What fractions mean, how to simplify them, and how to add, subtract, multiply, and divide them — the foundation for every topic that follows
Chapter 2 gave you tools for understanding the structure of whole numbers. You learned to factor, to find common ground between numbers, and to identify the building blocks hidden inside larger values. Chapter 3 uses that foundation to enter one of the most important and most misunderstood areas of arithmetic: fractions.
Fractions appear everywhere. They describe portions of a recipe, doses of medication, lengths in carpentry, rates in science, probabilities in statistics, and slopes in algebra. Despite their ubiquity, fractions are the single topic where student confidence most commonly breaks down. The reason is almost always the same: students were taught procedures without understanding, and when the procedures blur together in memory, there is nothing left to fall back on.
This chapter will not ask you to simply memorize rules. It will build each idea from the ground up, show you why the rule works, and then give you practice applying it. By the end of this chapter, you will understand what a fraction is, how to represent fractions in multiple ways, how to simplify them, and how to perform all four operations with them. Every idea here will carry directly into Chapter 4 on mixed numbers, Chapter 5 on decimals, Chapter 6 on ratios and percents, and the algebra chapters that follow.
Take your time in this chapter. Fractions reward careful thinking.
A recipe calls for 3/4 of a cup of flour. You want to make half of the recipe. How much flour do you need? While you are working in the kitchen, you also realize you have 1/3 of a cup of sugar remaining but need 5/6 of a cup. How much more sugar do you need to buy?
These two questions require multiplying a fraction by a whole number and subtracting fractions with different denominators. By the end of this chapter, both will be straightforward.
Fractions are not a detour from "real math." They are the foundation of nearly everything that follows in this textbook and in later mathematics. Ratios are comparisons written as fractions. Percent means "per hundred," which is a fraction. Slope in algebra is a fraction. Probability is a fraction. Division of algebraic expressions uses the same rules as division of numerical fractions. A student who is fluent and confident with fractions enters Algebra I prepared. A student who is not will struggle at every turn, not because algebra is harder, but because fractions keep appearing inside every algebraic procedure.
A fraction is a way of expressing a quantity that is part of a whole, a comparison between two quantities, or the result of a division. The word fraction comes from a Latin word meaning "to break." A fraction describes something that has been broken into equal parts.
When we write 3/4, we are communicating something precise: a whole has been divided into 4 equal parts, and we are considering 3 of them. But fractions are richer than just "parts of a pizza." They have at least five interpretations that you will use throughout this textbook.
This is the most familiar interpretation. Imagine a rectangle divided into 4 equal sections. If 3 sections are shaded, the shaded region represents 3/4 of the whole rectangle. The key word here is equal. If the 4 sections are not equal in size, the fraction 3/4 does not describe the shaded area accurately. Equal parts are the foundation of every fraction.
Draw a rectangle divided into 4 equal vertical sections. Shade 3 of them. Label the shaded region "3/4." The unshaded section represents the remaining 1/4.
A fraction can describe a portion of a group of objects, not just a portion of a single shape. If a bag contains 8 marbles and 3 of them are red, the fraction of marbles that are red is 3/8. Here, the "whole" is the set of 8 marbles, and 3 is the count of the particular type we are interested in.
The fraction 3/4 means exactly 3 ÷ 4. The fraction bar is a division symbol. A fraction and a division expression are two ways of writing the same thing: 3/4 = 3 ÷ 4 = 0.75. This interpretation will be developed fully in Section 3.3.
A fraction is a number, and like every number, it has a specific location on the number line. The number 3/4 sits three-fourths of the way between 0 and 1. This will be explored fully in Section 3.4.
A fraction can express a comparison between two quantities. If a class has 12 boys and 16 girls, the ratio of boys to total students is 12/28, which simplifies to 3/7. Ratios are studied in depth in Chapter 6, but recognizing that fractions and ratios share the same notation prepares you for that work.
