Find equivalent fractions. - SSAT Middle Level Quantitative

SECTION 1

Where Did Fractions Come From?

People have used fractions (numbers that represent parts of a whole) for thousands of years. Ancient farmers needed to split land evenly. Merchants needed to divide goods into fair portions. Over time, mathematicians realized that the same amount could be written in many different ways — and the idea of equivalent fractions (different fractions that name the same value) was born.

~1800 BCE • Egyptian Fractions

Ancient Egyptians wrote fractions using unit fractions (fractions with 1 on top), like ½ and ¼. They had to combine several unit fractions to express other amounts.

~500 BCE • Greek Ratios

Greek mathematicians like Euclid studied ratios and discovered that the same ratio could be expressed in many ways. For example, 2 : 4 is the same as 1 : 2.

~600 CE • Indian Notation

Mathematicians in India began writing fractions as one number over another, much like we do today. They developed rules for simplifying and comparing them.

~1200 CE • Fibonacci Spreads the Idea

The Italian mathematician Fibonacci introduced the fraction bar (the line between the numerator and denominator) to Europe, making it easier to work with equivalent fractions.

So here is the big question: if ½ and 2/4 look different, how do we know they represent the same amount? And how can we create or identify equivalent fractions whenever we need them? That is exactly what this lesson will teach you.

SECTION 2

Core Principles of Equivalent Fractions

Before we start finding equivalent fractions, let's make sure a few key ideas are solid. These four principles are the building blocks for everything else in this lesson.

What a Fraction Means

A fraction like ¾ means you divide something into 4 equal parts (the denominator) and take 3 of them (the numerator).

The Golden Rule

If you multiply (or divide) the numerator and the denominator by the same nonzero number, the value of the fraction does not change.

Simplest Form

A fraction is in simplest form (also called lowest terms) when the numerator and denominator share no common factor other than 1.

Cross-Multiplication Check

To test whether two fractions are equivalent, cross-multiply. If the two products are equal, the fractions are equivalent.

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KEY TAKEAWAY

Think of equivalent fractions like slicing a pizza. If you cut a pizza into 4 slices and eat 2, you ate the same amount as if you cut it into 8 slices and ate 4. The pieces are smaller, but you have more of them — the total amount of pizza is identical. That is what equivalent fractions are: different-looking fractions that represent the exact same quantity.

SECTION 3

Seeing Equivalent Fractions

The best way to understand equivalent fractions is to see them. The diagram below shows three rectangles that are exactly the same size. Each one is shaded to represent the same amount — but the number of pieces is different.

Each rectangle is the same total size. The shaded portion in every case covers exactly half the rectangle — whether it is 1 out of 2 pieces, 2 out of 4 pieces, or 4 out of 8 pieces. The bottom row shows how multiplying the numerator and denominator by 2 each time produces the next equivalent fraction.

Look at the top row. The shaded region in every rectangle is the same width. That is the visual proof that 1/2, 2/4, and 4/8 are all equivalent. The bottom row shows the rule in action: multiply both the top and bottom of 1/2 by 2 to get 2/4, and repeat to get 4/8.

SECTION 4

The Math Behind Equivalent Fractions

There are two main operations you can use: multiplying to build up to a larger equivalent fraction, and dividing to simplify down to a smaller one. There is also a handy test — cross-multiplication — to check whether two fractions are equivalent.

BUILDING UP (MULTIPLYING)

a / b = (a × n) / (b × n)

Here a is the numerator, b is the denominator, and n is any nonzero whole number you choose. You are really multiplying by n/n, which equals 1 — so the value does not change.

SIMPLIFYING (DIVIDING)

a / b = (a ÷ n) / (b ÷ n)

This time n must be a common factor (a number that divides evenly into both a and b). When you divide top and bottom by their greatest common factor (GCF), you reach simplest form in one step.

CROSS-MULTIPLICATION TEST

a / b = c / d ⟺ a × d = b × c

To check whether a/b and c/d are equivalent, multiply across the diagonals. If a × d equals b × c, the fractions are equivalent.

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Why Does This Work?

When you multiply by n/n, you are really multiplying by 1 (because any number divided by itself is 1). Multiplying a number by 1 never changes its value. That is the mathematical reason the Golden Rule works!

SECTION 5

Three Methods Side by Side

You now know the formulas, but how do you actually decide which method to use? The diagram below gives you a quick decision path, and the table after it compares all three methods.

This flowchart shows how to choose a method based on your goal. If you need a larger denominator, multiply. If you need to reduce, divide by the GCF. If you need to verify two fractions are equal, cross-multiply.

