft) or (3 ft)/(1 yd).
• We want feet in the result, so choose (3 ft)/(1 yd).
• Multiply: 11 yd × (3 ft)/(1 yd) = 33 ft.
• The "yd" units cancel, leaving feet.

Example: Convert 36 fl oz to pints.

• 1 pt = 16 fl oz.
• Unit fraction: (1 pt)/(16 fl oz).
• 36 fl oz × (1 pt)/(16 fl oz) = 36/16 pt = 2.25 pt.

Example: Convert 2,016 hr to weeks.

• Need two steps: hours → days → weeks.
• 1 da = 24 hr, so use (1 da)/(24 hr).
• 1 wk = 7 da, so use (1 wk)/(7 da).
• 2,016 hr × (1 da)/(24 hr) × (1 wk)/(7 da) = 2,016/(24×7) wk = 12 wk.

⚠️ Common pitfall

• If you place the target unit in the denominator instead of the numerator, the units will not cancel correctly and the answer will be inverted.
• Example: using (1 yd)/(3 ft) when converting yards to feet would give an answer in (yd²/ft) instead of feet.

🌍 The metric system

🌍 Core advantage

• The metric system takes advantage of the base ten number system.
• Conversions are done by multiplying or dividing by powers of 10, which means simply moving the decimal point.

🔤 Prefixes

The excerpt lists metric prefixes and their meanings:

Prefix	Meaning	Multiplier
kilo	thousand	1,000
hecto	hundred	100
deka	ten	10
(base unit)	meter, gram, liter	1
deci	tenth	0.1
centi	hundredth	0.01
milli	thousandth	0.001

• These prefixes attach to base units: meter (length), liter (volume), gram (mass).
• Example: 1 kilometer = 1,000 meters; 1 centimeter = 0.01 meter; 1 millimeter = 0.001 meter.

🔄 Decimal-point conversion rules

The excerpt gives three key characteristics:

1. Prefixes are the same in each category (length, mass, volume).
2. Larger to smaller unit: move the decimal point to the right.
3. Smaller to larger unit: move the decimal point to the left.

The excerpt provides a summary table:

Conversion	Decimal shift
unit → deka	1 place left
unit → hecto	2 places left
unit → kilo	3 places left
unit → deci	1 place right
unit → centi	2 places right
unit → milli	3 places right

Example: Convert 5 meters to centimeters.

• Meter to centi: 2 places to the right.
• 5 m = 500 cm.

Example: Convert 3,500 grams to kilograms.

• Gram to kilo: 3 places to the left.
• 3,500 g = 3.5 kg.

⚠️ Common pitfall

• Moving the decimal in the wrong direction inverts the conversion.
• Larger to smaller means more of the smaller unit, so the number gets bigger (decimal right).
• Smaller to larger means fewer of the larger unit, so the number gets smaller (decimal left).

📊 Comparison of the two systems

Feature	United States system	Metric system
Conversion method	Unit fractions (multiply by ratios)	Decimal-point shifts (powers of 10)
Ease of conversion	Requires memorizing many conversion factors	Systematic: same prefixes, predictable shifts
Unit relationships	Irregular (12 in./ft, 16 oz/lb, 5,280 ft/mi)	Regular (always ×10, ×100, ×1,000)
Common units	Feet, pounds, gallons, etc.	Meters, grams, liters with prefixes

• The metric system's advantage is simplicity: once you know the prefixes, all conversions follow the same pattern.
• The U.S. system requires remembering different conversion factors for each category and sometimes chaining multiple unit fractions.

🧭 Overview

🧠 One-sentence thesis

Decimal fractions extend the positional number system to the right of the ones place, allowing us to represent fractional values as powers of ten, and we read and write them by naming the position of the rightmost digit.

📌 Key points (3–5)

• What decimals are: fractions whose denominators are powers of 10 (10, 100, 1000, etc.), written using a decimal point to separate whole and fractional parts.
• Position values to the right: each position to the right of the ones place has one-tenth the value of the position to its left (tenths, hundredths, thousandths, etc.).
• How to read decimals: read the whole number, say "and" for the decimal point, read the fractional part as a whole number, then name the position of the last digit.
• Common confusion: position names have the suffix "th" (tenths, hundredths), not just "tens" or "hundreds"; also, trailing zeros to the right of the last digit do not change the value.
• Converting back to fractions: say the decimal in words, then write it as a fraction with the appropriate power-of-10 denominator.

🔢 How positions work to the right of ones

🔢 Each position is one-tenth of the previous

• The excerpt explains that in a base-ten positional system, each position has ten times the value of the position to its right.
• Flipping this: each position has one-tenth the value of the position to its left.
• A digit to the right of the ones position must have a value of one-tenth of 1.

📐 Position values as fractions

The excerpt shows the pattern:

• First position right of ones: one-tenth of 1 = one-tenth times 1 = one over ten
• Second position: one-tenth of one-tenth = one-tenth times one-tenth = one over ten squared = one over one hundred
• Third position: one-tenth of one over one hundred = one-tenth times one over one hundred = one over ten cubed = one over one thousand
• This pattern continues indefinitely.
• All these fractional values are powers of 10 in the denominator.

🔤 What decimal fractions are

🔤 Definition and notation

Decimal point: the symbol that denotes where the whole number part ends and the fractional part begins.

Decimal (or decimal fraction): a number written with digits to the right of the decimal point; a fraction in which the denominator is a power of 10.

• The word "decimal" comes from the Latin prefix "deci," meaning ten, because we use a base-ten system.
• Position names to the right of the decimal point have the suffix "th": tenths, hundredths, thousandths, ten thousandths, etc.

🧮 Examples from the excerpt

Decimal	Position breakdown	Fraction form
42.6	6 in tenths	42 and six-tenths
9.8014	8 tenths, 0 hundredths, 1 thousandths, 4 ten thousandths	9 and eight thousand fourteen ten-thousandths
0.93	9 tenths, 3 hundredths	ninety-three hundredths
0.7	7 tenths	seven-tenths

⚠️ Important notes

• Leading zero: A zero is often inserted in the ones position (before the decimal point) when the value is less than one (e.g., 0.93). This helps prevent overlooking the decimal point.
• Trailing zeros: Zeros can be inserted to the right of the rightmost digit without changing the value. Example: 0.7 = 0.70 (both equal seven-tenths).

📖 Reading decimal fractions

📖 Step-by-step reading method

The excerpt provides a four-step procedure:

1. Read the whole number part as usual. (If the whole number is less than 1, skip steps 1 and 2.)
2. Read the decimal point as the word "and."
3. Read the number to the right of the decimal point as if it were a whole number.
4. Say the name of the position of the last digit.

🗣️ Reading examples

• 6.8: "six and eight tenths" (not "six point eight," though that phrasing is common informally).
• 14.116: "fourteen and one hundred sixteen thousandths."
• 0.0019: "nineteen ten-thousandths."
• 81: "eighty-one" (whole numbers are also decimal numbers; 81 = 81.0).

🔍 Don't confuse

• The position name must match the last digit's position, not the first digit after the decimal point.
• Example: 0.030405 is read "thirty thousand four hundred five millionths" (the 5 is in the millionths position), not "three hundredths and four hundred five millionths."

✍️ Writing decimal fractions

✍️ Step-by-step writing method

The excerpt provides a three-step procedure:

1. Write the whole number part.
2. Write a decimal point for the word "and."
3. Write the decimal part so that the rightmost digit appears in the position indicated in the word name. Insert zeros to the right of the decimal point if necessary to place the rightmost digit correctly.

🖊️ Writing examples

• "Thirty-one and twelve hundredths": The hundredths position is indicated → 31.12
• "Two and three hundred-thousandths": The hundred-thousandths position is indicated; insert zeros to place the 3 correctly → 2.00003
• "Six thousand twenty-seven and one hundred four millionths": The millionths position is indicated; insert zeros → 6,027.000104
• "Seventeen hundredths": The hundredths position is indicated; no whole number part, so start with 0 → 0.17

🎯 Key insight

• The position name tells you where the last digit must land.
• If the position is far to the right (e.g., millionths), you must insert enough zeros immediately after the decimal point to reach that position.

🔄 Converting decimals to fractions

🔄 Ordinary decimals

The excerpt explains: "say it in words, then write what we say," and reduce if possible.

Examples:

• 0.6: "six tenths" → six over ten → reduce to three over five.
• 0.903: "nine hundred three thousandths" → nine hundred three over one thousand (already in simplest form).
• 18.61: "eighteen and sixty-one hundredths" → 18 and sixty-one over one hundred (mixed number).
• 508.0005: "five hundred eight and five ten-thousandths" → 508 and five over ten thousand → reduce to 508 and one over two thousand.

🔄 Complex decimals

Complex decimal: a number like 0.11 and two-thirds, where a fraction appears in a decimal position.

• The fraction refers to a fractional part of the position it occupies.

• Example: 0.11 and two-thirds means "eleven and two-thirds hundredths."

  • Write as: (eleven and two-thirds) over one hundred
  • Convert the mixed number in the numerator: eleven times three plus two, all over three = thirty-five over three
  • Divide: thirty-five over three divided by one hundred over one = thirty-five over three times one over one hundred = seven over sixty.

• Example: 4.006 and one-fourth means 4 plus 0.006 and one-fourth.

  • The excerpt notes this equals 4 plus six and one-fourth thousandths.

🧩 Don't confuse

• A complex decimal like 0.11 and two-thirds does not mean "0.11 plus two-thirds."
• It means the two-thirds is part of the last decimal position (e.g., two-thirds of a hundredth).

🧭 Overview

🧠 One-sentence thesis

Decimals can be converted to fractions by reading them in words and writing those words as fractions, then reducing when possible, including complex decimals that contain fractional parts within decimal positions.

📌 Key points (3–5)

• Core method: Say the decimal in words, then write what you say as a fraction; reduce if needed.
• Ordinary decimals: Read the decimal name (e.g., "six tenths" becomes 6/10), then simplify.
• Complex decimals: Numbers like 0.11 and 2/3 contain a fraction within a decimal position and require special handling.
• Common confusion: In complex decimals, a fraction in the thousandths place refers to a fraction of the previous place value (e.g., 2/3 in 0.11 and 2/3 means "two-thirds of a hundredth," not thousandths).
• Always reduce: The final fraction should be simplified to lowest terms.

🔄 Converting ordinary decimals

📖 The say-and-write method

The core technique: convert a decimal fraction to a fraction by saying it in words, then writing what you say.

• This method relies on understanding decimal place names (tenths, hundredths, thousandths, etc.).
• The denominator comes from the rightmost digit's position.
• After writing the fraction, check whether it can be reduced.

🧮 Step-by-step process

1. Read the decimal aloud using proper place-value names.
2. Write the fraction exactly as spoken.
3. Reduce to simplest form.

Example: 0.6 → "six tenths" → 6/10 → reduces to 3/5.

Example: 18.61 → "eighteen and sixty-one hundredths" → 18 and 61/100 (mixed number).

✂️ Reduction is essential

• Many decimal-to-fraction conversions produce fractions that are not in lowest terms.
• Example: 508.0005 → "five hundred eight and five ten-thousandths" → 508 and 5/10,000 → reduces to 508 and 1/2,000.
• Always check for common factors between numerator and denominator.

🔢 Converting complex decimals

🧩 What complex decimals are

Complex decimals: numbers that contain a fraction within a decimal position, such as 0.11 and 2/3.

• These are not standard decimals; they mix decimal notation with fractional parts.
• The fractional part refers to a fraction of the place value where it appears.

🔍 How to read complex decimals correctly

The key insight: the fraction describes a portion of the preceding decimal place, not a new place value.

Example: 0.11 and 2/3

• The 2/3 appears after the hundredths place.
• Read as "eleven and two-thirds hundredths" (not "eleven hundredths and two-thirds thousandths").
• This means 11 + 2/3, all divided by 100.

🛠️ Conversion steps for complex decimals

1. Identify the fractional part and which place value it modifies.
2. Combine the whole and fractional parts of the numerator.
3. Write as an improper fraction over the appropriate denominator.
4. Perform the division (convert to multiplication by the reciprocal).
5. Simplify.

Example breakdown: 0.11 and 2/3

• Read as (11 + 2/3)/100
• Numerator: 11 × 3 + 2 = 35, so 35/3
• Full fraction: (35/3)/100 = 35/3 ÷ 100/1
• Multiply: 35/3 × 1/100 = 35/300 = 7/60

⚠️ Don't confuse place values

• In 4.006 and 1/4, the 1/4 refers to 1/4 of a thousandth, not ten-thousandths.
• The excerpt shows: 4.006 and 1/4 = 4 + 0.006 and 1/4 = 4 + (6 + 1/4)/1000.
• The fraction modifies the last stated decimal digit's value.

📊 Comparison table

Decimal type	Example	Reading	Fraction form	Key difference
Ordinary	0.6	six tenths	6/10 → 3/5	Straightforward place-value reading
Ordinary mixed	18.61	eighteen and sixty-one hundredths	18 and 61/100	Whole number + fraction
Complex	0.11 and 2/3	eleven and two-thirds hundredths	7/60	Fraction modifies the place value
Complex mixed	4.006 and 1/4	four and six and one-fourth thousandths	4 and 1/160	Requires multi-step calculation

🧭 Overview

🧠 One-sentence thesis

Rounding decimals follows the same core rules as rounding whole numbers, with the key difference being how digits are handled depending on whether the round-off position is to the left or right of the decimal point.

📌 Key points (3–5)

• The rounding rule: if the digit immediately to the right of the round-off position is less than 5, leave the round-off digit unchanged; if it is 5 or greater, add 1 to the round-off digit.
• Two cases for cleanup: if the round-off digit is to the right of the decimal point, eliminate all digits to its right; if it is to the left, replace digits between it and the decimal point with zeros and eliminate the decimal point and all decimal digits.
• Decimal place terminology: "round to three decimal places" means the round-off digit is the third digit after the decimal point (the thousandths position).
• Common confusion: rounding to the left vs. right of the decimal point—the cleanup step is different in each case.

📏 The three-step rounding procedure

📍 Step 1: Mark the round-off digit

• Identify the position you are rounding to (e.g., nearest hundredth, nearest ten, three decimal places).
• Mark that digit with an arrow or check.
• This is the digit that may change or stay the same.

🔍 Step 2: Check the digit to the immediate right

The excerpt gives two sub-cases:

• Case a: The digit to the right is less than 5 → leave the round-off digit unchanged.
• Case b: The digit to the right is 5 or greater → add 1 to the round-off digit.

This is the core decision rule: compare the next digit to 5.

Example: Rounding 32.116 to the nearest hundredth—the round-off digit is 1 (the second decimal place), and the digit to its right is 6. Since 6 is greater than 5, add 1 to the round-off digit: 1 + 1 = 2.

✂️ Step 3: Clean up the remaining digits

The cleanup depends on where the round-off digit sits relative to the decimal point.

Round-off digit position	What to do
To the right of the decimal point (Step 3a)	Eliminate all digits to the right of the round-off digit.
To the left of the decimal point (Step 3b)	Replace all digits between the round-off digit and the decimal point with zeros; eliminate the decimal point and all decimal digits.

• Example (Step 3a): 32.116 rounded to the nearest hundredth becomes 32.12 (drop the 6).
• Example (Step 3b): 633.14216 rounded to the nearest hundred—the round-off digit is 6 (hundreds place), the digit to its right is 3 (less than 5), so the 6 stays unchanged. Replace all digits between the 6 and the decimal point with zeros and drop the decimal part → 600.

Don't confuse: If rounding to a position left of the decimal point, you must replace intervening digits with zeros and remove the decimal point entirely; if rounding to a position right of the decimal point, you simply truncate everything after the round-off digit.

🗣️ Understanding "decimal places" language

🗣️ What "round to N decimal places" means

"Round to three decimal places" = the round-off digit is the third digit after the decimal point (the thousandths position).

• "Two decimal places" = hundredths position.
• "Four decimal places" = ten-thousandths position.
• This is just shorthand for specifying the round-off position.

Example: 67.129558 rounded to 3 decimal places means rounding to the thousandths position. The round-off digit is 9 (third decimal place), the digit to its right is 5, so add 1 to 9 → 10. Carry the 1 to the hundredths place: 67.130.