Students who think fractions only mean "part of a pizza" become confused when fractions appear in division, on number lines, or in algebraic expressions. Understanding all five interpretations gives you flexibility. When one interpretation does not fit a context, another one will.
Denominator — the bottom number. It names the type of piece by indicating how many equal parts the whole has been divided into.
Numerator — the top number. It counts how many of those parts are being considered.
In the fraction 5/8, the denominator is 8 (the whole has been divided into 8 equal parts) and the numerator is 5 (we are counting 5 of those parts). Although the numerator is written on top, the denominator should be understood first. The denominator establishes what kind of pieces you are working with. Once you know the pieces are eighths, the numerator tells you how many eighths you have. Think of it this way: you would not say "I have 5 ... of something" before knowing what the something is. The denominator names the unit; the numerator counts it.
A unit fraction is a fraction with a numerator of 1, such as 1/2, 1/3, 1/8, and 1/100. Every fraction can be understood as a multiple of the corresponding unit fraction: 3/4 = 3 × (1/4). This means "three copies of one-fourth." This interpretation is especially helpful when multiplying fractions by whole numbers.
| Type | Condition | Examples | Value |
|---|---|---|---|
| Proper fraction | Numerator < Denominator | 1/2, 3/5, 7/10 | Between 0 and 1 |
| Improper fraction | Numerator ≥ Denominator | 5/3, 8/8, 11/4 | Greater than or equal to 1 |
| Unit fraction | Numerator = 1 | 1/2, 1/7, 1/100 | Between 0 and 1 |
The word "improper" does not mean wrong or forbidden. Improper fractions are perfectly valid and often more useful than mixed numbers when performing operations. The fraction 11/4 is as legitimate as the fraction 3/4. Chapter 4 will examine the relationship between improper fractions and mixed numbers in full detail.
Any nonzero number divided by itself equals 1: 4/4 = 1, 9/9 = 1, 100/100 = 1. This fact is central to creating equivalent fractions in Section 3.5.
The expression a/b means a ÷ b. This is not a metaphor or approximation. It is a definition. A fraction is a division expression written in a convenient form.
6/2 = 6 ÷ 2 = 3 3/5 = 3 ÷ 5 = 0.6 7/4 = 7 ÷ 4 = 1.75
The division interpretation explains several things that otherwise seem arbitrary. It explains why any number over itself equals 1: n/n = n ÷ n = 1. It explains why any number over 1 equals itself: n/1 = n ÷ 1 = n. It explains how to convert fractions to decimals (divide numerator by denominator), and it explains why fractions appear in algebraic formulas.
The division interpretation is the most natural one for sharing situations. If you have 3 granola bars to share equally among 4 people, each person receives 3 ÷ 4 = 3/4 of a granola bar. Dividing 3 things equally into 4 groups gives each group three one-fourth pieces.
Draw 3 rectangles side by side, representing 3 granola bars. Divide each rectangle into 4 equal sections. Shade one section from each bar and group them together. Each person's share consists of three sections each of size 1/4, which totals 3/4.
a. 5 ÷ 8 = 5/8 (does not simplify to a whole number)
b. 12 ÷ 4 = 12/4 = 3 (simplifies to a whole number)
c. 1 ÷ 3 = 1/3 (a unit fraction)
Students sometimes write 5 ÷ 8 as 8/5, reversing the numerator and denominator. Remember: the dividend (the number being divided) goes on top. 5 ÷ 8 = 5/8, not 8/5. Always place the number being divided in the numerator.
Every fraction has a unique location on the number line. Proper fractions live between 0 and 1. Improper fractions live at 1 or beyond. To place 3/4 on the number line: locate 0 and 1, divide the segment into 4 equal parts (denominator is 4), count 3 parts from 0 (numerator is 3), and mark the point.
Draw a number line from 0 to 1. Mark four equally spaced tick marks at 1/4, 2/4, 3/4, and 4/4. Label each mark. Place a bold filled circle at 3/4.