Method	When to Use It	Example
Multiply top & bottom	You need a fraction with a specific larger denominator (e.g., for adding fractions).	3/5 = (3 × 4)/(5 × 4) = 12/20
Divide by GCF	You want the simplest form of a fraction.	12/20 → GCF is 4 → (12 ÷ 4)/(20 ÷ 4) = 3/5
Cross-multiply	You want to check whether two fractions are equivalent without simplifying.	3/5 and 12/20 → 3 × 20 = 60, 5 × 12 = 60 → equal ✓

Comparison of three methods for working with equivalent fractions

SECTION 6

Worked Example: Step by Step

Let's work through a complete problem together. Imagine you see this question on the SSAT:

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Problem

Which fraction is equivalent to 6/9?

Finding an Equivalent Fraction for 6/9

1
Step 1 — Identify the Numerator and Denominator

The fraction is 6/9. The numerator (top number) is 6 and the denominator (bottom number) is 9.
2
Step 2 — Find the GCF of 6 and 9

List the factors of each number. Factors of 6: 1, 2, 3, 6. Factors of 9: 1, 3, 9. The largest number that appears in both lists is 3. So the GCF is 3.

GCF = 3
3
Step 3 — Divide Numerator and Denominator by the GCF

Divide the top by 3: 6 ÷ 3 = 2. Divide the bottom by 3: 9 ÷ 3 = 3. The simplified fraction is 2/3.

6/9 = 2/3
4
Step 4 — Verify with Cross-Multiplication

Check: does 6 × 3 equal 9 × 2? Compute: 6 × 3 = 18 and 9 × 2 = 18. The products match, so 6/9 and 2/3 are equivalent. ✓

6 × 3 = 18 = 9 × 2 ✓ Confirmed equivalent

On the SSAT, if the answer choices were (A) 2/3, (B) 3/4, (C) 4/6, (D) 1/3, (E) 5/9, you would pick (A) 2/3. Notice that (C) 4/6 is also equivalent to 2/3, so on a real test you would double-check which answer appears. Always simplify first and then match.

SECTION 7

Common Mistakes and How to Avoid Them

Even if you understand the rules, small slip-ups can cost you points. Here are the most common mistakes students make with equivalent fractions — and how to steer clear of them.

Mistake	Why It Happens	How to Fix It
Adding the same number to top and bottom	Students think 1/2 = 2/3 by adding 1 to both. Adding does NOT keep the value the same.	Always multiply or divide — never add or subtract the same number.
Multiplying only the numerator	A student writes 2/5 = 4/5 by doubling only the top. The denominator must also be multiplied.	Whatever you do to the top, you must also do to the bottom.
Dividing by a non-factor	Trying to simplify 5/8 by dividing by 2, which does not divide evenly into 5.	Before dividing, check that the number divides evenly into BOTH the numerator and the denominator.
Not simplifying fully	Writing 4/8 = 2/4 but stopping there instead of continuing to 1/2.	Use the GCF to reach simplest form in one step. Check that the numerator and denominator share no common factor other than 1.

Common mistakes and fixes for equivalent fractions

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REMEMBER THIS

Think of equivalent fractions like resizing a photo. When you resize a photo correctly, you change the width and the height by the same factor, so the picture still looks right. If you only stretched the width, the image would look squished and wrong. Fractions work the same way — always apply the same operation to both the numerator and the denominator.

SECTION 8

Connecting to Bigger Ideas

Equivalent fractions are not just a standalone skill. They are the foundation for many topics you will see later in math. The table below shows how this concept grows into more advanced ideas.

What You Learn Now	Where It Leads
Multiplying top and bottom by the same number	Finding a common denominator to add and subtract fractions
Simplifying fractions using the GCF	Reducing algebraic fractions in pre-algebra and algebra
Cross-multiplication to check equivalence	Solving proportions (e.g., "If 3 apples cost $2, how much do 12 apples cost?")
Understanding that many fractions can represent the same value	Ratios, rates, percents, and probability

How equivalent fractions connect to future math topics

On the SSAT, equivalent fractions show up in many forms. You might be asked to simplify, to find a missing numerator or denominator, or to compare two fractions. Mastering this skill now will make those problems feel automatic.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Which of the following fractions is equivalent to 1/3? (A) 2/5 (B) 3/9 (C) 4/6 (D) 2/4 (E) 5/12

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The correct answer is (B) 3/9. To check, simplify 3/9 by dividing both the numerator and denominator by their GCF of 3: 3 ÷ 3 = 1 and 9 ÷ 3 = 3, giving 1/3. You can also cross-multiply: 1 × 9 = 9 and 3 × 3 = 9, which are equal. (A) 2/5 is not equivalent because 1 × 5 ≠ 3 × 2. (C) 4/6 simplifies to 2/3, not 1/3. (D) 2/4 simplifies to 1/2. (E) 5/12 cannot be simplified to 1/3 because 5 × 3 ≠ 12 × 1.