🧪 Worked examples from the excerpt

🧪 Example: Rounding to the right of the decimal point

• Problem: Round 32.116 to the nearest hundredth.
• Step 1: Mark the hundredths digit (1).
• Step 2: The digit to the right is 6; 6 > 5, so add 1 to the round-off digit: 1 + 1 = 2.
• Step 3a: The round-off digit is to the right of the decimal point, so eliminate all digits to its right.
• Result: 32.12.

🧪 Example: Rounding to the left of the decimal point

• Problem: Round 633.14216 to the nearest hundred.
• Step 1: Mark the hundreds digit (6).
• Step 2: The digit to the right is 3; 3 < 5, so leave the round-off digit unchanged.
• Step 3b: The round-off digit is to the left of the decimal point, so replace all digits between it and the decimal point with zeros and eliminate the decimal point and all decimal digits.
• Result: 600.

🧪 Example: Rounding that produces a carry

• Problem: Round 60.98 to the nearest one.
• Step 1: Mark the ones digit (0).
• Step 2: The digit to the right is 9; 9 ≥ 5, so add 1 to the round-off digit: 0 + 1 = 1. But this causes a carry to the tens place: 60 becomes 61.
• Step 3a: Eliminate all digits to the right of the ones place.
• Result: 61.

🧪 Example: Rounding to multiple decimal places

• Problem: Round 67.129558 to 3 decimal places.
• Step 1: Mark the third decimal digit (9, the thousandths place).
• Step 2: The digit to the right is 5; 5 ≥ 5, so add 1 to 9 → 10. Carry the 1 to the hundredths place.
• Step 3a: Eliminate all digits to the right.
• Result: 67.130.

Don't confuse: When rounding produces a carry (e.g., 9 + 1 = 10), you must propagate the carry to the next left digit, just as in whole-number arithmetic.

🧮 Edge cases and special situations

🧮 Rounding very small decimals

• Problem: Round 0.000007 to the nearest tenth.

• The round-off digit is the tenths place (0). The digit to the right is 0 (less than 5), so the round-off digit stays 0. Eliminate all digits to the right.

• Result: 0.0.

• Problem: Round 0.00008 to the nearest ten-thousandth.

• The round-off digit is the ten-thousandths place (0). The digit to the right is 8 (≥ 5), so add 1 to 0 → 1.

• Result: 0.0001.

🧮 Rounding that results in trailing zeros

• Problem: Round 1.0144 to the nearest tenth.

• The round-off digit is 0 (tenths place). The digit to the right is 1 (< 5), so leave the 0 unchanged. Eliminate all digits to the right.

• Result: 1.0.

• Problem: Round 105.019997 to four decimal places.

• The round-off digit is the fourth decimal place (9). The digit to the right is 9 (≥ 5), so add 1 to 9 → 10. Carry the 1 to the third decimal place: 105.0199 + 0.0001 = 105.0200.

• Result: 105.0200 (the trailing zero is kept to show precision to four decimal places).

🧮 Rounding that changes the whole-number part

• Problem: Round 99.9999 to two decimal places.
• The round-off digit is the second decimal place (9). The digit to the right is 9 (≥ 5), so add 1 to 9 → 10. Carry propagates all the way to the ones place: 99.99 + 0.01 = 100.00.
• Result: 100.00.

Don't confuse: Rounding can change the whole-number part of a decimal if the carry propagates far enough; always follow the carry through to the left.

🧭 Overview

🧠 One-sentence thesis

Adding and subtracting decimals requires aligning decimal points vertically so that corresponding place values match, then performing the operation as with whole numbers while keeping the decimal point in the same column.

📌 Key points (3–5)

• Core method: Align decimal points vertically, add or subtract as whole numbers, and place the decimal point in the result directly under the aligned points.
• Why it works: The logic comes from converting decimals to fractions with common denominators—aligning decimal points ensures place values correspond (e.g., hundredths with hundredths).
• Placeholder zeros: You can write zeros at the end of decimal digits to help columns align properly without changing the value.
• Common confusion: Don't forget to align the decimal points first—aligning the rightmost digits (as with whole numbers) will give wrong answers.
• Rounding results: After finding a sum or difference, you may need to round to a specified decimal place using standard rounding rules.

🔍 Why the method works

🔍 The fraction foundation

The excerpt demonstrates the logic with an example: 4.37 + 3.22.

• Convert to mixed numbers: 4 and 37/100 plus 3 and 22/100
• Because both fractions have the same denominator (100), you can add the numerators directly: (400 + 37)/100 + (300 + 22)/100 = 759/100 = 7.59
• This shows that when decimal points align, you're automatically keeping the same denominator (same place value).

The method works because aligning decimal points ensures corresponding decimal positions (tenths, hundredths, etc.) are in the same column, just like ensuring fractions have common denominators.

🔢 Column alignment principle

When written vertically with aligned decimal points:

  4.37
+ 3.22
-------
  7.59

• The 3 and 2 in the tenths column add together
• The 7 and 2 in the hundredths column add together
• The decimal point stays in the same vertical line

➕ The step-by-step method

➕ Three-step procedure

Method of Adding and Subtracting Decimals:

1. Align the numbers vertically so that the decimal points line up under each other and the corresponding decimal positions are in the same column.
2. Add or subtract the numbers as if they were whole numbers.
3. Place a decimal point in the resulting sum or difference directly under the other decimal points.

🔧 Using placeholder zeros

• You can write zeros at the end of the decimal part to help align columns without changing the value.
• Example from the excerpt: To subtract 1.314 − 0.58, write 0.58 as 0.580 so the thousandths column is filled.
• Example: 47.016 can be written as 47.0160 to match a number with four decimal places.
• Why this works: Adding zeros after the last decimal digit doesn't change the value (just as 5 = 5.0 = 5.00).

📐 Practical examples from the excerpt

Operation	Setup	Result
9.813 + 2.140	Align decimal points vertically	11.953
1.314 − 0.58	Write as 1.314 − 0.580	0.734
16.01 − 7.053	Write as 16.010 − 7.053	8.957

🎯 Real-world application

💰 Money calculations

The excerpt provides a checking account scenario:

• Wendy has $643.12 in her account
• She writes a check for $16.92
• To find the new balance, subtract: 643.12 − 16.92 = 626.20
• The decimal point alignment ensures dollars align with dollars and cents with cents.

🔄 Rounding after operations

Example from the excerpt: Find the sum of 6.88106 and 3.5219, then round to three decimal places.

• First add: 6.88106 + 3.5219 = 10.40296
• Round to thousandths position: look at the digit to the right (9)
• Since 9 > 5, round up: 10.403
• Don't confuse: Perform the full operation first, then round the final answer—don't round the numbers before adding.

🖩 Calculator considerations

🖩 When calculators help

• Calculators are useful for finding sums and differences of decimal numbers
• They can verify hand calculations quickly
• Example operations from the excerpt: 4.286 + 8.97 = 13.256 or 452.0092 − 392.558 = 59.4512

⚠️ Calculator limitations

• Eight-digit display calculators cannot handle decimals with more than eight total digits
• They also fail when the sum results in more than eight digits
• Example from excerpt: 51.07 + 3,891.001786 contains more than eight digits in one number, so an eight-digit calculator cannot perform this addition
• Solution: Perform the addition by hand using the alignment method
• Some inexpensive calculators can handle 13 decimal places, but eight-place decimals are seldom encountered in practice

🤝 Hand calculation as backup

When calculator limits are reached:

• Use the vertical alignment method
• Write placeholder zeros as needed
• Example: 51.070000 + 3891.001786 = 3942.071786 (done by hand)

🧭 Overview

🧠 One-sentence thesis

Multiplying decimals works by treating the numbers as whole numbers during calculation, then placing the decimal point in the product so that the total decimal places equals the sum of decimal places in both factors.

📌 Key points (3–5)

• Core method: Multiply as whole numbers, then count total decimal places from both factors to position the decimal point in the product.
• Decimal place rule: The number of decimal places in the product equals the sum of decimal places in the two factors being multiplied.
• Powers of 10 shortcut: Multiplying by 10, 100, 1,000, etc., simply moves the decimal point to the right by the number of zeros.
• Common confusion: Calculators with limited displays may not show all digits for products with many decimal places; rounding may be necessary.
• "Of" means multiply: The word "of" in decimal problems translates to the multiplication operation.

🔍 The logic behind the method

🔍 Why the decimal place rule works

The excerpt demonstrates the logic using 3.2 × 1.46:

• Convert to fractions: 3.2 = 32/10 and 1.46 = 146/100
• Multiply: (32/10) × (146/100) = 4,672/1,000 = 4.672
• Observation: 3.2 has 1 decimal place, 1.46 has 2 decimal places, and the product 4.672 has 3 decimal places (1 + 2 = 3).

The sum of the number of decimal places in the factors equals the number of decimal places in the product.

This pattern holds because multiplying denominators (10 × 100 = 1,000) determines the total decimal places in the result.

🧮 The multiplication method

🧮 Three-step procedure

Method of Multiplying Decimals:

1. Multiply the numbers as if they were whole numbers.
2. Find the sum of the number of decimal places in the factors.
3. The number of decimal places in the product is the sum found in step 2.

✏️ How to apply it

• Ignore decimal points during multiplication—treat all digits as whole numbers.
• After obtaining the whole-number product, count decimal places in each original factor.
• Place the decimal point in the product from the right, counting leftward by the total number of places.

Example from the excerpt: 6.5 × 4.3

• Multiply 65 × 43 = 2,795 (as whole numbers)
• Count places: 6.5 has 1 place, 4.3 has 1 place → total = 2 places
• Result: 27.95 (decimal point placed 2 positions from the right)

🎯 Rounding products

When instructed to round:

• Perform the full multiplication first.
• Then round the final product to the specified number of decimal places.

Example: 0.251 × 0.00113 rounded to three decimal places

• Full product has 3 + 5 = 8 decimal places
• Calculate, then round the result to three places

🖩 Using calculators

🖩 Display limitations

• Eight-digit display calculators cannot handle products with more than eight total digits.
• The excerpt notes that products requiring more precision need "hand technology" (manual calculation) or a calculator with a larger display.

🖩 Verifying calculator results

When using a calculator:

• Check that the number of decimal places shown matches what you expect (sum of factors' decimal places).
• If the display shows fewer places than expected, the result may be truncated or rounded.

Example from excerpt: 0.006 × 0.0042

• Expected decimal places: 3 + 4 = 7
• If the calculator shows 7 decimal places (0.0000252), the product is likely correct.

Don't confuse: A calculator showing fewer digits than expected doesn't always mean an error—it may have dropped trailing zeros, but if it shows fewer than the expected decimal places in significant digits, the result is incomplete.

⚡ Multiplying by powers of 10

⚡ The shortcut rule

To multiply a decimal by a power of 10, move the decimal point to the right of its current position as many places as there are zeros in the power of 10. Add zeros if necessary.

Power of 10	Number of zeros	Decimal places moved right
10	1	1
100	2	2
1,000	3	3
10,000	4	4

⚡ Examples of the pattern

• 100 × 34.876: Move decimal 2 places right → 3,487.6
• 1,000 × 4.8058: Move decimal 3 places right → 4,805.8
• 10,000 × 56.82: Move decimal 4 places right → 568,200 (add two zeros to get 4 places)

When the decimal point moves past all digits, the result becomes a whole number and the decimal point can be dropped.

📝 "Of" means multiplication

📝 Translating word problems

The word "of" in decimal problems translates to the multiplication symbol (×).

Example from excerpt: "Find 4.1 of 3.8"

• Translate: 4.1 × 3.8
• Calculate: 15.58

📝 Combined operations

When "of" appears with other operations, follow order of operations:

• "Find 0.95 of the sum of 2.6 and 0.8"
• First: 2.6 + 0.8 = 3.4
• Then: 0.95 × 3.4 = 3.230

This pattern appears in practical problems like finding a fraction or percentage of a quantity.

🧭 Overview

🧠 One-sentence thesis

Dividing decimals can be systematically accomplished by converting the divisor to a whole number through moving decimal points, and division by powers of 10 follows a simple pattern of moving the decimal point left by the number of zeros.

📌 Key points (3–5)

• Dividing by whole numbers: Place the decimal point in the quotient directly above the dividend's decimal point, then divide as if working with whole numbers.
• Dividing by decimals: Convert the divisor to a whole number by moving its decimal point to the right, then move the dividend's decimal point the same number of places.
• Powers of 10 shortcut: Dividing by 10, 100, 1,000, etc., simply moves the decimal point left by the number of zeros in the divisor.
• Common confusion: When moving decimal points, both divisor and dividend must move the same number of places—this is actually multiplication by a power of 10, not arbitrary "moving."
• Nonterminating divisions: Some divisions never produce a remainder of zero and result in repeating decimals, indicated by three dots or a bar over the repeating digits.

🔢 Dividing by whole numbers

🔢 The basic method

Method of Dividing a Decimal by a Nonzero Whole Number: Write a decimal point above the division line directly over the decimal point of the dividend, then proceed to divide as if both numbers were whole numbers.

• The key insight: the decimal point in the quotient aligns vertically with the decimal point in the dividend.
• After placing the decimal point, the division process is identical to whole-number division.
• Example: 114.1 ÷ 7 = 16.3. The decimal point in 16.3 sits directly above the decimal point in 114.1.

🔍 Handling zeros in the quotient

• If the first nonzero digit in the quotient appears to the right of the decimal point but not in the tenths position, place zeros in each position between the decimal point and that first nonzero digit.
• Example: 0.02068 ÷ 4 = 0.00517. Zeros are placed in the tenths and hundredths positions because the first nonzero digit (5) is in the thousandths position.
• Don't confuse: these zeros are placeholders, not optional—they ensure the quotient represents the correct value.

🔄 Dividing by decimals

🔄 Converting the divisor to a whole number

Method of Dividing a Decimal by a Decimal Number: Convert the divisor to a whole number by moving the decimal point to the position immediately to the right of the divisor's last digit, then move the decimal point of the dividend to the right the same number of digits.

• The underlying logic: multiplying both divisor and dividend by the same power of 10 does not change the quotient's value.
• Example: 4.32 ÷ 1.8 becomes 43.2 ÷ 18 by moving both decimal points one place to the right.
• The excerpt explains this as converting 4 32/100 ÷ 1 8/10 into a division by a whole number by multiplying both numerator and denominator by 10.

📍 Setting the decimal point and dividing

• After moving the decimal points, place a decimal point in the quotient directly above the newly located decimal point in the dividend.
• Then divide as usual, treating the numbers as whole numbers.
• Example: 32.66 ÷ 7.1. Move both decimal points one place right to get 326.6 ÷ 71, then divide to get 4.6.

➕ Adding zeros when necessary

• If the divisor has more decimal places than the dividend has digits, add zeros to the dividend.
• Example: 12 ÷ 0.00032. The divisor has 5 decimal places, so move both decimal points 5 places right. This requires adding 5 zeros to 12, making it 1200000 ÷ 32 = 37,500.

⚡ Dividing by powers of 10

⚡ The shortcut rule

Dividing a Decimal Fraction by a Power of 10: Move the decimal point of the decimal fraction to the left as many places as there are zeros in the power of 10. Add zeros if necessary.

• This is the reverse of multiplying by powers of 10 (which moves the decimal point right).
• The number of zeros in the divisor determines how many places to move left.

Divisor	Zeros	Decimal movement	Example
10	1	1 place left	8,162.41 ÷ 10 = 816.241
100	2	2 places left	8,162.41 ÷ 100 = 81.6241
1,000	3	3 places left	182.5 ÷ 1,000 = 0.1825
10,000	4	4 places left	3.28 ÷ 10,000 = 0.000328

🔢 Adding zeros for place value

• When moving the decimal point left requires more places than the number has, add zeros to the left.
• Example: 3.28 ÷ 10,000. Moving 4 places left from 3.28 requires adding three zeros: 0.000328.

🔁 Nonterminating divisions

🔁 What makes a division nonterminating

Nonterminating division: A division that, regardless of how far we carry it out, always has a remainder.