An improper fraction like 7/4 belongs between 1 and 2 on the number line. Since 4/4 = 1 and 8/4 = 2, divide the segment from 1 to 2 into 4 equal parts and count 3 parts past 1, because 7/4 = 4/4 + 3/4 = 1 + 3/4.
Draw a number line from 0 to 2. Mark eighths from 0 to 2. Label 4/4 = 1 and place a bold filled circle at 7/4, which falls three-fourths of the way between 1 and 2.
First, it confirms that fractions are real numbers with specific positions. Second, it makes comparison immediate: the number further right is larger, so 3/4 > 1/2. Third, it connects fractions to decimals: 1/2 and 0.5 mark the exact same point on the number line. They are different names for the same location.
Number line: 0 — 1/5 — 2/5 — 3/5 — 4/5 — 1
● ●
2/5 3/5
Two fractions are equivalent if they represent the same quantity, even though they are written with different numerators and denominators. The fractions 1/2, 2/4, 3/6, 4/8, and 50/100 all mark the same point on the number line: the midpoint between 0 and 1.
Draw a stack of three identical rectangles. Divide the first into 2 equal parts and shade 1. Divide the second into 4 equal parts and shade 2. Divide the third into 6 equal parts and shade 3. The shaded region is the same size in all three rectangles, showing that 1/2 = 2/4 = 3/6.
To make a larger equivalent fraction: multiply both numerator and denominator by the same nonzero whole number.
1/2 = (1×3)/(2×3) = 3/6
To make a smaller equivalent fraction: divide both numerator and denominator by the same nonzero whole number.
6/10 = (6÷2)/(10÷2) = 3/5
Both operations multiply the fraction by a form of 1 (such as 3/3 or 2/2), which leaves the value unchanged.
To check whether 3/4 and 9/12 are equivalent, multiply across: 3 × 12 = 36 and 4 × 9 = 36. The cross products are equal, so the fractions are equivalent.
2/5 = (2×2)/(5×2) = 4/10
2/5 = (2×3)/(5×3) = 6/15
2/5 = (2×10)/(5×10) = 20/50
3/8 = (3×5)/(8×5) = 15/40Reducing a fraction means writing it as an equivalent fraction with the smallest possible numerator and denominator. A fraction in this form is in lowest terms or simplest form. A fraction is in lowest terms when the GCF of its numerator and denominator is 1.
The most efficient method: divide both numerator and denominator by their Greatest Common Factor in one step. If you do not immediately see the GCF, divide by any common factor and repeat until no common factors remain.
48 = 2⁴ × 3
72 = 2³ × 3²
GCF(48, 72) = 2³ × 3 = 24
48/72 = (48÷24)/(72÷24) = 2/3
105 = 3 × 5 × 7
140 = 2² × 5 × 7
GCF(105, 140) = 5 × 7 = 35
105/140 = (105÷35)/(140÷35) = 3/4
A fraction is not fully reduced just because you divided by some common factor. You must keep dividing until the only common factor remaining is 1. After reducing, verify your result by checking whether the GCF of the new numerator and denominator is 1.
Reduce each fraction to lowest terms. Show the GCF you used.
Cancellation is a shortcut for reducing fractions during multiplication. Instead of multiplying first and then reducing the result, you reduce before multiplying by dividing any numerator and any denominator by a common factor they share. The same answer is reached with smaller intermediate numbers.
Cancellation works because the commutative and associative properties of multiplication allow you to rearrange factors in the numerator and denominator, and dividing paired factors by their GCF is equivalent to multiplying by a form of 1.
You may cancel a numerator with a denominator when they share a common factor.
You may cancel across different fractions in a multiplication problem.
You may not cancel a numerator with another numerator, or a denominator with another denominator.
You may not cancel in addition or subtraction — only in multiplication.