Question 2: BASIC CALCULATION

What value of n makes the following equation true? 4/7 = n/21 (A) 8 (B) 10 (C) 12 (D) 14 (E) 16

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The correct answer is (C) 12. Notice that the denominator changed from 7 to 21, which means it was multiplied by 3 (because 7 × 3 = 21). By the Golden Rule, you must also multiply the numerator by 3: 4 × 3 = 12. You can verify with cross-multiplication: 4 × 21 = 84 and 7 × 12 = 84. They match, confirming n = 12. (A) 8 would mean multiplying by 2, but 7 × 2 = 14, not 21. (B) 10 gives 10/21, which does not simplify to 4/7. (D) 14 gives 14/21 = 2/3. (E) 16 gives 16/21, which also does not simplify to 4/7.

Question 3: INTERMEDIATE

Simplify the fraction 18/24 to its simplest form. (A) 2/3 (B) 9/12 (C) 6/8 (D) 3/4 (E) 4/6

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The correct answer is (D) 3/4. First, find the GCF of 18 and 24. The factors of 18 are 1, 2, 3, 6, 9, 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The greatest common factor is 6. Divide both: 18 ÷ 6 = 3 and 24 ÷ 6 = 4, giving 3/4. Check with cross-multiplication: 18 × 4 = 72 and 24 × 3 = 72. ✓ (A) 2/3 ≠ 18/24 because 2 × 24 = 48 but 3 × 18 = 54. (B) 9/12 is an equivalent fraction but is not in simplest form. (C) 6/8 simplifies to 3/4 but is not in simplest form either. (E) 4/6 simplifies to 2/3, which is a different value.

Question 4: APPLIED

Maria ate 6 out of 15 candies in a bag. Jake ate 4 out of 10 candies in a different bag. Did they eat the same fraction of their bag? (A) Yes, because both fractions simplify to 1/3 (B) Yes, because both fractions simplify to 2/5 (C) No, Maria ate 2/5 and Jake ate 1/3 (D) No, Maria ate 1/3 and Jake ate 2/5 (E) Yes, because 6 − 4 = 2 and 15 − 10 = 5

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The correct answer is (B) Yes, because both fractions simplify to 2/5. Simplify Maria's fraction: 6/15. The GCF of 6 and 15 is 3. So 6 ÷ 3 = 2 and 15 ÷ 3 = 5, giving 2/5. Simplify Jake's fraction: 4/10. The GCF of 4 and 10 is 2. So 4 ÷ 2 = 2 and 10 ÷ 2 = 5, giving 2/5. Both simplify to 2/5, so they ate the same fraction. (A) is wrong because neither simplifies to 1/3. (C) and (D) assign the wrong simplified fractions. (E) gives a coincidentally correct-looking answer but uses subtraction, which is not a valid method for comparing fractions.

Question 5: CRITICAL THINKING

A student claims that 5/8 and 15/24 are equivalent fractions, and that 5/8 and 20/36 are also equivalent. Which statement is correct? (A) Both claims are true (B) Only the first claim is true (C) Only the second claim is true (D) Both claims are false (E) You cannot tell without a calculator

Show answer

The correct answer is (B) Only the first claim is true. Check the first claim using cross-multiplication: 5 × 24 = 120 and 8 × 15 = 120. They match, so 5/8 = 15/24. ✓ Now check the second claim: 5 × 36 = 180 and 8 × 20 = 160. Since 180 ≠ 160, these fractions are NOT equivalent. ✗ You can also simplify 20/36: the GCF of 20 and 36 is 4, so 20 ÷ 4 = 5 and 36 ÷ 4 = 9, giving 5/9, which is not the same as 5/8. (A) is wrong because the second claim fails. (C) is the reverse of the correct answer. (D) is wrong because the first claim is valid. (E) is wrong because cross-multiplication can be done by hand.

SUMMARY

Lesson Summary

Summary

Equivalent fractions are different fractions that represent the same value. You create them by using the Golden Rule: multiply or divide both the numerator and the denominator by the same nonzero number. To build up a fraction, multiply; to simplify, divide by the greatest common factor (GCF).

To check whether two fractions are equivalent, use cross-multiplication: if a × d = b × c, then a/b = c/d. Remember: never add or subtract the same number to top and bottom — that changes the value. This skill is essential on the SSAT and builds the foundation for adding fractions, solving proportions, and working with ratios and percents.