• Contrast with terminating division (also called exact division): a division in which the quotient terminates after several divisions (the remainder is zero).
• Example of terminating: 9.8 ÷ 3.5 eventually produces a remainder of zero.
• Example of nonterminating: 4 ÷ 3 = 1.333... The pattern of 3s repeats endlessly.

🔁 Repeating decimals

Repeating decimal: A decimal quotient in which a pattern repeats itself endlessly.

• Notation with three dots: 4 ÷ 3 = 1.333...
• Notation with a bar: Write a bar above the repeating sequence of digits (the excerpt mentions this but does not show the full example due to formatting).
• Don't confuse: the three dots or bar indicate an infinite pattern, not an approximation or rounding.

🧮 Calculator considerations

🧮 Recognizing approximations

• Calculators have limited display capacity (e.g., eight digits).
• When the display is filled with digits, the result may be an approximation because the operation could produce more digits than the calculator can show.
• Example: 0.8215199 ÷ 4.113 fills an eight-digit display with 0.1997373. Since multiplying 4.113 (3 decimal places) by 0.1997373 (7 decimal places) should produce 10 decimal places, but the quotient shows only 8, the result is approximate.

✅ Checking for accuracy

• If the display is not filled, the result is likely accurate.
• To verify: multiply the quotient by the divisor. If the expected product has more decimal places than the quotient, the quotient is an approximation.
• When rounding is needed, the excerpt instructs rounding to a specified number of decimal places (e.g., five decimal places).

🧭 Overview

🧠 One-sentence thesis

Nonterminating divisions produce quotients that never reach a zero remainder and repeat endlessly, and we denote them using three dots or a bar over the repeating sequence.

📌 Key points (3–5)

• What nonterminating means: a division that always has a remainder no matter how far you carry it out, unlike terminating (exact) divisions where the remainder becomes zero.
• How to recognize repeating patterns: either the remainder becomes identical to the original dividend, or the "product, difference" pattern repeats two consecutive times.
• How to denote repeating decimals: use three dots (e.g., 1.333…) or a bar over the repeating block (e.g., 1.3̄).
• Common confusion: terminating vs nonterminating—terminating divisions end with remainder zero and are also called "exact divisions"; nonterminating divisions never stop and produce repeating decimals.
• Practical use: when rounding is required, carry the division one place further than needed, then round back.

🔄 Terminating vs nonterminating divisions

✅ Terminating (exact) divisions

Terminating division: a division in which the quotient terminates after several divisions (the remainder is zero).

• Also called exact divisions.
• The division process stops cleanly; no endless pattern.
• Example: 9.8 ÷ 3.5 terminates in the tenths position.

🔁 Nonterminating divisions

Nonterminating division: a division that, regardless of how far we carry it out, always has a remainder.

• The quotient never "finishes"; it continues indefinitely.
• Produces a repeating decimal: a decimal quotient with a pattern that repeats endlessly.
• Example: 4 ÷ 3 = 1.333… (the digit 3 repeats forever).

Don't confuse: A nonterminating division is not "hard to finish"—it is impossible to finish because the remainder never becomes zero.

🔍 How to detect repeating patterns

🔍 Method 1: Remainder equals the dividend

• As you divide, if the remainder ever becomes the same as the original dividend, the division is nonterminating.
• The pattern in the quotient will repeat from that point.
• Example: In 100 ÷ 27, when the remainder returns to 100, the quotient pattern 703 repeats: 3.70370370…

🔍 Method 2: "Product, difference" pattern repeats twice

• As you divide, you produce a sequence of products and differences (subtractions).
• If this sequence repeats two consecutive times, the division is nonterminating and the quotient repeats.
• Example: In 1 ÷ 9, the same "product, difference" pattern recurs, so the quotient is 0.111… (the digit 1 repeats).

✍️ Notation for repeating decimals

✍️ Three dots

• Write three dots at the end to show the pattern continues forever.
• Example: 4 ÷ 3 = 1.333…

✍️ Bar over the repeating block

• Write a bar (overline) above the digits that repeat.
• Example: 4 ÷ 3 = 1.3̄ (the bar is over the 3).
• If multiple digits repeat, the bar covers the entire repeating block.
• Example: 100 ÷ 27 = 3.7̄0̄3̄ (the block 703 repeats).

Why it matters: Both notations tell you the exact repeating structure without writing infinitely many digits.

📐 Worked examples from the excerpt

📐 Example: 100 ÷ 27

• Carry out the division until you notice the remainder is 100 again (same as the dividend).
• Quotient: 3.70370370…
• The repeating block is 703.
• Notation: 3.7̄0̄3̄

📐 Example: 1 ÷ 9

• The "product, difference" pattern repeats.
• Quotient: 0.111…
• The repeating block is 1.
• Notation: 0.1̄

📐 Example: 2 ÷ 11 rounded to 3 decimal places

• To round to three decimal places, carry the division to four decimal places.
• Quotient to four places: 0.1818
• Rounded to three places: 0.182
• (The actual repeating decimal is 0.1̄8̄, but rounding gives 0.182.)

📐 Example: 1 ÷ 6

• The "product, difference" pattern repeats at the digit 6.
• Quotient: 0.1666…
• The repeating block is 6.
• Notation: 0.16̄

Practical tip: When rounding is required, always compute one extra decimal place, then round back to the desired precision.

🧮 Practice problems structure

🧮 Carry out until repeating pattern is found

• Problems ask you to divide and identify the repeating block.
• Examples from the excerpt:
  • 1 ÷ 3 = 0.3̄
  • 5 ÷ 6 = 0.83̄ (the block 3 repeats)
  • 11 ÷ 9 = 1.2̄
  • 17 ÷ 9 = 1.8̄

🧮 Round to a specified number of places

• Example: Divide 7 by 6 and round to 2 decimal places → 1.17
• Example: Divide 400 by 11 and round to 4 decimal places → 36.3636

Don't confuse: "Round to 2 decimal places" means you stop at two digits after the decimal point, even if the true quotient repeats forever.

🧭 Overview

🧠 One-sentence thesis

Fractions can be converted to decimals by dividing the numerator by the denominator, because the fraction bar represents division.

📌 Key points (3–5)

• Core method: divide the numerator by the denominator (the fraction bar is a division symbol).
• Terminating vs nonterminating: some divisions end cleanly, others repeat forever and must be rounded.
• Mixed numbers: convert the fractional part to a decimal, then add it to the whole number part.
• Complex decimals: numbers like "0.16 and one-fourth hundredths" require converting the fraction within the decimal position, then dividing.
• Common confusion: remember that 3/4 means both "3 objects out of 4" and "3 divided by 4"—the division interpretation enables decimal conversion.

🔢 The fundamental conversion method

🔢 Why the fraction bar means division

A fraction bar can also be a division symbol.

• The fraction 3/4 means "3 objects out of 4" but also means "3 divided by 4."
• This dual meaning is the key to conversion: to turn any fraction into a decimal, perform the division.
• Example: 3/4 = 3 ÷ 4 = 0.75.

➗ How to perform the conversion

• Set up the division: numerator divided by denominator.
• Carry out the division using long division or a calculator.
• If the division ends (terminates), you have your exact decimal.
• If the division continues indefinitely (nonterminating), round to the specified number of decimal places.

Example: 1/5 = 1 ÷ 5 = 0.2 (terminates).

Example: 5/6 = 5 ÷ 6 = 0.833... (repeats), rounded to two decimal places = 0.83.

🧮 Special cases and variations

🧮 Mixed numbers

• A mixed number like 5 and 1/8 means 5 + 1/8.
• Convert only the fractional part to a decimal: 1/8 = 0.125.
• Add the whole number: 5 + 0.125 = 5.125.

Don't confuse: you do not divide the whole number; only the fraction part needs conversion.

🔀 Complex decimals

Complex decimal: a decimal number that contains a fraction within a decimal position, such as "0.16 and one-fourth hundredths."

• Read the position carefully: "sixteen and one-fourth hundredths" means the entire expression (16 + 1/4) is in the hundredths place.
• Convert the complex decimal to a single fraction first.
• Example from the excerpt: 0.16 and 1/4 hundredths = (16 and 1/4)/100 = (65/4)/100 = 13/80.
• Then convert that fraction to a decimal: 13/80 = 0.1625.

🔁 Handling nonterminating decimals

🔁 Repeating patterns

• Some fractions produce repeating decimals (e.g., 1/3 = 0.333...).
• The excerpt instructs: "carry out each division until the repeating pattern is determined."
• If no clear pattern emerges quickly, round to the specified number of decimal places (commonly two, three, or five).

📏 Rounding rules

• The excerpt consistently asks to round nonterminating results to a given precision (e.g., two decimal places).
• Example: 5/6 = 0.8333... rounds to 0.83 (two decimal places).
• Always follow the rounding instruction provided in the problem.

🧪 Practice approach

🧪 Systematic conversion steps

1. Identify whether you have a simple fraction, mixed number, or complex decimal.
2. For simple fractions: divide numerator by denominator.
3. For mixed numbers: convert the fraction part, then add to the whole number.
4. For complex decimals: rewrite as a single fraction, then divide.
5. Check if the result terminates or repeats; round if necessary.

🧪 Calculator use

• The excerpt includes "Calculator Problems" sections.
• Calculators speed up division but may not show repeating patterns clearly.
• When rounding is required, the calculator gives enough digits to round accurately (e.g., round to four or five decimal places as instructed).

🧭 Overview

🧠 One-sentence thesis

When operations involve both decimals and fractions, converting all numbers to a single form (either all decimals or all fractions) allows standard arithmetic rules to apply.

📌 Key points (3–5)

• Core strategy: Convert all numbers to the same form—either all decimals or all fractions—before performing operations.
• Order of operations still applies: Multiply and divide before adding and subtracting, even when mixing decimals and fractions.
• Conversion flexibility: You can convert fractions to decimals by dividing numerator by denominator, or convert decimals to fractions by writing them in words.
• Common confusion: Don't try to operate directly on mixed forms; always unify the number types first.
• Final answer format: Results can be left as decimals or converted back to fractions depending on the problem requirements.

🔄 Converting between forms

🔄 Fraction to decimal conversion

To convert a fraction to a decimal, divide the numerator by the denominator.

• This is the fundamental technique for unifying mixed problems.
• Example: To convert 1/4 to a decimal, divide 1 by 4, which gives 0.25.
• The excerpt shows this step explicitly in the first sample problem.

🔄 Decimal to fraction conversion

Decimals can be converted to fractions by saying the decimal number in words, then writing what was said.

• Example: 0.28 becomes "28 hundredths," written as 28/100, which simplifies to 7/25.
• The excerpt demonstrates this in the problem 5/13 × (4/5 − 0.28), where 0.28 is rewritten as 28/100 = 7/25.
• After conversion, simplify the fraction if possible.

🧮 Performing mixed operations

🧮 Choosing which form to use

The excerpt shows two main approaches:

Approach	When to use	Example from excerpt
Convert to decimals	When fractions are simple to convert	0.38 × 1/4: convert 1/4 to 0.25
Convert to fractions	When decimals are simple to express as fractions	5/13 × (4/5 − 0.28): convert 0.28 to 7/25

• The choice is flexible; either method works as long as you are consistent within a single problem.
• Don't confuse: You must pick one form and stick with it for all numbers in that calculation.

🧮 Applying order of operations

• Multiply and divide first, then add and subtract.
• Example: In 1.85 + 3/8 × 4.1, first convert 3/8 to 0.375, then multiply 0.375 × 4.1 = 1.5375, and finally add 1.85 + 1.5375 = 3.3875.
• Parentheses override the standard order, just as with pure decimal or pure fraction problems.

🧮 Multi-step problems

The excerpt includes a complex example:

• Problem: 0.125 × 1 1/3 + 1/16 − 0.1211

• Steps:

  1. Convert 0.125 to a fraction: 125/1000 = 1/8
  2. Convert 1 1/3 to an improper fraction: 4/3
  3. Multiply: 1/8 × 3/4 = 3/32
  4. Convert 1/16 to the same denominator: 2/32
  5. Add: 3/32 + 2/32 = 5/32
  6. Convert 5/32 to decimal: 0.15625
  7. Subtract: 0.15625 − 0.1211 = 0.03515
  8. Convert back to fraction: 3515/100000 = 703/20000

• Notice the excerpt shows flexibility: converting back and forth as needed to simplify each step.

📝 Practice problem patterns

📝 Addition and subtraction

• Example: 3/5 + 1.6 = 0.6 + 1.6 = 2.2 (or 2 1/5 in mixed number form)
• Example: 5/8 − 0.513 = 0.625 − 0.513 = 0.112
• The key is aligning the forms before performing the operation.

📝 Multiplication and division

• Example: 0.22 × 1/4 = 0.22 × 0.25 = 0.055
• Example: 3/5 × 8.4 (convert 3/5 to 0.6, then 0.6 × 8.4)
• Example: 1/25 × 3.19 = 0.04 × 3.19 = 0.1276
• Division follows the same principle: unify forms, then divide.

📝 Expressions with parentheses

• Example: 15/16 × (7/10 − 0.5) = 15/16 × (0.7 − 0.5) = 15/16 × 0.2 = 0.1875
• Example: 0.2 × (7/20 + 1.1143) requires converting 7/20 to 0.35, then adding inside parentheses first.
• Don't confuse: Parentheses always take priority, but you still must unify number forms inside the parentheses before operating.

📝 Exponents and roots

• Example: 0.5 × 1/4 + (0.3)² = 0.5 × 0.25 + 0.09 = 0.125 + 0.09 = 0.215 (the excerpt shows 0.615, suggesting a different calculation path)
• Example: (3/8)² − 0.000625 + (1.1)² = 0.140625 − 0.000625 + 1.21 = 1.35
• Exponents and roots are evaluated before multiplication and division, following standard order of operations.

⚠️ Common pitfalls

⚠️ Forgetting to convert

• You cannot add, subtract, multiply, or divide a decimal and a fraction directly.
• Always convert first, then operate.
• Example: 3/10 + 0.7 requires converting 3/10 to 0.3 (or 0.7 to 7/10) before adding.

⚠️ Inconsistent conversions

• Once you choose to work in decimals or fractions, convert all numbers in that step to the same form.
• Switching back and forth mid-calculation increases error risk.

⚠️ Ignoring order of operations

• Even after unifying number forms, you must still multiply/divide before adding/subtracting.
• Example: 8.91 + 1/5 × 1.6 is not (8.91 + 1/5) × 1.6; it is 8.91 + (1/5 × 1.6) = 8.91 + 0.32 = 9.23.

🧭 Overview

🧠 One-sentence thesis

This chapter equips learners to work with ratios, rates, proportions, and percents—including conversions, solving for missing factors, and applying these tools to real-world problems.

📌 Key points (3–5)

• Ratios vs rates: distinguish between pure-number comparisons (ratios) and comparisons involving different units (rates).
• Proportions: describe proportional relationships and solve for unknown values in proportions, including those with rates.
• Percent conversions: understand percents as ratios and convert fluently among fractions, decimals, and percents (including fractions of one percent).
• Common confusion: denominate numbers (with units) vs pure numbers (without units)—only like denominate or pure numbers can be compared by subtraction.
• Applications: use the five-step method to solve proportion problems and distinguish base, percent, and percentage in percent problems.

📐 Ratios and Rates

📐 Denominate vs pure numbers

Denominate numbers: numbers together with some specified unit.
Pure numbers: numbers that exist purely as numbers and do not represent amounts of quantities.

• Denominate numbers carry units (e.g., 8 miles, 5 gallons).
• Pure numbers have no units (e.g., 8, 0.07, 21 5/8).
• Like denominate numbers: same unit (e.g., 8 miles and 3 miles).
• Unlike denominate numbers: different units (e.g., 8 miles and 5 gallons).

🔢 Comparing by subtraction

• Subtraction tells how much more one number is than another.
• You can subtract only if both numbers are like denominate or both are pure.
• Example: 8 miles − 3 miles = 5 miles → "8 miles is 5 miles more than 3 miles."
• Don't confuse: comparing 8 miles and 5 gallons by subtraction makes no sense because the units differ.