(5̶/1̶2̶) × (8̶/1̶5̶) → (1/3) × (2/3) = 2/9
÷5 ÷4 ÷4 ÷5
Cancellation only works in multiplication. You cannot cancel in 2/3 + 1/2 by removing the 2's. In addition and subtraction, every number stays where it is until you have a common denominator. Incorrect cancellation in addition is one of the most frequent and damaging errors in fraction work.
Use cancellation to simplify before multiplying. Show the factors you cancel.
One of the most important rules in all of mathematics is this: you cannot divide by zero. The denominator of any fraction can never be zero.
To understand why, recall what division means. If 6/3 = 2, it is because 3 × 2 = 6. Division and multiplication are inverse operations. Every division problem corresponds to a multiplication problem. Now ask: what would 6/0 mean? It would require finding a number k such that 0 × k = 6. But 0 × k = 0 for every possible value of k. No such k exists. The expression 6/0 has no answer. We say it is undefined.
The case 0/0 is different but equally problematic. Here, 0 × k = 0 is true for every value of k, so any number would satisfy the condition. Because the result is not unique, 0/0 is called indeterminate. Both cases are unusable for our purposes.
| Expression | Value | Reason |
|---|---|---|
0/5 | 0 | Zero divided by any nonzero number is zero. |
5/1 | 5 | Any number divided by 1 is itself. |
n/n (n≠0) | 1 | Any nonzero number divided by itself is 1. |
5/0 | Undefined | No number times 0 gives 5. |
0/0 | Indeterminate | Every number times 0 gives 0; the answer is not unique. |
In algebra, you will often work with fractions that have variables in the denominator. Whenever you see a variable in a denominator, one of your first responsibilities is to note that the variable cannot equal zero. This habit begins here.
You can only add fractions directly when they have the same denominator. This is not an arbitrary rule. It reflects what fractions mean. Adding 2/7 + 3/7 means: 2 sevenths plus 3 sevenths equals 5 sevenths. You add the numerators and keep the denominator. This is exactly like adding units in any context: 2 apples + 3 apples = 5 apples. What you cannot do is add 2/7 + 3/5 directly, any more than you can add 2 sevenths and 3 fifths without first converting them to a common unit.
Add the numerators. Keep the denominator. Reduce the result if possible.
a/c + b/c = (a+b)/c
1. Find the Least Common Denominator (LCD) = LCM of the denominators.
2. Rewrite each fraction as an equivalent fraction with the LCD.
3. Add the numerators. Keep the LCD as denominator.
4. Reduce the result to lowest terms.
3/10 + 4/10 = (3+4)/10 = 7/101/4 = (1×3)/(4×3) = 3/12
2/6 = (2×2)/(6×2) = 4/12
3/12 + 4/12 = 7/121/3 = 4/12, 1/4 = 3/12, 1/6 = 2/124/12 + 3/12 + 2/12 = 9/12GCF(9, 12) = 3 → 9/12 = 3/4Students sometimes add both numerators and denominators: 1/3 + 1/4 = 2/7. This is incorrect. The denominator must become a common denominator, not a sum. The correct answer is 7/12 ≈ 0.583, which is larger than either addend, as a sum should be. The result 2/7 ≈ 0.286 is smaller than 1/3, which is impossible for a sum of two positive fractions.
Subtraction of fractions follows the same logic as addition. You need a common denominator. Once the fractions share a denominator, subtract the numerators and keep the denominator.
Find a common denominator, convert, subtract numerators, keep the denominator, reduce.
a/c - b/c = (a-b)/c
7/9 - 2/9 = (7-2)/9 = 5/95/6 = 20/24, 3/8 = 9/2420/24 - 9/24 = 11/247/10 = 21/30, 2/15 = 4/3021/30 - 4/30 = 17/30After finding a common denominator, double-check your equivalent fractions before subtracting. A small error in converting leads to a wrong answer even if the subtraction itself is correct. Write out the multiplication clearly: 7/10 = (7×3)/(10×3) = 21/30.
Multiply numerators together. Multiply denominators together. Reduce the result.
(a/b) × (c/d) = (a×c)/(b×d)
No common denominator is needed. Apply cancellation before multiplying to keep numbers small.
Multiplication with fractions means "of." The expression (2/3) × (3/4) means "two-thirds of three-fourths." The result will be less than three-fourths, because you are taking only part of it. This surprises many students: multiplying two fractions less than 1 gives a product smaller than either factor. One-half of one-half is one-fourth. Take half of something, then half of that. You end up with a quarter of the original.
Draw a square. Shade the left half to represent 1/2. Now shade the lower portion of the shaded half to represent 1/2 of that half. The doubly-shaded region is one-fourth of the whole square, confirming (1/2) × (1/2) = 1/4.
3 and 9: divide both by 3 → 1 and 3
8 and 4: divide both by 4 → 2 and 1
(1/1) × (2/3) = 2/3
(3/5) × (30/1)
Cancel: 30 and 5 share factor 5 → 6 and 1
= (3/1) × (6/1) = 18
Division asks: how many times does one quantity fit into another? How many 2's fit in 8? The answer is 4. How many one-fourths fit in three-fourths? The answer is 3. The procedure for dividing fractions is called Keep-Change-Flip (KCF): keep the first fraction, change the division to multiplication, and flip the second fraction (write its reciprocal).
(a/b) ÷ (c/d) = (a/b) × (d/c)
Keep the first fraction. Change ÷ to ×. Flip the second fraction.
This works because multiplying by d/c is the inverse of multiplying by c/d. A number and its reciprocal always multiply to 1.
(3/4) × (5/2) = 15/8(4/1) × (3/1) = 12(5/6) × (1/3) = 5/18Students sometimes flip the first fraction instead of the divisor. Always flip only the second fraction, the one you are dividing by. In (3/5) ÷ (2/7), flip 2/7 to get 7/2, not the other way around.
The reciprocal of a fraction is the fraction with its numerator and denominator swapped. The reciprocal of a/b is b/a.
Defining property: a fraction and its reciprocal always multiply to 1.
(3/4) × (4/3) = 12/12 = 1
| Number | Written As | Reciprocal | Verify |
|---|---|---|---|
| 3/4 | 3/4 | 4/3 | (3/4) × (4/3) = 1 |
| 5 (whole number) | 5/1 | 1/5 | 5 × (1/5) = 1 |
| 1/7 (unit fraction) | 1/7 | 7/1 = 7 | (1/7) × 7 = 1 |
| 0 | 0/1 | 1/0 = undefined | Zero has no reciprocal |
Every division problem can be rewritten as multiplication by the reciprocal. This is the deeper reason why Keep-Change-Flip works: a ÷ b = a × (1/b), and (a/b) ÷ (c/d) = (a/b) × (d/c). The reciprocal "undoes" the original fraction.
Reciprocal of 7/9 = 9/7
Reciprocal of 6 = 1/6 (since 6 = 6/1)
Reciprocal of 1/4 = 4 (since 1/4 → 4/1 = 4)
Reciprocal of 5/1 = 1/5
A fraction can be positive, negative, or zero, just like any other number. The sign of a fraction follows the same rules for dividing signed numbers that you learned in Chapter 1.
| Division | Sign | Example |
|---|---|---|
| Positive ÷ Positive | Positive | 3/4 > 0 |
| Negative ÷ Negative | Positive | (−3)/(−4) = 3/4 > 0 |
| Negative ÷ Positive | Negative | (−3)/4 < 0 |
| Positive ÷ Negative | Negative | 3/(−4) < 0 |
For any fraction, the negative sign can appear in three equivalent positions:
−(3/4) = (−3)/4 = 3/(−4)
All three represent negative three-fourths. The standard convention is to write the negative sign in front of the entire fraction: −(3/4).
Placing the negative sign in the denominator is unusual in practice and can cause confusion. Prefer −(3/4) or (−3)/4 over 3/(−4).