➗ Comparing by division

• Division tells how many times larger or smaller one number is than another.
• Works for both like and unlike denominate numbers.
• Like denominate: units cancel out.
  • Example: (8 miles) ÷ (2 miles) = 4 → "8 miles is 4 times as large as 2 miles."
• Unlike denominate: units remain in the result (a rate).
  • Example: (30 miles) ÷ (2 gallons) = 15 miles per 1 gallon → "the car gets 15 miles per gallon."

🧮 Proportions

🧮 What proportions are

Proportion: a statement that two ratios (or rates) are equal.

• Used to describe relationships where one quantity scales with another.
• Learners will be able to find the missing factor in a proportion.

🔧 Proportions involving rates

• Proportions can include rates (comparisons of unlike units).
• Example context: if 30 miles uses 2 gallons, how many gallons for 45 miles? (The excerpt does not give the full method, but the objective is to work with such problems.)

🛠️ Applications of Proportions

🛠️ Five-step method

• The chapter teaches a structured five-step approach to solve proportion problems.
• (The excerpt does not detail the steps, only that learners should be able to apply them.)

📊 Percent

📊 Relationship between ratios and percents

• Percents are a special kind of ratio: parts per hundred.
• Understanding this connection allows conversions between fractions, decimals, and percents.

🔄 Conversions

• Learners should be able to convert:
  • Fractions ↔ decimals ↔ percents.
  • Fractions of one percent (e.g., 1/2 of 1%) ↔ decimals.
• Example context: 0.5% = 0.005 as a decimal.

🧩 Base, percent, and percentage

• Base: the whole or reference amount.
• Percent: the rate (parts per hundred).
• Percentage: the actual part (result of applying the percent to the base).
• Learners must distinguish these three concepts to solve percent problems correctly.
• Don't confuse: "percent" (the rate) with "percentage" (the resulting amount).

🎯 Fractions of One Percent

🎯 Meaning and conversions

Fraction of one percent: a percent value that is itself a fraction (e.g., 1/2%, 3/4%).

• These represent very small proportions.
• Learners should understand what such values mean and how to convert them to decimals or fractions.
• Example: 1/2% = 0.5% = 0.005 as a decimal.

🧭 Overview

🧠 One-sentence thesis

Ratios and rates are two distinct ways to compare quantities by division—ratios compare like or pure numbers while rates compare unlike quantities—and understanding this distinction is essential for solving proportion problems.

📌 Key points (3–5)

• Two types of numbers: denominate numbers (with units) vs pure numbers (without units); denominate numbers can be "like" (same units) or "unlike" (different units).
• Two ways to compare: subtraction shows "how much more," while division shows "how many times larger/smaller."
• Ratio vs rate distinction: ratios compare pure numbers or like denominate numbers (e.g., 8 miles to 2 miles); rates compare unlike denominate numbers (e.g., 30 miles to 2 gallons).
• Common confusion: when dividing like quantities, units cancel out; when dividing unlike quantities, units remain and create a rate.
• Simplification: both ratios and rates can be expressed as fractions and reduced to simpler equivalent forms.

🔢 Types of numbers

🔢 Denominate numbers

Denominate numbers: numbers together with some specified unit.

• These are measurements or quantities with labels attached.
• Example: 8 miles, 5 gallons, 12 people.

🔢 Like vs unlike denominate numbers

Like denominate numbers: denominate numbers with the same units. Unlike denominate numbers: denominate numbers with different units.

Type	Example	Can compare by subtraction?
Like	8 miles and 3 miles	Yes
Unlike	8 miles and 5 gallons	No

🔢 Pure numbers

Pure numbers: numbers that exist purely as numbers and do not represent amounts of quantities.

• Examples: 8, 254, 0.07, fractions like 2/5.
• No units attached; they stand alone as abstract quantities.

⚖️ Two ways to compare quantities

➖ Comparison by subtraction

Comparison by subtraction: indicates how much more one number is than another.

• Rule: Numbers can be compared by subtraction if and only if they are both pure numbers or both like denominate numbers.
• Example: 8 miles − 3 miles = 5 miles means "8 miles is 5 miles more than 3 miles."
• Don't confuse: comparing 8 miles and 5 gallons by subtraction makes no sense because the units are different.

➗ Comparison by division

Comparison by division: indicates how many times larger or smaller one number is than another.

• Works for all types: pure numbers, like denominate numbers, and unlike denominate numbers.
• Example: 36 ÷ 4 = 9 means "36 is 9 times as large as 4."
• The result depends on whether units are alike or different (see next section).

🎯 Ratios and rates defined

🎯 What is a ratio

Ratio: a comparison, by division, of two pure numbers or two like denominate numbers.

• When dividing like quantities, the units cancel out.
• Example: (8 miles) / (2 miles) = 4 (no units remain).
• Interpretation: "8 miles is 4 times as large as 2 miles."
• Can be written as a/b or "a to b."

🎯 What is a rate

Rate: a comparison, by division, of two unlike denominate numbers.

• When dividing unlike quantities, the units do not cancel—they remain in the answer.
• Example: (30 miles) / (2 gallons) = 15 miles per 1 gallon.
• Interpretation: "for every 2 gallons, the car goes 30 miles," which simplifies to "15 miles per gallon."
• Example: (4 televisions) / (12 people) reduces to (1 television) / (3 people), meaning "for every 1 television, there are 3 people."

🎯 Key distinction

Comparison type	What you're comparing	Units in result	Example
Ratio	Pure or like denominate	Units cancel	8 miles / 2 miles = 4
Rate	Unlike denominate	Units remain	30 miles / 2 gallons = 15 miles/gallon

🧮 Working with ratios and rates

🧮 Expressing as fractions

• Both ratios and rates can be written as fractions: a/b.
• The ratio "3 to 2" becomes 3/2.
• The rate "5 books to 4 people" becomes (5 books) / (4 people).

🧮 Reducing to simplest form

• Since ratios and rates are fractions, they can be reduced.
• Example: the ratio 30/2 reduces to 15/1, so "30 to 2" is equivalent to "15 to 1."
• Example: the rate (4 televisions) / (12 people) reduces to (1 television) / (3 people)—both rates mean the same thing.
• Don't confuse: when reducing a ratio of like quantities, the final answer has no units; when reducing a rate, the units stay in the simplified form.

🧮 Reading notation

• a/b is read as "a to b."
• b/a is read as "b to a."
• Example: 9/5 is "9 to 5"; (25 miles) / (2 gallons) is "25 miles to 2 gallons."

🧭 Overview

🧠 One-sentence thesis

A proportion states that two ratios or rates are equal, and solving proportions allows us to find an unknown quantity by using cross products and division.

📌 Key points (3–5)

• What a proportion is: a statement that two ratios or rates are equal.
• How to solve for the unknown: use cross products to create a multiplication statement, then divide the product by the known factor.
• Form matters for rates: when setting up proportions with unlike units, the same unit type must appear in corresponding positions (numerator with numerator, denominator with denominator).
• Common confusion: incorrectly placing unit types on opposite sides produces a numerically wrong cross product and a meaningless comparison ("hooks to hooks as poles to poles").
• Reading proportions: "3 is to 5 as 12 is to 20" corresponds to the fraction form 3/5 = 12/20.

📖 What is a proportion

📖 Definition and structure

Proportion: a statement that two ratios or rates are equal.

• A proportion compares two fractions or rates and asserts they are equivalent.
• It can involve pure numbers (ratios) or denominate numbers (rates).
• Example: 3/5 = 12/20 is read "3 is to 5 as 12 is to 20."
• Example with rates: "10 items to 5 dollars as 2 items to 1 dollar" is written as (10 items)/(5 dollars) = (2 items)/(1 dollar).

🗣️ How to read and write proportions

Fractional form	Sentence form
3/8 = 6/16	3 is to 8 as 6 is to 16
(2 people)/(1 window) = (10 people)/(5 windows)	2 people are to 1 window as 10 people are to 5 windows
15/4 = 75/20	15 is to 4 as 75 is to 20

• The word "as" separates the two ratios or rates.
• Each ratio is expressed with "is to" or "are to."

🔍 Finding the missing factor

🔍 The cross-product method

• Many practical problems give three numbers and one unknown (represented by a letter like x).
• Two fractions are equal if and only if their cross products are equal.
• Example: if 3/5 = 12/20, then 3 · 20 = 5 · 12 (both equal 60).
• In a proportion with an unknown, the cross product always produces: (number) · (letter) = (number) · (number).
• This is a missing factor statement.

➗ Solving for the unknown

Finding the missing factor: x = (product) ÷ (known factor)

• Step 1: Write the cross product equation.
• Step 2: Identify the product (the result of multiplying two known numbers).
• Step 3: Divide the product by the known factor (the number multiplied by x).

Example: Solve x/4 = 20/16.

• Cross product: 16 · x = 4 · 20
• Simplify: 16 · x = 80
• Divide: x = 80 ÷ 16 = 5
• Check: 5/4 = 20/16 (both simplify to 5/4).

Example: Solve 5/x = 20/16.

• Cross product: 5 · 16 = 20 · x
• Simplify: 80 = 20 · x
• Divide: x = 80 ÷ 20 = 4
• Check: 5/4 = 20/16.

Example: Solve 16/3 = 64/x.

• Cross product: 16 · x = 64 · 3
• Simplify: 16 · x = 192
• Divide: x = 192 ÷ 16 = 12
• Check: 16/3 = 64/12.

Example: Solve 9/8 = x/40.

• Cross product: 9 · 40 = 8 · x
• Simplify: 360 = 8 · x
• Divide: x = 360 ÷ 8 = 45
• Check: 9/8 = 45/40 (both simplify to 9/8).

🧩 Where the unknown can appear

The unknown can be in any of the four positions:

• x/4 = 20/16 means 16 · x = 4 · 20
• 4/x = 16/20 means 4 · 20 = 16 · x
• 5/4 = x/16 means 5 · 16 = 4 · x
• 5/4 = 20/x means 5 · x = 4 · 20

Regardless of position, the result is always a multiplication statement with one unknown factor.

⚠️ Proportions involving rates

⚠️ Correct form for rates

• A rate compares two unlike denominate numbers (e.g., miles and gallons, people and windows).
• When setting up a proportion with rates, unit types must align:
  • (unit type 1)/(unit type 2) = (unit type 1)/(unit type 2), or
  • (unit type 2)/(unit type 1) = (unit type 2)/(unit type 1)
• Both forms produce a cross product of the type: (unit type 1) · (unit type 2) = (unit type 1) · (unit type 2).

Correct examples:

• (2 hooks)/(3 poles) = (4 hooks)/(6 poles)
• (3 poles)/(2 hooks) = (6 poles)/(4 hooks)
• (40 miles)/(2 gallons) = (80 miles)/(4 gallons)

🚫 Incorrect form and why it fails

Incorrect: (2 hooks)/(3 poles) = (6 poles)/(4 hooks)

Two reasons this is wrong:

1. Numerically wrong cross product: 2 · 4 ≠ 3 · 6 (8 ≠ 18).
2. Meaningless comparison: the cross product would say "hooks are to hooks as poles are to poles," which makes no sense in the context of a rate.

Don't confuse: Placing the same unit type on opposite sides (numerator on one side, denominator on the other) breaks the comparison structure and produces an invalid proportion.

🛠️ Setting up proportion problems

🛠️ The five-step method (introduction)

The excerpt introduces a systematic approach for solving proportion problems:

1. Identify the unknown quantity by careful reading and represent it with a letter (there will be only one unknown).
2. Identify the three specified numbers given in the problem.
3. Determine which comparisons are to be made and set up the proportion with correct unit alignment.

(The excerpt cuts off before completing the five-step method, but these first three steps are the foundation.)

📝 Translating sentences to proportions

Example: "5 hats are to 4 coats as x hats are to 24 coats."

• Proportion: 5/4 = x/24
• Cross product: 5 · 24 = 4 · x
• Solve: 120 = 4 · x, so x = 30.

Example: "1 spacecraft is to 7 astronauts as 5 spacecraft are to x astronauts."

• Proportion: 1/7 = 5/x
• Cross product: 1 · x = 7 · 5
• Solve: x = 35.

Example: "18 calculators are to 90 calculators as x students are to 150 students."

• Proportion: 18/90 = x/150
• Cross product: 18 · 150 = 90 · x
• Solve: 2700 = 90 · x, so x = 30.

✅ Checking if a proportion is true

To verify a proportion, compute both cross products and check if they are equal.

Example: Is 3/16 = 12/64 true?

• Cross products: 3 · 64 = 192 and 16 · 12 = 192.
• Since 192 = 192, the proportion is true.

Example: Is 1/9 = 3/30 true?

• Cross products: 1 · 30 = 30 and 9 · 3 = 27.
• Since 30 ≠ 27, the proportion is false.

Example: Is (33 miles)/(1 gallon) = (99 miles)/(3 gallons) true?

• Cross products: 33 · 3 = 99 and 1 · 99 = 99.
• Since 99 = 99, the proportion is true.

🧭 Overview

🧠 One-sentence thesis

Proportion problems can be systematically solved by identifying the unknown quantity, setting up equivalent ratios from the given information, and solving through cross-multiplication to find practical answers in real-world contexts.

📌 Key points (3–5)

• The five-step method: a structured approach to solving proportion problems—identify the unknown, list the three known numbers, set up the proportion, solve by cross-multiplication, and interpret the result with units.
• Step 1 is critical: many problems fail because students skip determining what the unknown quantity is and representing it with a letter.
• Suspending units during computation: when setting up proportions with rates or ratios, drop the units temporarily to simplify the algebra, then restore them in the final interpretation.
• Common confusion: distinguishing which quantities to compare—always match corresponding types (e.g., inches to miles on one side, inches to miles on the other; not inches to inches and miles to miles unless the structure demands it).
• Wide applicability: the method works for maps, mixtures, shadows, ratios of groups, and rates of events.

🛠️ The Five-Step Method

🛠️ What the method is

The five-step method for solving proportion problems:

1. By careful reading, determine what the unknown quantity is and represent it with some letter. There will be only one unknown in a problem.
2. Identify the three specified numbers.
3. Determine which comparisons are to be made and set up the proportion.
4. Solve the proportion (using cross-multiplication).
5. Interpret and write a conclusion in a sentence with the appropriate units of measure.

• The excerpt emphasizes that Step 1 is extremely important: "Many problems go unsolved because time is not taken to establish what quantity is to be found."
• The excerpt states: "When solving an applied problem, always begin by determining the unknown quantity and representing it with a letter."

🔍 Why Step 1 matters

• Without a clear unknown, you cannot set up the proportion correctly.
• The excerpt notes that there will be only one unknown in a problem.
• Example: if the problem asks "How many miles are represented by 8 inches?" the unknown is miles, so you write "Let x = number of miles represented by 8 inches."

📋 The three specified numbers (Step 2)

• These are the concrete values given in the problem.
• They form the basis of the two ratios you will compare.
• Example: "On a map, 2 inches represents 25 miles. How many miles are represented by 8 inches?" → the three numbers are 2 inches, 25 miles, and 8 inches.

⚖️ Setting up the proportion (Step 3)

• Identify which quantities correspond to each other.
• Write two ratios that should be equal.
• The excerpt advises: "Proportions involving ratios and rates are more readily solved by suspending the units while doing the computations."
• Example: 2 inches to 25 miles → 2/25; 8 inches to x miles → 8/x. So the proportion is 2/25 = 8/x.
• Don't confuse: the structure must match—if the first ratio is "inches over miles," the second must also be "inches over miles," not "miles over inches."

🔢 Solving by cross-multiplication (Step 4)

• Cross-multiply: if a/b = c/d, then a·d = b·c.
• Solve the resulting equation for the unknown letter.
• Example: 2/25 = 8/x → 2·x = 8·25 → 2x = 200 → x = 100.

📝 Interpreting the result (Step 5)

• Translate the numerical answer back into the context of the problem.
• Always include the appropriate units of measure.
• Example: "If 2 inches represents 25 miles, then 8 inches represents 100 miles."

🗺️ Map and scale problems

🗺️ Map scale example

Problem: On a map, 2 inches represents 25 miles. How many miles are represented by 8 inches?

• Step 1: Let x = number of miles represented by 8 inches.
• Step 2: The three numbers are 2 inches, 25 miles, 8 inches.
• Step 3: Comparisons: 2 inches to 25 miles → 2/25; 8 inches to x miles → 8/x. Proportion: 2/25 = 8/x.
• Step 4: Cross-multiply: 2x = 200 → x = 100.
• Step 5: 8 inches represents 100 miles.

📐 Blueprint/model scale example

Problem: A model is built to 2/15 scale. If a particular part of the model measures 6 inches, how long is the actual structure?

• The excerpt does not show the full solution, but the principle is the same: set up the ratio of model size to actual size.
• The answer given is 45 inches.

🧪 Mixture and solution problems

🧪 Acid solution example

Problem: An acid solution is composed of 7 parts water to 2 parts acid. How many parts of water are there in a solution composed of 20 parts acid?

• Step 1: Let n = number of parts of water.
• Step 2: The three numbers are 7 parts water, 2 parts acid, 20 parts acid.
• Step 3: Comparisons: 7 parts water to 2 parts acid → 7/2; n parts water to 20 parts acid → n/20. Proportion: 7/2 = n/20.
• Step 4: Cross-multiply: 7·20 = 2·n → 140 = 2n → n = 70.
• Step 5: 7 parts water to 2 parts acid indicates 70 parts water to 20 parts acid.

🧪 Alloy example

Problem: An alloy contains 3 parts of nickel to 4 parts of silver. How much nickel is in an alloy that contains 44 parts of silver?

• The answer given is 33 parts.
• The setup follows the same pattern: 3/4 = n/44.

🌞 Shadow and height problems

🌞 Shadow example with a girl

Problem: A 5-foot girl casts a 3 and 1/3-foot shadow at a particular time of the day. How tall is a person who casts a 3-foot shadow at the same time of the day?

• Step 1: Let h = height of the person.
• Step 2: The three numbers are 5 feet (height of girl), 3 and 1/3 feet (length of shadow), 3 feet (length of shadow).
• Step 3: Comparisons: 5-foot girl is to 3 and 1/3 foot shadow → 5/(3 and 1/3); h-foot person is to 3-foot shadow → h/3. Proportion: 5/(3 and 1/3) = h/3.
• Step 4: Cross-multiply: 5·3 = (3 and 1/3)·h → 15 = (10/3)·h. Divide 15 by 10/3: h = 15 ÷ (10/3) = 15 · (3/10) = 9/2 = 4 and 1/2.
• Step 5: A person who casts a 3-foot shadow at this particular time of the day is 4 and 1/2 feet tall.

🌞 Why shadows work

• At the same time of day, the sun's angle is constant, so the ratio of height to shadow length is the same for all objects.
• This is a classic similar-triangles application of proportions.

👥 Ratio and rate problems

👥 Men to women ratio example

Problem: The ratio of men to women in a particular town is 3 to 5. How many women are there in the town if there are 19,200 men in town?

• Step 1: Let x = number of women in town.
• Step 2: The three numbers are 3, 5, 19,200.
• Step 3: Comparisons: 3 men to 5 women → 3/5; 19,200 men to x women → 19,200/x. Proportion: 3/5 = 19,200/x.
• Step 4: Cross-multiply: 3x = 19,200·5 → 3x = 96,000 → x = 32,000.
• Step 5: There are 32,000 women in town.

🏆 Wins to losses rate example

Problem: The rate of wins to losses of a particular baseball team is 9/2. How many games did this team lose if they won 63 games?

• Step 1: Let n = number of games lost.
• Step 2: Since 9/2 means 9 wins to 2 losses, the three numbers are 9 (wins), 2 (losses), 63 (wins).
• Step 3: Comparisons: 9 wins to 2 losses → 9/2; 63 wins to n losses → 63/n. Proportion: 9/2 = 63/n.
• Step 4: Cross-multiply: 9n = 2·63 → 9n = 126 → n = 14.
• Step 5: This team had 14 losses.

🎲 Odds example

Problem: The odds for a particular event occurring are 11 to 2. (For every 11 times the event does occur, it will not occur 2 times.) How many times does the event occur if it does not occur 18 times?

• The answer given is: The event occurs 99 times.
• The setup: 11/2 = x/18 → 11·18 = 2x → 198 = 2x → x = 99.

🧮 Comparison table of problem types

Problem type	What is compared	Example unknown	Key insight
Map/scale	Distance on map to real distance	Real miles for given map inches	Scale ratio stays constant
Mixture/solution	Parts of one substance to parts of another	Parts of water for given parts of acid	Composition ratio stays constant
Shadow/height	Height to shadow length	Height of person for given shadow	Sun angle is constant at same time
Group ratio	Number in one group to number in another	Number of women for given number of men	Population ratio stays constant
Win/loss rate	Wins to losses	Losses for given wins	Performance ratio stays constant

⚠️ Common pitfalls

⚠️ Skipping Step 1

• The excerpt warns: "Step 1 is extremely important. Many problems go unsolved because time is not taken to establish what quantity is to be found."
• Always write "Let [letter] = [description of unknown]" before setting up the proportion.

⚠️ Mismatching comparisons

• Don't confuse: if the first ratio is "inches to miles," the second must also be "inches to miles," not "miles to inches."
• Example: 2 inches to 25 miles → 2/25, so 8 inches to x miles → 8/x. Not 25/2 = x/8.

⚠️ Forgetting units in Step 5

• The final answer must include the appropriate units of measure.
• Example: not just "100," but "100 miles."

⚠️ Suspending vs restoring units

• The excerpt advises suspending units during computation to simplify algebra.
• But you must restore them in the final interpretation.
• Example: during Step 4, work with 2/25 = 8/x (no units); in Step 5, write "8 inches represents 100 miles" (units restored).

🧭 Overview

🧠 One-sentence thesis

Percent is a ratio that compares a number to 100, and it can be converted to and from fractions and decimals using systematic techniques.

📌 Key points (3–5)

• What percent means: "for each hundred" or "for every hundred," derived from Latin "per centum."
• Percent as a ratio: any ratio comparing a number to 100 can be written as a percent using the % symbol.
• Three interchangeable forms: fractions, decimals, and percents can all be converted to one another.
• Common confusion: moving the decimal point—to convert decimal to percent, move 2 places right and add %; to convert percent to decimal, move 2 places left and drop %.
• Why it matters: percents provide a standardized way to express ratios and proportions for easy comparison.

🔢 What percent means

📖 Definition and origin

Percent means "for each hundred," or "for every hundred." The symbol % is used to represent the word percent.

• The word comes from Latin: "per" (for each/for every) + "centum" (hundred).
• A percent is fundamentally a ratio where the second term is always 100.
• Example: The ratio 26 to 100 can be written as 26%, read as "twenty-six percent."

🧮 How ratios become percents

• Any ratio with 100 as the denominator is a percent.
• The ratio 165 to 100 = 165% (one hundred sixty-five percent).
• Percents can exceed 100%—this simply means the numerator is larger than 100.
• Example: 25 cents is 25/100 of a dollar, which means 25 cents is 25% of one dollar.

✏️ Writing percents as fractions

• Drop the % symbol and write the number over 100.
• 38% = 38/100
• 210% = 210/100 or the mixed number 2 10/100 or the decimal 2.1
• Don't confuse: a percent greater than 100 is not an error; it represents a value greater than the whole.

🔄 Converting between forms

🔄 The three forms

The excerpt shows that fractions, decimals, and percents are three ways to express the same relationship:

Form	Example	Represents
Fraction	3/5	Three parts out of five
Decimal	0.6	Six tenths
Percent	60%	Sixty per hundred

🔀 Decimal to percent

• Move the decimal point 2 places to the right and affix the % symbol.
• Example: 0.75 = 75/100 = 75%
• Example: 5.64 = 564%
• The underlying logic: multiplying by 100 (to get "per hundred") shifts the decimal point right.

🔀 Percent to decimal

• Drop the % symbol and move the decimal point 2 places to the left.
• Example: 12% = 12/100 = 0.12
• Example: 461% = 4.61
• Example: 0.09% = 0.0009
• Don't confuse: small percents (less than 1%) become very small decimals.

🔁 Converting fractions

➡️ Fraction to percent

Two-step process:

1. Convert the fraction to a decimal (divide numerator by denominator).
2. Convert the decimal to a percent (move decimal point 2 places right, add %).

Example: 3/5 → divide 3 by 5 → 0.6 → 60%

Alternative reasoning: 3/5 = 6/10 = 60/100 = 60%

⬅️ Percent to fraction

• Drop the % sign and write the number "over" 100.
• Reduce if possible.
• Example: 42% = 42/100 = 21/50 (reduced)
• Alternative: convert percent to decimal first (42% = 0.42), then write as fraction (42/100 = 21/50).

🔁 Repeating decimals and percents

• Some fractions produce repeating decimals, which become repeating percents.
• Example: 3/11 converts to 27.27% (with repeating digits).
• Example: 6/11 converts to 54.54% (with repeating digits).
• The excerpt uses notation like "27.27%" to indicate the repeating pattern.

📋 Conversion technique summary

The excerpt provides a systematic table of techniques:

Starting form	To decimal	To percent	To fraction
Fraction	Divide numerator by denominator	Convert to decimal first, then move point 2 right + %	Already a fraction
Decimal	Already a decimal	Move point 2 right + %	Read the decimal, write as fraction, reduce
Percent	Move point 2 left, drop %	Already a percent	Drop %, write over 100, reduce

🎯 Key pattern

• The decimal point always moves 2 places because percent means "per hundred" (100 = 10²).
• Direction matters: decimal → percent (right), percent → decimal (left).
• Example: 0.55 → 55% (right); 15% → 0.15 (left).

🧭 Overview

🧠 One-sentence thesis

Fractions of one percent (like ½% or ⅗%) represent portions of 1% and can be converted to fractions or decimals by multiplying the fractional part by 1/100, with special care needed for nonterminating decimals.

📌 Key points (3–5)

• What fractions of one percent mean: percents like ½% or ⅗% represent a fraction of 1%, not yet reaching a full 1%.
• Basic conversion method: multiply the fraction by 1/100 (since "percent" means "for each hundred" and "of" means "times").
• Nonterminating fraction challenge: fractions like ⅔ produce nonterminating decimals, so conversions require rounding to a specified number of decimal places.
• Common confusion: when converting to decimals with a target precision, round the fractional part before moving the decimal point two places left (not after).
• Why it matters: these conversions are essential for working with very small percentages in practical calculations.

🔢 Understanding fractions of one percent

🔢 What they represent

Fractions of one percent: percents where 1% has not been attained, such as ½%, ⅗%, ⅝%, or 7/11%.

• These are not "half of something" or "three-fifths of something" in general—they are specifically fractions of 1%.
• The key insight: ½% = ½ of 1%, ⅗% = ⅗ of 1%, etc.
• Example: ½% means "half of one percent," which is much smaller than ½ (which would be 50%).

🧮 The conversion logic

• "Percent" means "for each hundred," so 1% = 1/100.
• "Of" means multiplication.
• Therefore: ½% = ½ × 1/100 = 1/200.
• The pattern applies to any fraction: (fraction)% = (fraction) × 1/100.

🔄 Converting to fractions

🔄 The multiplication method

The excerpt shows the systematic approach:

Percent	Interpretation	Calculation	Result
½%	½ of 1%	½ × 1/100	1/200
⅗%	⅗ of 1%	⅗ × 1/100	3/500
⅝%	⅝ of 1%	⅝ × 1/100	5/800
7/11%	7/11 of 1%	7/11 × 1/100	7/1100

📐 Mixed number fractions

• For mixed numbers like 3⅓%, treat the whole number and fraction together.
• Example from the excerpt: ⅔% = ⅔ × 1/100 = 1/150 (after simplification: 2/3 × 1/100 = 2/300 = 1/150).
• The excerpt shows 3⅓% converts to 1/30.

🔁 Converting to decimals (nonterminating case)

⚠️ The nonterminating challenge

• Some fractions (like ⅔ = 0.6666...) produce decimals that never end.
• These cannot be expressed exactly as decimals.
• The excerpt states: "it is customary to express the fraction as a rounded decimal with at least three decimal places."

📏 The two-step procedure

To convert a nonterminating fraction of 1% to a decimal:

1. Convert the fraction to a rounded decimal.
2. Move the decimal point two digits to the left and remove the percent sign.

Critical detail: The rounding must account for the final precision needed.

🎯 Precision planning

The excerpt emphasizes planning ahead:

• If you want a three-place final decimal, and moving the decimal point accounts for two places, round the fraction to one place first (2 + 1 = 3).
• If you want a four-place final decimal, round the fraction to two places first (2 + 2 = 4).

Example from the excerpt:

• Convert ⅔% to a three-place decimal:
  1. ⅔ = 0.6666... → round to one place: 0.7
  2. 0.7% → move decimal two places left: 0.007
• Result: ⅔% = 0.007 (to three decimal places)

🧪 More complex examples

The excerpt provides:

• 5 4/11% to four places:
  1. 4/11 = 0.3636... → round to two places: 5.36%
  2. Move decimal: 0.0536
• 28 5/9% to ten-thousandths (four places):
  1. 5/9 = 0.5555... → round to two places: 28.56%
  2. Move decimal: 0.2856

Don't confuse: The rounding happens to the fractional part before you apply the percent conversion, not to the final answer. This ensures the correct precision in the final result.

🔍 Common patterns and practice

🔍 Terminating vs nonterminating

Type	Example	Method	Result
Terminating	⅝%	⅝ × 1/100 = 0.625 × 0.01	0.00625 (exact)
Nonterminating	⅔%	Round ⅔ first, then shift decimal	0.007 (approximate)

💡 Why the two-place shift

• Removing the percent sign is equivalent to dividing by 100.
• Dividing by 100 moves the decimal point two places to the left.
• Example: 0.7% becomes 0.007 (added zeros to locate the decimal correctly).

🧭 Overview

🧠 One-sentence thesis

Every percent problem involves three components—base (the total supply or starting amount), percent (the rate per 100), and percentage (the part or portion)—and finding any one requires translating the word problem into a multiplication equation and solving for the missing value.

📌 Key points (3–5)

• Three components in every percent problem: base (source/total), percent (rate per 100), and percentage (part/portion).
• All problems translate to one equation: (percentage) = (percent) × (base).
• Three problem types: finding the percentage (missing product), finding the percent (missing factor), or finding the base (missing factor).
• Common confusion: distinguishing base from percentage—the base is the whole or starting amount; the percentage is the part or result.
• Real-world applications: commissions, salary increases, discounts, exam scores, and solution concentrations all use the same three-part structure.

🧩 The three core components

🧩 Percentage (the part)

Percentage: the missing product P in a percent problem; it means "part" or "portion."

• The percentage represents a particular part of the base.
• In the equation P = 30% × 50, P is the percentage—the portion of 50 being considered.
• Example: "What number is 30% of 50?" asks for the percentage.
• Don't confuse: percentage is not the percent; it is the numerical result (the part), not the rate.

🧩 Percent (the rate)

Percent: the missing factor Q; it means "per 100" or "part of 100."

• The percent indicates what part of the base is being taken or considered.
• In the equation 15 = Q × 50, Q is the percent.
• Specifically, if 50 were divided into 100 equal parts, Q tells how many of those parts equal 15.
• Example: "15 is what percent of 50?" asks for the percent.

🧩 Base (the whole or starting amount)

Base: the missing factor B; meanings include "source of supply" or "starting place."

• The base indicates the total supply or the amount from which a part is taken.
• In the equation 15 = 30% × B, B is the base—the total from which 15 represents 30%.
• Example: "15 is 30% of what number?" asks for the base.
• Don't confuse: the base is the whole; the percentage is the part of that whole.

🔢 Finding the percentage (missing product)

🔢 The setup

• Problem type: "What number is [percent] of [base]?"
• Equation form: P = (percent) × (base)
• Method: Convert the percent to a decimal, then multiply.

🔢 Basic examples

Problem	Equation	Calculation	Result
What is 30% of 50?	P = 30% × 50	P = 0.30 × 50	P = 15
What is 36% of 95?	P = 36% × 95	P = 0.36 × 95	P = 34.2

🔢 Real-world scenarios

Commission calculation

• A salesperson earns 12% commission on an $8,400 sale.
• "What part of $8,400?" → percentage problem.
• P = 12% × 8,400 = 0.12 × 8,400 = $1,008.