1/4 = 2/8−5/8 + 2/8 = (−5+2)/8 = −3/8−(4/5) × (3/2) = −(12/10) = −6/5
Reviewing common errors is as important as practicing procedures. Recognizing an error pattern is the first step to correcting it permanently.
The denominator is not a quantity being added. Find the LCD, convert both fractions, then add numerators only. Correct answer: 4/12 + 3/12 = 7/12.
Cancellation only applies in multiplication. In addition and subtraction, every number stays until you have a common denominator. Changing 2/3 to 1/3 changes its value entirely.
GCF(6, 8) = 2. The correct final answer is 3/4. A fraction is not fully simplified until the GCF of numerator and denominator equals 1.
Flip only the divisor, the second fraction. The first fraction stays unchanged. Correct: (3/5) × (7/2) = 21/10.
When numerators are equal, the larger denominator means smaller pieces, giving a smaller fraction. 3/5 > 3/8 because fifths are larger pieces than eighths. Draw both on a number line to confirm.
Draw two equal rectangles side by side. Divide the first into 5 equal parts and shade 3. Divide the second into 8 equal parts and shade 3. The shaded region in the first rectangle is visibly larger, confirming 3/5 > 3/8.
Using the product of denominators works but gives larger numbers. Using the LCD (LCM of the denominators) keeps numbers smaller and reduces the amount of simplification at the end.
Improper fractions are perfectly valid. In multiplication and division, it is often easier to work with improper fractions than mixed numbers. Converting prematurely can create confusion.
−3/4 and 3/4 are different numbers separated by a distance of 3/2 on the number line. Track your signs carefully through every step. A dropped negative sign changes the answer entirely.
Each problem contains an error. Identify what went wrong and write the correct answer.
Answers may be placed in the answer key.
Reduce to lowest terms. Show the GCF used.
Note: Apply multiplication before addition, as covered in Chapter 11. Answers may be placed in the answer key.
Solve each problem using the Read → Represent → Solve → Check → Explain method.
Answers may be placed in the answer key.
Each item shows a student's completed work. Find the error, explain what went wrong, and give the correct solution.
What went wrong? What is the correct answer?
Is the student finished? What should the next step be? What is the correct fully reduced answer?
Identify the error. What is the correct answer?
The student has partially correct reasoning but reached a wrong answer. What step contains the error?
What conceptual error was made? Which fraction is actually larger, and why?
Find the sign error. What is the correct answer?
3/7 with a denominator of 35.60/84 to lowest terms using the GCF method.45/75 to lowest terms.5/9 + 2/37/10 − 3/8(5/6) × (9/20)(7/12) ÷ (14/9)−3/4 + 1/2(−5/6) × (−3/10)11/7, 8, and 1/5.9/0 has no value.Answers may be placed in the answer key.
Chapter 3 gave you the full toolkit for working with fractions: what they mean, how to simplify them, and how to add, subtract, multiply, and divide them. These skills do not pause for the rest of this textbook. They are in constant use from this point forward.
Chapter 4 extends fractions into mixed numbers, which combine a whole number and a fraction into a single value. You will convert between improper fractions and mixed numbers, and you will perform all four operations with mixed numbers using exactly the fraction skills you practiced here.
Chapter 5 will show you that decimals are fractions in disguise. Every terminating decimal is a fraction with a power of ten in the denominator. Moving fluently between fractions and decimals depends on the equivalent fraction work from this chapter.
Chapter 6 on ratios and percents will treat percent as a fraction with denominator 100, and proportions as pairs of equivalent fractions set equal to each other.
In the algebra chapters, you will encounter rational expressions — fractions with variables in the numerator or denominator. Adding, subtracting, multiplying, and dividing rational expressions follows exactly the same logic you applied to numerical fractions here. Every step you practiced has an algebraic counterpart waiting ahead. Take the time now to be fluent.