Increase in performance

• A typist increases speed by 110% from 16 words per minute.
• P = 110% × 16 = 1.10 × 16 = 17.6 words per minute increase.
• New speed: 16 + 17.6 = 33.6 words per minute.

Salary raise

• A student earns $125/month and receives a 4% raise.
• New salary is 100% + 4% = 104% of original.
• P = 104% × 125 = 1.04 × 125 = $130.

Discount pricing

• An item marked $24.95 is on sale at 15% off.
• Sale price is 100% − 15% = 85% of marked price.
• P = 85% × 24.95 = 0.85 × 24.95 = $21.21 (rounded to two decimal places for money).

🔍 Finding the percent (missing factor)

🔍 The setup

• Problem type: "[Percentage] is what percent of [base]?"
• Equation form: (percentage) = Q × (base)
• Method: Recall that (missing factor) = (product) ÷ (known factor), so Q = (percentage) ÷ (base), then convert the decimal result to a percent.

🔍 Basic examples

Problem	Equation	Calculation	Result
15 is what percent of 50?	15 = Q × 50	Q = 15 ÷ 50 = 0.3	Q = 30%
4.32 is what percent of 72?	4.32 = Q × 72	Q = 4.32 ÷ 72 = 0.06	Q = 6%

🔍 Real-world scenarios

Exam score

• A student answers 125 out of 160 questions correctly.
• 125 = Q × 160 → Q = 125 ÷ 160 = 0.78125 → 78% (rounded to two decimal places).

Solution concentration

• A bottle contains 80 ml HCl and 30 ml water (total 110 ml).
• 80 = Q × 110 → Q = 80 ÷ 110 ≈ 0.727272... → approximately 73% HCl.
• The symbol "≈" means "approximately."

Salary increase

• A woman earned $19,200 five years ago and now earns $42,000.
• 42,000 = Q × 19,200 → Q = 42,000 ÷ 19,200 = 2.1875 → 219% (rounded to two decimal places).
• This means her salary increased by 219% of the original.

🎯 Finding the base (missing factor)

🎯 The setup

• Problem type: "[Percentage] is [percent] of what number?"
• Equation form: (percentage) = (percent) × B
• Method: Convert percent to decimal, then B = (percentage) ÷ (percent).

🎯 Basic examples

Problem	Equation	Calculation	Result
15 is 30% of what number?	15 = 30% × B	B = 15 ÷ 0.30	B = 50
56.43 is 33% of what number?	56.43 = 33% × B	B = 56.43 ÷ 0.33	B = 171

🎯 Real-world scenarios

Solution volume

• 15 ml of water represents 2% of an HCl solution.
• "Total supply" → base.
• 15 = 2% × B → B = 15 ÷ 0.02 = 750 ml total solution.

Sales tax calculation

• A sales tax of 6½% results in $2.99 tax on an item.
• "Price" is the starting place → base.
• 2.99 = 6.5% × B → 2.99 = 0.065 × B → B = 2.99 ÷ 0.065 = $46.00.

Original price before discount

• An item is priced at $20.40 after a 15% discount.
• The new price is 100% − 15% = 85% of the original.
• "Original price" is the starting place → base.
• 20.40 = 85% × B → 20.40 = 0.85 × B → B = 20.40 ÷ 0.85 = $24.00.

Commission and sale amount

• A salesman earns 18¼% commission and makes $152.39 on a sale.
• 152.39 = 18.25% × B → 152.39 = 0.1825 × B → B = 152.39 ÷ 0.1825 ≈ $835 (rounded to nearest dollar).

🔄 Common patterns and tips

🔄 Translating word problems

• "What number is..." → looking for percentage (product).
• "...is what percent of..." → looking for percent (factor).
• "...is [percent] of what number?" → looking for base (factor).
• All translate to: (percentage) = (percent) × (base).

🔄 Percent conversions in context

• Increase by X%: new amount is (100% + X%) of original.
  • Example: 4% raise → 104% of original salary.
• Decrease by X%: new amount is (100% − X%) of original.
  • Example: 15% discount → 85% of marked price.

🔄 Rounding rules

• For money: round to two decimal places.
• For percents: round as specified in the problem (often two decimal places or nearest whole percent).
• Use "≈" to indicate approximate values when rounding.

🔄 Don't confuse

• Base vs. percentage: base is the whole or starting amount; percentage is the part or result.
• Percent vs. percentage: percent is the rate (e.g., 30%); percentage is the numerical result (e.g., 15).
• Original vs. new amounts: when calculating increases or decreases, identify which is the base (usually the original) and which is the percentage (usually the new or the change).

🧭 Overview

🧠 One-sentence thesis

After completing this chapter, students should be able to estimate arithmetic results using rounding, clustering, the distributive property, and fraction-rounding techniques, and understand when and why estimation is valuable.

📌 Key points (3–5)

• Four estimation techniques: rounding, clustering, mental arithmetic via the distributive property, and rounding fractions.
• Why estimate: to have an expected value before computing and to check if a result is reasonable afterward.
• Rounding for estimation is flexible: convenience guides rounding choices, not strict rounding rules, so different people may get different estimated results.
• Common confusion: estimation rounding vs. formal rounding—estimation prioritizes convenience (e.g., rounding 26 to 20 instead of 30 when dividing 80, because 80 ÷ 20 is easier).
• Scope: applies to addition, subtraction, multiplication, division, and fractions.

🎯 Purpose of estimation

🎯 When and why to estimate

• Before computing: to have an idea of what value to expect.
• After computing: to check if the result is reasonable.
• Estimation helps catch errors and builds number sense.

🗣️ Common estimation language

Common words used in estimation are about, near, and between.

• These words signal approximate rather than exact values.
• Example: "The sum is about 8,500" means an estimated, not precise, answer.

🔢 Estimation by rounding

🔢 The rounding technique

The rounding technique estimates the result of a computation by rounding the numbers involved in the computation to one or two nonzero digits.

• Focus on simplifying numbers to make mental arithmetic easier.
• Example: To estimate 2,357 + 6,106, round 2,357 to 2,400 and 6,106 to 6,100, then add 2,400 + 6,100.

🧩 Convenience over strict rules

• Key difference from formal rounding: estimation rounding uses convenience as the guide, not hard-and-fast rounding rules.
• Example: For 80 ÷ 26, you might round 26 to 20 (not 30) because 80 is more conveniently divided by 20.
• Don't confuse: this is not "wrong" rounding—it's a different goal (ease of calculation vs. precision).

🔄 Results may vary

• Because convenience is subjective, different people may round differently and get different estimated results.
• This variation is expected and acceptable in estimation.

🧮 Other estimation techniques (overview)

🧮 Clustering

• Used when adding more than two numbers that are close together (cluster around a similar value).
• The excerpt states students should understand the concept and be able to estimate sums using clustering.

🧮 Mental arithmetic with the distributive property

• Uses the distributive property to obtain exact results mentally (not just estimates).
• Example context: breaking down multiplication into easier parts.

🧮 Rounding fractions

• Applies rounding to fractions to estimate sums of two or more fractions.
• Simplifies fraction arithmetic by approximating fractions to convenient values.

📋 Summary of learning objectives

Technique	What students should be able to do
Estimation by rounding	Understand why estimation matters; estimate addition, subtraction, multiplication, or division results using rounding
Estimation by clustering	Understand clustering; estimate sums when numbers cluster together
Distributive property	Understand the property; use it for exact mental multiplication
Rounding fractions	Estimate sums of fractions by rounding them

• All techniques aim to make arithmetic faster, easier, and more intuitive.
• Estimation is a practical skill for everyday and academic math.

🧭 Overview

🧠 One-sentence thesis

Estimation by rounding helps you quickly predict the result of a computation by rounding numbers to one or two nonzero digits, making mental arithmetic easier and letting you check whether an exact answer is reasonable.

📌 Key points (3–5)

• What estimation is: the process of determining an expected value of a computation before or after doing the exact calculation.
• How rounding for estimation differs from standard rounding: convenience guides the rounding (e.g., rounding 26 to 20 instead of 30 if dividing into 80), so results may vary between people.
• The technique: round each number to one or two nonzero digits, perform the simpler computation, then use words like "about," "near," or "between" to describe the result.
• Common confusion: estimation results can vary because different people may round differently based on what feels convenient—there is no single "correct" estimate.
• Why it matters: estimation gives you a quick sense of what answer to expect and helps you spot errors in exact calculations.

🔢 What estimation means

🔢 Definition and purpose

Estimation: the process of determining an expected value of a computation.

• Estimation is not the exact answer; it is an approximation that tells you what ballpark the answer should be in.
• Common words used: "about," "near," "between."
• Two main uses:
  • Before computing: get an idea of what value to expect.
  • After computing: check if the exact result is reasonable.

Example: If you estimate 2,357 + 6,106 as "about 8,500" and your exact answer is 8,463, the estimate confirms your work is reasonable.

🧭 Convenience-based rounding

• The excerpt emphasizes that estimation rounding does not always follow strict rounding rules.
• Instead, you round to whatever is convenient for mental math.
• Example: To estimate 80 ÷ 26, you might round 26 to 20 (not 30) because 80 ÷ 20 is easier to compute mentally.

Don't confuse: This is different from formal rounding (e.g., rounding to the nearest ten). Estimation prioritizes speed and simplicity.

➕ Estimating addition and subtraction

➕ Addition by rounding

• Round each addend to one or two nonzero digits, then add the rounded numbers.
• Example from the excerpt:
  • Estimate 2,357 + 6,106.
  • Round: 2,357 is near 2,400; 6,106 is near 6,100.
  • Estimate: 2,400 + 6,100 = 8,500.
  • Exact answer: 8,463 (close to the estimate).

➖ Subtraction by rounding

• Round each number to one or two nonzero digits, then subtract.
• Example from the excerpt:
  • Estimate 5,203 − 3,015.
  • Round: 5,203 is near 5,200; 3,015 is near 3,000.
  • Estimate: 5,200 − 3,000 = 2,200.
  • Exact answer: 2,188.
• The excerpt notes you could also round 5,203 to 5,000 (one nonzero digit) for a quicker but less accurate estimate of 2,000.

Key insight: More nonzero digits → more accurate estimate, but slower mental math. Fewer nonzero digits → faster, but less precise.

✖️ Estimating multiplication

✖️ Rounding factors

• Round each factor to one nonzero digit, then multiply.
• The excerpt reminds you: to multiply numbers ending in zeros, multiply the nonzero parts and then affix the total number of zeros.

✖️ Examples

Problem	Rounded form	Estimate	Exact answer
73 · 46	70 · 50	3,500	3,358
87 · 4,316	90 · 4,000	360,000	375,492

• Example: 73 · 46 → round to 70 and 50 → 70 · 50 = 3,500 (multiply 7 · 5 = 35, then add two zeros).

➗ Estimating division

➗ Rounding dividend and divisor

• Round both the dividend (number being divided) and divisor to one or two nonzero digits, choosing values that divide evenly if possible.
• Example from the excerpt:
  • Estimate 153 ÷ 17.
  • Round: 153 is close to 150; 17 is close to 15.
  • Estimate: 150 ÷ 15 = 10.
  • Exact answer: 9.

➗ Larger numbers

• Example: 742,000 ÷ 2,400.
• Round: 742,000 → 700,000 (one nonzero digit); 2,400 → 2,000.
• Estimate: 700,000 ÷ 2,000 = 350.
• Exact answer: 309.16.

Tip: The excerpt shows that rounding to convenient divisors (like 15 into 150, or 2,000 into 700,000) makes mental division much easier.

🔢 Estimating with decimals

🔢 Addition of decimals

• Round each decimal to a whole number or simple decimal with one or two nonzero digits.
• Example from the excerpt:
  • Estimate 53.82 + 41.6.
  • Round: 53.82 → 54; 41.6 → 42.
  • Estimate: 54 + 42 = 96.
  • Exact answer: 95.42.

🔢 Multiplication of decimals

• Round each decimal factor to one or two nonzero digits.
• Example from the excerpt:
  • Estimate (31.28)(14.2).
  • Round: 31.28 → 30; 14.2 → 15.
  • Estimate: 30 · 15 = 450 (3 · 15 = 45, then affix one zero).
  • Exact answer: 444.176.

🔢 Percentages

• Convert the percentage to a decimal, then round both numbers.
• Example from the excerpt:
  • Estimate 21% of 5.42.
  • 21% = 0.21 → round to 0.2; 5.42 → round to 5.
  • Estimate: (0.2)(5) = 1.
  • Exact answer: 1.1382.

⚠️ Important reminders

⚠️ Results may vary

• The excerpt repeatedly states: "Results may vary."
• Because rounding is done for convenience, different people may round differently and get different estimates.
• Example: One person might round 137.88 to 138; another might round it to 140. Both are valid if they make the mental math easier.

⚠️ Comparing estimate to exact value

• After estimating, the excerpt recommends finding the exact value and comparing.
• This comparison helps you:
  • Verify your exact calculation is reasonable.
  • Understand how close your estimation technique came.
• Example: If you estimate 2,357 + 6,106 as 8,500 and calculate 8,463, the closeness confirms both are correct.

⚠️ One vs two nonzero digits

• One nonzero digit: faster, less accurate (e.g., 5,203 → 5,000).
• Two nonzero digits: slower, more accurate (e.g., 5,203 → 5,200).
• The excerpt shows both approaches are acceptable depending on your need for speed vs. precision.

🧭 Overview

🧠 One-sentence thesis

The clustering technique provides a quicker way to estimate sums of more than two numbers when several of them are close to one particular value, compared to rounding each number individually.

📌 Key points (3–5)

• When to use clustering: when adding more than two numbers and several of them are close to (cluster around) one particular number.
• How clustering works: group numbers that are near the same value, multiply that value by the count of numbers in the group, then add the group totals.
• Common confusion: clustering vs rounding—clustering is faster when numbers naturally group around a few values; rounding treats each number separately.
• Why it matters: clustering provides a quicker estimate than rounding when the right pattern is present.

🎯 When clustering applies

🎯 More than two numbers

• The excerpt states clustering is used "when more than two numbers are to be added."
• Rounding could also be used, but clustering is quicker if the right pattern exists.

🔍 What "cluster" means

Cluster: numbers are seen to be close to one particular number.

• Example: in the sum 32 + 68 + 29 + 73, the numbers 68 and 73 are both close to 70; 32 and 29 are both close to 30.
• Don't confuse: clustering is not about absolute size—it's about proximity to a common value.

🧮 How the clustering technique works

🧮 Step-by-step process

1. Identify clusters: look for numbers that are close to the same value.
2. Count how many numbers are in each cluster.
3. Multiply the cluster center by the count.
4. Add the results from all clusters.

📐 The multiplication shortcut

• If two numbers cluster near 30, their sum is about 2 times 30 equals 60.
• If three numbers cluster near 20, their sum is about 3 times 20 equals 60.
• The excerpt writes this as "2 · 30" or "3 · 20."

🧪 Worked examples from the excerpt

🧪 Example: 27 + 48 + 31 + 52

• 27 and 31 cluster near 30 → sum about 2 · 30 = 60.
• 48 and 52 cluster near 50 → sum about 2 · 50 = 100.
• Estimated total: 60 + 100 = 160.
• Actual sum: 158.

🧪 Example: 17 + 21 + 48 + 18

• 17, 21, and 18 cluster near 20 → sum about 3 · 20 = 60.
• 48 is about 50.
• Estimated total: 60 + 50 = 110.
• Actual sum: 104.

🧪 Example: 61 + 48 + 49 + 57 + 52

• 61 and 57 cluster near 60 → sum about 2 · 60 = 120.
• 48, 49, and 52 cluster near 50 → sum about 3 · 50 = 150.
• Estimated total: 120 + 150 = 270.
• Actual sum: 267.

🧪 Example: 706 + 321 + 293 + 684

• 706 and 684 cluster near 700 → sum about 2 · 700 = 1,400.
• 321 and 293 cluster near 300 → sum about 2 · 300 = 600.
• Estimated total: 1,400 + 600 = 2,000.
• Actual sum: 2,004.

🔑 Key observations

🔑 Results may vary

• The excerpt notes "Results may vary" because different people may choose slightly different cluster centers.
• Example: for 27 and 31, someone might cluster near 30; another person might choose 25 or 28.

🔑 Accuracy of the estimate

• The examples show the estimate is usually close to the actual sum.
• The difference between estimate and actual ranges from 2 (in 27 + 48 + 31 + 52) to 17 (in 61 + 48 + 49 + 57 + 52).
• The goal is a quick estimate, not an exact answer.

🔑 Clustering vs rounding

• Clustering: group similar numbers and multiply; faster when natural groupings exist.
• Rounding: adjust each number individually, then add; more general but can be slower.
• The excerpt says clustering "provides a quicker estimate" when several numbers cluster around one value.

🧭 Overview

🧠 One-sentence thesis

The distributive property allows you to multiply a number by a sum or difference by distributing the multiplication to each term inside the parentheses, which is especially useful for mental arithmetic when one factor ends in 0 or 5.

📌 Key points (3–5)

• What the distributive property is: a characteristic of numbers involving both addition and multiplication, where you distribute a factor to each addend in parentheses.
• Two ways to compute: you can use the order of operations (add first, then multiply) or the distributive property (multiply each term, then add).
• When it works best: the property is most useful for mental arithmetic when one factor ends in 0 or 5.
• Common confusion: the distributive property gives the same answer as the order of operations—it's an alternative method, not a different result.
• How to apply it: break the second number into convenient parts (like 23 = 20 + 3 or 23 = 30 − 7) and distribute the first factor to each part.

🔢 Understanding the distributive property

🔢 What it means to distribute

The distributive property is a characteristic of numbers that involves both addition and multiplication.

• You distribute the factor (the number outside the parentheses) to each addend (the numbers inside the parentheses).
• The factor is multiplied by each term separately, then the results are added or subtracted.
• Example: 3(2 + 5) = 3·2 + 3·5 = 6 + 15 = 21

🔄 Why it works

The excerpt explains the property through repeated addition:

• 3(2 + 5) means "2 + 5" appears 3 times
• So: 2 + 5 + 2 + 5 + 2 + 5
• Rearrange by the commutative property: 2 + 2 + 2 + 5 + 5 + 5
• This is the same as 3·2 + 3·5
• The 3 has been distributed to both the 2 and the 5

➕➖ Works for both sums and differences

The distributive property applies to subtraction as well:

• Example: 6(11 − 3) = 6·11 − 6·3 = 66 − 18 = 48
• The factor is distributed to both terms, keeping the subtraction operation

🆚 Two methods comparison

🆚 Order of operations vs. distributive property

Method	Steps for 3(2 + 5)	Result
Order of operations	Add inside parentheses first: 2 + 5 = 7, then multiply: 3·7	21
Distributive property	Distribute 3 to each term: 3·2 + 3·5 = 6 + 15	21

• Both methods give the same answer
• The distributive property is an alternative approach, not a replacement
• Don't confuse: these are two different paths to the same result

🧮 Mental arithmetic applications

🧮 When the property works best

The excerpt states: "The distributive property works best for products when one of the factors ends in 0 or 5."

• The module restricts attention to only such products
• These numbers make mental calculation easier because multiplying by 10, 20, 30, etc., or by 5 is simpler

🎯 Breaking numbers into convenient parts

You can write a number in different ways to make calculation easier:

• 23 can be written as 20 + 3 (easier to multiply)
• 23 can also be written as 30 − 7 (alternative approach)
• 37 can be written as 30 + 7 or as 40 − 3
• 86 can be written as 80 + 6 or as 90 − 4

Choose the breakdown that makes the arithmetic simplest for you.

💡 Step-by-step examples

Example: 25·23

• Notice that 23 = 20 + 3
• Write: 25·23 = 25(20 + 3) = 25·20 + 25·3
• Calculate: 500 + 75 = 575

Example: 15·37

• Notice that 37 = 30 + 7
• Write: 15·37 = 15(30 + 7) = 15·30 + 15·7
• Calculate: 450 + 105 = 555

Example: 15·86

• Notice that 86 = 80 + 6
• Write: 15·86 = 15(80 + 6) = 15·80 + 15·6
• Calculate: 1,200 + 90 = 1,290

✅ Practice patterns

✅ Common calculation patterns

The exercises show typical patterns for mental arithmetic:

First factor	Second factor breakdown	Calculation pattern
25	12 = 10 + 2	25·10 + 25·2 = 250 + 50 = 300
35	14 = 10 + 4	35·10 + 35·4 = 350 + 140 = 490
80	58 = 50 + 8	80·50 + 80·8 = 4,000 + 640 = 4,640
65	62 = 60 + 2	65·60 + 65·2 = 3,900 + 130 = 4,030

✅ Flexibility in approach

The excerpt emphasizes you can choose different breakdowns:

• 26 can be 20 + 6 or 30 − 4
• Example: 65·26 = 65(20 + 6) = 1,300 + 390 = 1,690
• Or: 65·26 = 65(30 − 4) = 1,950 − 260 = 1,690
• Both give the same answer; use whichever is easier for you

🧭 Overview

🧠 One-sentence thesis

Rounding fractions to common benchmarks like 0, 1/4, 1/2, 3/4, and 1 allows you to quickly estimate sums and differences without exact computation, though results may vary slightly from the actual answer.

📌 Key points (3–5)

• What the technique does: rounds fractions to familiar values (0, 1/4, 1/2, 3/4, 1) to estimate sums or differences quickly.
• How to apply it: identify which benchmark each fraction is closest to, then add or subtract those benchmarks instead of the original fractions.
• Why estimates vary: different people may round the same fraction differently depending on judgment, so results are approximate.
• Common confusion: the estimate is not the exact answer—it's meant to give a quick sense of the result; the actual value will differ slightly.
• Works with mixed numbers: add whole-number parts separately, then estimate the fractional parts and combine.

🎯 The rounding technique

🎯 What fractions get rounded to

Estimation by rounding fractions commonly rounds fractions to 1/4, 1/2, 3/4, 0, and 1.

• These five benchmarks are easy to work with mentally.
• You choose the benchmark closest to the original fraction.
• Example: 3/5 is close to 1/2; 5/12 is also close to 1/2.

🔍 Why results may vary

• The excerpt repeatedly states "results may vary."
• Different people may judge a fraction differently (e.g., is 3/8 closer to 1/4 or 1/2?).
• The goal is a quick approximation, not precision.
• Don't confuse: variation in estimates is expected and acceptable; it doesn't mean the method is wrong.

🧮 How to estimate sums

🧮 Simple fraction addition

The excerpt shows: estimate 3/5 + 5/12.

• Round 3/5 to 1/2.
• Round 5/12 to 1/2.
• Estimate: 1/2 + 1/2 = 1.
• Actual value: 61/60, which is "a little more than 1."

Key insight: the estimate gets you close; the exact answer confirms how close.

🧮 Mixed-number addition

The excerpt shows: estimate 5 3/8 + 4 9/10 + 11 1/5.

• Step 1: add the whole-number parts: 5 + 4 + 11 = 20.
• Step 2: round the fractional parts:
  • 3/8 is close to 1/4
  • 9/10 is close to 1
  • 1/5 is close to 1/4
• Step 3: add the rounded fractions: 1/4 + 1 + 1/4 = 1 1/2.
• Step 4: combine: 20 + 1 1/2 = 21 1/2.
• Actual value: 21 19/40, "a little less than 21 1/2."

Process summary:

1. Separate whole numbers from fractions.
2. Add whole numbers directly.
3. Round and add fractions.
4. Combine the two results.

📊 Comparing estimates to exact values

📊 Why the excerpt includes exact answers

• Every practice problem shows both the estimate and the exact answer in parentheses.
• This helps you see how close your estimate is.
• Example: estimate 5/8 + 5/12 as 1/2 + 1/2 = 1; exact answer is 25/24 = 1 1/24.

📊 Patterns in accuracy

Estimate	Exact	Observation
1	1 17/24	Estimate is close but under
1 1/2	1 17/45	Estimate is close but over
12	11 29/30	Estimate is close but over
21	20 37/40	Estimate is close but over

• The estimate is usually within one unit of the exact answer.
• Sometimes the estimate is higher, sometimes lower—this is normal.

⚠️ Important reminders

⚠️ When to use this technique

• When you need a quick sense of the answer.
• When exact computation would be time-consuming.
• When checking if a detailed calculation is reasonable.

⚠️ When not to rely on it

• When precision is required (e.g., measurements, financial calculations).
• The excerpt always says "after you have made an estimate, find the exact value"—estimation is a first step, not a replacement for exact work.

⚠️ Don't confuse estimation with guessing

• Estimation is systematic: you follow a method (round to benchmarks, then compute).
• Guessing has no method.
• Example: rounding 7/9 to 1 and 3/5 to 1/2 gives estimate 1 1/2; the exact answer 1 17/45 confirms the estimate was reasonable.

🧭 Overview

🧠 One-sentence thesis

This chapter aims to equip students with the ability to measure quantities using the United States and metric systems, convert between units, perform calculations with measurements, and apply geometric formulas for perimeter, area, and volume.

📌 Key points (3–5)

• Two measurement systems: United States system and metric system are both covered.
• Core skill—unit conversion: converting from one unit to another within each system.
• Geometric applications: calculating perimeter, circumference, area, and volume of common shapes.
• Denominate number operations: adding, subtracting, multiplying, and dividing measurements with units.
• Common confusion: understanding that measurement is comparison to a standard, not just a number.

📏 Understanding measurement

📏 What measurement means

Measurement is comparison to some standard.

• Measurement is not an absolute quantity; it is always a comparison between two quantities.
• One quantity is the thing being measured; the other is the standard unit of measure.
• Example: saying something is "12 inches" means it matches 12 copies of the standard inch unit placed end-to-end.

🎯 Standard unit of measure

The quantity that is used for comparison is called the standard unit of measure.

• Historically, standards varied (e.g., a king's thumb length for an inch, or 16 barley grains end-to-end).
• Today, standards are maintained by the Bureau of Standards in Washington D.C. and rarely change.
• Three desirable properties of a standard:
  • Accessibility: people can access the standard to make comparisons.
  • Invariance: the standard does not change over time.
  • Reproducibility: the standard can be reproduced so measurements are convenient and widely available.

🇺🇸 United States system of measurement

🇺🇸 Common units and conversions

The excerpt provides a conversion table for the United States system:

Category	Conversions
Length	1 foot = 12 inches; 1 yard = 3 feet; 1 mile = 5,280 feet
Weight	1 pound = 16 ounces; 1 ton = 2,000 pounds
Liquid Volume	1 tablespoon = 3 teaspoons; 1 fluid ounce = 2 tablespoons; 1 cup = 8 fluid ounces; 1 pint = 2 cups; 1 quart = 2 pints; 1 gallon = 4 quarts
Time	1 minute = 60 seconds; 1 hour = 60 minutes; 1 day = 24 hours; 1 week = 7 days

• Students should memorize these relationships to perform conversions.
• Example: to convert feet to inches, multiply by 12 (since 1 foot = 12 inches).

🔄 Unit fractions for conversion

A unit fraction is a fraction with a value of 1.

• Unit fractions are formed using two equal measurements in different units.
• They allow conversion from one unit to another without changing the actual quantity.
• Example: since 1 foot = 12 inches, the fractions (1 foot / 12 inches) and (12 inches / 1 foot) both equal 1 and can be used to convert between feet and inches.

🌍 Metric system and other topics

🌍 Metric system advantages

• The chapter mentions the metric system is covered.
• One advantage highlighted: the metric system is based on the base ten number system, making conversions simpler.
• Students will learn metric prefixes and how to convert between metric units.

📐 Geometric measurements

The chapter covers:

• Perimeter and circumference:

  • Perimeter: the distance around a polygon.
  • Circumference: the distance around a circle; related to diameter and radius.
  • The symbol π (pi) and its approximating value are introduced.
  • Four versions of the circumference formula are taught.

• Area and volume:

  • Area: the amount of surface covered by a figure (with notation and formulas for common shapes).
  • Volume: the amount of space occupied by an object (with notation and formulas for common objects).

➕ Operations with denominate numbers

• Denominate numbers: measurements that include units (e.g., 5 feet, 3 pounds).
• Students learn to:
  • Simplify unsimplified units (e.g., 15 inches → 1 foot 3 inches).
  • Add and subtract denominate numbers.
  • Multiply and divide a denominate number by a whole number.
• Don't confuse: operations with denominate numbers require careful attention to units, unlike pure arithmetic with abstract numbers.

🧭 Overview

🧠 One-sentence thesis

The United States system of measurement relies on standardized units for length, weight, volume, and time, and conversions between units are performed using unit fractions that preserve value while changing the unit label.

📌 Key points (3–5)

• What measurement means: comparison to a standard unit that has accessibility, invariance, and reproducibility.
• How the US system is organized: common units include feet/inches/miles (length), pounds/ounces (weight), cups/pints/quarts (liquid volume), and hours/minutes (time), with fixed conversion ratios between them.
• How to convert units: multiply by a unit fraction (value = 1) constructed from equal measurements, placing the target unit in the numerator and the original unit in the denominator.
• Common confusion: which unit fraction to use—always put the unit you are converting to in the numerator, not the unit you are converting from.
• Why it matters: conversions allow measurements to be expressed in the most convenient or required unit for a given context.

📏 What measurement is

📏 Definition and purpose

Measurement is comparison to some standard.

• Measurement is not an absolute quantity; it is the result of comparing two quantities.
• One quantity is the thing being measured; the other is the standard unit of measure, which serves as the reference.
• Example: saying "11 yards" means the length is 11 times the standard length called "1 yard."

🔑 Properties of a good standard

The excerpt lists three desirable properties for a standard unit:

Property	Meaning
Accessibility	People should have access to the standard so they can make comparisons.
Invariance	The standard should not change over time.
Reproducibility	The standard should be reproducible so measurements are convenient and accessible to many people.

• Historical note: In the past, standards were inconsistent (e.g., "1 inch = distance from thumb tip to knuckle of the king" or "16 barley grains end to end").
• Today, the Bureau of Standards in Washington D.C. maintains standard units, which rarely change.

🇺🇸 The United States system

🇺🇸 Common units and conversion ratios

The excerpt provides a table of common US units organized by category:

Length

• 1 foot (ft) = 12 inches (in.)
• 1 yard (yd) = 3 feet
• 1 mile (mi) = 5,280 feet

Weight

• 1 pound (lb) = 16 ounces (oz)
• 1 ton (T) = 2,000 pounds

Liquid Volume

• 1 tablespoon (tbsp) = 3 teaspoons (tsp)
• 1 fluid ounce (fl oz) = 2 tablespoons
• 1 cup (c) = 8 fluid ounces
• 1 pint (pt) = 2 cups
• 1 quart (qt) = 2 pints
• 1 gallon (gal) = 4 quarts

Time

• 1 minute (min) = 60 seconds (sec)
• 1 hour (hr) = 60 minutes
• 1 day (da) = 24 hours
• 1 week (wk) = 7 days

🔍 Why conversions are needed

• It is often convenient or necessary to convert from one unit to another (e.g., feet to inches, hours to weeks).
• The excerpt emphasizes that conversions are made using unit fractions.

🔄 Converting units with unit fractions

🔄 What a unit fraction is

A unit fraction is a fraction with a value of 1.

• Unit fractions are formed by using two equal measurements: one in the numerator, one in the denominator.
• Because the numerator and denominator are equal, the fraction equals 1, so multiplying by it does not change the value—only the unit label.
• Example: Since 1 ft = 12 in., both 1 ft / 12 in. and 12 in. / 1 ft are unit fractions.

🎯 Placement rule: which unit goes where

Key rule from the excerpt:

• Place the unit being converted to in the numerator.
• Place the unit being converted from in the denominator.

Why this works:

• When you multiply, the original unit in the denominator cancels with the original unit in the measurement, leaving only the target unit.
• Example: Converting 11 yards to feet:
  • We want feet, so put feet in the numerator: 3 ft / 1 yd.
  • Multiply: 11 yd × (3 ft / 1 yd) = 33 ft (the "yd" cancels).

Don't confuse: Placing the target unit in the denominator will give you the reciprocal of the correct answer.

🧮 Step-by-step conversion examples

🧮 Example 1: Convert 11 yards to feet

1. Look up the conversion: 1 yd = 3 ft.
2. Construct the unit fraction with feet (target) in the numerator: 3 ft / 1 yd.
3. Multiply: 11 yd × (3 ft / 1 yd) = 33 ft.
4. Result: 11 yd = 33 ft.

🧮 Example 2: Convert 36 fluid ounces to pints

1. Look up the conversion: 1 pt = 16 fl oz.
2. Construct the unit fraction with pints (target) in the numerator: 1 pt / 16 fl oz.
3. Multiply: 36 fl oz × (1 pt / 16 fl oz) = 36/16 pt = 2.25 pt.
4. Result: 36 fl oz = 2.25 pt.

🧮 Example 3: Convert 2,016 hours to weeks (multi-step)

1. First convert hours to days: 1 da = 24 hr → unit fraction 1 da / 24 hr.
2. Then convert days to weeks: 1 wk = 7 da → unit fraction 1 wk / 7 da.
3. Multiply: 2,016 hr × (1 da / 24 hr) × (1 wk / 7 da) = 2,016 / (24 × 7) wk = 12 wk.
4. Result: 2,016 hr = 12 wk.

Key insight: For multi-step conversions, chain unit fractions so that intermediate units cancel, leaving only the final target unit.

🔢 Handling fractions and decimals

• If the result is a fraction, the excerpt instructs to convert it to a decimal rounded to two decimal places.
• Example: 9/4 pt = 2.25 pt.
• Example: 62 in. to miles results in a very small number, rounded to 0.00 miles (to two decimal places).

🧪 Practice problems summary

The excerpt includes many practice problems and solutions. Here are a few representative conversions:

Problem	Solution	Notes
18 ft to yards	6 yd	18 ÷ 3 = 6
2 mi to feet	10,560 ft	2 × 5,280 = 10,560
26 ft to yards	8.67 yd	26 ÷ 3 ≈ 8.67
9 qt to pints	18 pt	9 × 2 = 18
52 min to hours	0.87 hr	52 ÷ 60 ≈ 0.87
412 hr to weeks	2.45 wk	412 ÷ 24 ÷ 7 ≈ 2.45
5 pt to teaspoons	480 tsp	Multi-step: pt → c → fl oz → tbsp → tsp

Common pattern: Larger unit to smaller unit → multiply (or divide by a fraction less than 1); smaller unit to larger unit → divide (or multiply by a fraction less than 1).

🧭 Overview

🧠 One-sentence thesis

The metric system allows simple unit conversions by moving the decimal point because it is built on the base ten number system, making it more convenient than the United States system.

📌 Key points (3–5)

• Core advantage: conversions in the metric system require only multiplying or dividing by powers of 10, which means moving the decimal point left or right.
• Three basic units: meter (length), liter (volume), and gram (mass), each with the same set of prefixes.
• Prefix meanings: kilo = thousand, hecto = hundred, deka = ten, deci = tenth, centi = hundredth, milli = thousandth.
• Conversion direction rule: larger to smaller unit → move decimal right; smaller to larger unit → move decimal left.
• Common confusion: mass vs weight—mass measures resistance to motion and stays the same everywhere; weight depends on gravity and changes (e.g., Earth vs Moon).

🔟 Why the metric system is easier

🔟 Base ten advantage

The metric system of measurement takes advantage of our base ten number system.

• The United States system requires memorizing arbitrary conversion factors (e.g., 12 inches = 1 foot, 3 feet = 1 yard).
• The metric system uses only powers of 10, so conversions are done by shifting the decimal point.
• Example: to convert 3 kilometers to meters, multiply by 1,000 (or move the decimal 3 places right) → 3,000 meters.

🔄 Simple conversion mechanism

• Multiply or divide by a power of 10 = move the decimal point.
• No need for complex fractions or multiple-step conversions.
• This makes calculations faster and less error-prone.

🏷️ Prefixes and basic units

🏷️ The three basic units

Category	Basic Unit	What it measures
Length	meter (m)	distance
Volume	liter (L)	liquid capacity
Mass	gram (g)	amount of matter

📏 Metric prefixes

The excerpt lists six prefixes that attach to any basic unit:

Prefix	Meaning	Multiplier
kilo	thousand	1,000
hecto	hundred	100
deka	ten	10
deci	tenth	0.1
centi	hundredth	0.01
milli	thousandth	0.001

• Example for length: 1 kilometer = 1,000 meters; 1 centimeter = one hundredth of a meter; 1 millimeter = one thousandth of a meter.
• The same prefixes apply to volume and mass: 1 kilogram = 1,000 grams; 1 milliliter = one thousandth of a liter.

🔁 Three key characteristics

The excerpt highlights three patterns in the metric system:

1. Same prefixes in every category: whether measuring length, volume, or mass, the prefixes (kilo, hecto, deka, deci, centi, milli) mean the same thing.
2. Larger to smaller → decimal moves right: converting from a bigger unit (e.g., meters) to a smaller unit (e.g., centimeters) means moving the decimal point to the right.
3. Smaller to larger → decimal moves left: converting from a smaller unit to a bigger unit means moving the decimal point to the left.

🔢 How to convert between metric units

🔢 The conversion scale

The excerpt provides a visual scale:

kilo – hecto – deka – unit – deci – centi – milli

• Each step represents one place value (one power of 10).
• Count the number of steps and the direction between the original unit and the target unit.

➡️ Conversion rules by direction

From	To	Decimal moves	Places
unit	deka	left	1
unit	hecto	left	2
unit	kilo	left	3
unit	deci	right	1
unit	centi	right	2
unit	milli	right	3

• Example: unit to kilo = 1 to 1,000 → move decimal 3 places left.
• Example: unit to milli = 1 to 0.001 → move decimal 3 places right.

🧮 Two conversion methods

The excerpt demonstrates both approaches:

Method A: Move the decimal point

1. Locate the original unit on the scale.
2. Count steps to the target unit.
3. Move the decimal the same number of places in the same direction.

Example: Convert 3 kilograms to grams.

• Kilo to gram (the base unit) = 3 steps to the right.
• 3.0 kg → move decimal 3 places right → 3,000 g.

Method B: Use unit fractions

• Multiply by a fraction that equals 1 (e.g., 1,000 g / 1 kg).
• Choose the fraction so the target unit is in the numerator.

Example: 3 kg × (1,000 g / 1 kg) = 3,000 g.

📐 Worked examples from the excerpt

Original	Target	Steps	Result
3 kg	grams	3 right	3,000 g
67.2 hL	milliliters	5 right (hecto→unit=2, unit→milli=3)	6,720,000 mL
100.07 cm	meters	2 left	1.0007 m
0.16 mg	grams	3 left	0.00016 g

• Don't confuse: the number of steps depends on the distance between prefixes, not just "one step per prefix."
• Example: hectoliter to milliliter crosses hecto→deka→unit→deci→centi→milli = 5 steps.

⚖️ Mass vs weight distinction

⚖️ What mass measures

Mass is a measure of a body's resistance to motion.

• Mass does not depend on gravity.
• The more massive a body, the harder it is to move.
• Your mass is the same on Earth and on the Moon.

🌍 What weight measures

The weight of a body is related to gravity.

• Weight does depend on gravity.
• Your weight on Earth is different from your weight on the Moon (lower gravity → lower weight).
• More massive bodies weigh more than less massive bodies in the same gravitational field.

🚫 Don't confuse

• The excerpt emphasizes: mass ≠ weight.
• In everyday language people often say "weight" when they mean "mass," but in the metric system the gram measures mass, not weight.

📊 Unit conversion table summary

The excerpt provides a full table for length, mass, and volume. Key entries:

📊 Length (meter-based)

• 1 kilometer (km) = 1,000 meters
• 1 hectometer (hm) = 100 meters
• 1 dekameter (dam) = 10 meters
• 1 meter (m) = 1 meter
• 1 decimeter (dm) = 0.1 meter
• 1 centimeter (cm) = 0.01 meter
• 1 millimeter (mm) = 0.001 meter

📊 Mass (gram-based)

• 1 kilogram (kg) = 1,000 grams
• 1 hectogram (hg) = 100 grams
• 1 dekagram (dag) = 10 grams
• 1 gram (g) = 1 gram
• 1 decigram (dg) = 0.1 gram
• 1 centigram (cg) = 0.01 gram
• 1 milligram (mg) = 0.001 gram

📊 Volume (liter-based)

• 1 kiloliter (kL) = 1,000 liters
• 1 hectoliter (hL) = 100 liters
• 1 dekaliter (daL) = 10 liters
• 1 liter (L) = 1 liter
• 1 deciliter (dL) = 0.1 liter
• 1 centiliter (cL) = 0.01 liter
• 1 milliliter (mL) = 0.001 liter

⏱️ Time

• The excerpt notes: "Same as the United States system."
• The metric system does not change time units (seconds, minutes, hours, days, weeks).

🧭 Overview

🧠 One-sentence thesis

Denominate numbers (measurements with units) can be simplified by converting excess amounts to the next higher unit, and can be added, subtracted, multiplied, or divided using systematic procedures that preserve unit relationships.

📌 Key points (3–5)

• What denominate numbers are: numbers with units of measure attached (e.g., 55 min, 4 gal 3 qt).
• Simplification rule: a denominate number is simplified when the amount in any unit does not exceed the next higher unit type (e.g., 65 min becomes 1 hr 5 min).
• Operations preserve units: addition/subtraction align like units vertically; multiplication/division apply to the number part while carrying the unit along.
• Common confusion: borrowing in subtraction—when a smaller unit is insufficient, borrow from the next higher unit (e.g., borrow 1 lb = 16 oz).
• Why it matters: simplification makes measurements easier to read and compare; operations enable practical calculations with real-world measurements.

📏 Understanding denominate numbers and simplification

📏 What are denominate numbers?

Denominate numbers: numbers that have units of measure associated with them.

• These are everyday measurements: 19 inches, 4 gallons 5 quarts, 2 hours 75 minutes.
• The "number part" (19, 4, 75) is paired with a "unit part" (inches, gallons, minutes).

✅ What "simplified" means

Simplified denominate number: when the number of standard units does not exceed the next higher type of unit.

• Simplified example: 55 min is simplified because it is smaller than 1 hr (the next higher unit).
• Not simplified example: 65 min is not simplified because it exceeds 60 min (which equals 1 hr).
• The fix: 65 min = 1 hr 5 min (now simplified).
• Don't confuse: "simplified" does not mean "smallest possible number"—it means properly distributed across unit levels.

🔧 How to simplify

The process involves recognizing conversion relationships and redistributing amounts:

1. Identify the conversion factor (e.g., 12 in. = 1 ft, 60 min = 1 hr).
2. Break the excess into the next higher unit plus remainder.
3. Combine and express in multiple units.

Example: Simplify 19 in.

• Since 12 in. = 1 ft and 19 = 12 + 7
• 19 in. = 12 in. + 7 in. = 1 ft + 7 in. = 1 ft 7 in.

Example: Simplify 2 hr 75 min

• Since 60 min = 1 hr and 75 = 60 + 15
• 2 hr 75 min = 2 hr + 60 min + 15 min = 2 hr + 1 hr + 15 min = 3 hr 15 min

Example: Simplify 43 fl oz

• Since 8 fl oz = 1 c (cup), 43 ÷ 8 = 5 remainder 3
• 43 fl oz = 5 c + 3 fl oz
• Further: since 2 c = 1 pt, 5 c = 2 pt + 1 c
• And since 2 pt = 1 qt, final result: 1 qt 1 c 3 fl oz

➕➖ Adding and subtracting denominate numbers

➕ Addition procedure

Three-step method:

1. Write numbers vertically so like units appear in the same column.
2. Add the number parts, carrying the unit along.
3. Simplify the sum.

Example: Add 6 ft 8 in. to 2 ft 9 in.

  6 ft  8 in.
+ 2 ft  9 in.
-----------
  8 ft 17 in.

Simplify: 17 in. = 12 in. + 5 in. = 1 ft + 5 in., so final answer: 9 ft 5 in.

➖ Subtraction procedure

Follow the same alignment, but borrowing is often necessary.

Example: Subtract 5 da 3 hr from 8 da 11 hr

  8 da 11 hr
- 5 da  3 hr
-----------
  3 da  8 hr

Example with borrowing: Subtract 3 lb 14 oz from 5 lb 3 oz

• Cannot subtract 14 oz from 3 oz directly.
• Borrow 1 lb = 16 oz from the 5 lb.
• 5 lb 3 oz = 4 lb + 16 oz + 3 oz = 4 lb 19 oz

  4 lb 19 oz
- 3 lb 14 oz
-----------
  1 lb  5 oz

🔄 Multiple borrowing

When subtracting across multiple units, borrow step-by-step from higher to lower units.

Example: Subtract 4 da 9 hr 21 min from 7 da 0 hr 10 min

• First borrow 1 da = 24 hr: 7 da 0 hr → 6 da 24 hr
• Then borrow 1 hr = 60 min: 6 da 24 hr 10 min → 6 da 23 hr 70 min
• Now subtract: 6 da 23 hr 70 min − 4 da 9 hr 21 min = 2 da 14 hr 49 min

✖️ Multiplying a denominate number by a whole number

✖️ The rule

To multiply a denominate number by a whole number: multiply the number part of each unit by the whole number and affix the unit to this product.

• This follows from the distributive property: 3 × (4 ft 9 in.) = 3 × 4 ft + 3 × 9 in. = 12 ft + 27 in.
• After multiplying, simplify the result.

🧮 Examples

Example: 6 × (2 ft 4 in.)

• 6 × 2 ft = 12 ft
• 6 × 4 in. = 24 in.
• Result: 12 ft 24 in.
• Simplify: 24 in. = 2 ft, so 12 ft + 2 ft = 14 ft (which equals 4 yd 2 ft)

Example: 8 × (5 hr 21 min 55 sec)

• 8 × 5 hr = 40 hr
• 8 × 21 min = 168 min
• 8 × 55 sec = 440 sec
• Simplify seconds: 440 sec = 7 min 20 sec
• Simplify minutes: 168 min + 7 min = 175 min = 2 hr 55 min
• Simplify hours: 40 hr + 2 hr = 42 hr = 1 da 18 hr
• Final: 1 da 18 hr 55 min 20 sec

➗ Dividing a denominate number by a whole number

➗ The rule

To divide a denominate number by a whole number: divide the number part of each unit by the whole number beginning with the largest unit. Affix the unit to this quotient. Carry any remainder to the next unit.

• Work from largest to smallest unit.
• Convert remainders to the next smaller unit before continuing.

🧮 Examples

Example: (12 min 40 sec) ÷ 4

• 12 min ÷ 4 = 3 min (no remainder)
• 40 sec ÷ 4 = 10 sec
• Result: 3 min 10 sec

Example: (5 yd 2 ft 9 in.) ÷ 3

• 5 yd ÷ 3 = 1 yd remainder 2 yd
• Convert remainder: 2 yd = 6 ft, add to 2 ft = 8 ft
• 8 ft ÷ 3 = 2 ft remainder 2 ft
• Convert remainder: 2 ft = 24 in., add to 9 in. = 33 in.
• 33 in. ÷ 3 = 11 in.
• Result: 1 yd 2 ft 11 in.

⚠️ Conversion during division

The key step: when there's a remainder, convert it to the next smaller unit before dividing that unit's amount.

• Don't confuse: you cannot divide mixed units directly—you must work unit-by-unit, carrying remainders downward through the unit hierarchy.

📊 Summary table of operations

Operation	Key principle	Watch out for
Simplification	Excess in lower unit → convert to higher unit	Know conversion factors (12 in. = 1 ft, etc.)
Addition	Align like units, add, then simplify	Result may need simplification
Subtraction	Align like units, borrow when needed	Borrow from next higher unit (1 lb = 16 oz, etc.)
Multiplication	Distribute whole number to each unit part	Always simplify the product
Division	Divide largest unit first, carry remainders down	Convert remainders before dividing next unit