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Equivalent
Fractions

Master how to find fractions that look different but represent the same value: a key skill for the SSAT Quantitative section.

SSAT Middle Level

What Is a Fraction?

A fraction represents a part of a whole. It is written as one number over another, separated by a line.

\(\frac{\color{#818cf8}3}{\color{#fbbf24}4}\) Numerator parts you have Denominator total equal parts

What Are Equivalent Fractions?

Equivalent fractions are fractions that name the same amount, even though they use different numerators and denominators.

\(\frac{1}{2}\) \(\frac{2}{4}\) \(\frac{4}{8}\) = Same value!

Why Equivalent Fractions Matter

Comparing
You need a common denominator to compare or order fractions.
Adding & Subtracting
You can only add fractions once they share the same denominator.
Simplifying
Reducing a fraction to lowest terms means finding the simplest equivalent fraction.

The Golden Rule

Multiply or divide both the numerator and denominator by the same nonzero number, and the fraction's value stays the same.

\(a\) \(b\) = \(a \times n\) \(b \times n\) \(n \neq 0\) \(\times n\) \(\times n\)

Visual Proof of the Rule

\(\frac{1}{3}\) × 2 \(\frac{2}{6}\)

Multiply Up vs. Divide Down

Multiply Up
Multiply both parts by the same number to get an equivalent fraction with a larger denominator.

\(\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}\)
Divide Down (Simplify)
Divide both parts by a common factor to get an equivalent fraction with a smaller denominator.

\(\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}\)

The Cross-Multiplication Test

To check if two fractions are equivalent, cross-multiply. If the cross-products are equal, the fractions are equivalent.

\(a\) \(b\) \(c\) \(d\) \(a \times d\) \(b \times c\) If \(a \times d\) = \(b \times c\) they're equivalent!

How to Find Equivalent Fractions

1
Identify your goal. Do you need a specific denominator, or are you simplifying to lowest terms?
2
Choose your multiplier or divisor. If you need denominator 20 from 5: \(20 \div 5 = 4\), so multiply by 4.
3
Apply to both numerator and denominator. Multiply (or divide) top and bottom by the same number.
4
Verify. Cross-multiply to confirm, or simplify back to check you get the original.

Example 1: Multiply Up

Find a fraction equivalent to \(\frac{3}{7}\) with denominator 28.

Step: \(28 \div 7 = 4\), so multiply both by 4 3 7 \(\times 4\) \(\times 4\) = 12 28

Example 2: Simplify Down

Simplify \(\frac{18}{24}\) to its lowest terms.

\(\text{GCF}(18, 24) = 6\), so divide both by 6 18 24 \(\div 6\) \(\div 6\) = 3 4

Example 3: Find the Missing Number

Solve: \(\frac{5}{8} = \frac{?}{40}\)

1
Find the multiplier: \(40 \div 8 = 5\)
2
Apply to the numerator: \(5 \times 5 = 25\)
3
Result: \(\frac{5}{8} = \frac{25}{40}\)

Example 4: Two-Step Simplify

Simplify \(\frac{36}{60}\) to lowest terms.

1
Find a common factor: Both 36 and 60 are divisible by 6.
2
Divide: \(\frac{36 \div 6}{60 \div 6} = \frac{6}{10}\)
3
Still not lowest terms! 6 and 10 are both divisible by 2.
4
Divide again: \(\frac{6 \div 2}{10 \div 2} = \frac{3}{5}\). Done!

Common Mistakes to Avoid

✕ Adding Instead of Multiplying
Adding the same number to top and bottom changes the value. \(\frac{1}{2} \neq \frac{1+3}{2+3} = \frac{4}{5}\). You must multiply, not add.
✕ Only Changing One Part
Multiplying just the numerator (or just the denominator) changes the fraction's value. Always do both.
✕ Not Fully Simplifying
Stopping too early, like writing \(\frac{6}{10}\) instead of \(\frac{3}{5}\). Always check if numerator and denominator share a common factor.

Summary Cheat Sheet

Multiply Up Divide Down Cross-Multiply Check
Purpose Create equivalent with larger denominator Simplify to smaller denominator Verify two fractions are equivalent
Operation \(\times n\) to both \(\div n\) to both Compare \(a \times d\) vs. \(b \times c\)
Example \(\frac{2}{3} = \frac{8}{12}\) \(\frac{8}{12} = \frac{2}{3}\) \(2 \times 12 = 3 \times 8\) ✓
When to Use Finding common denominators Simplifying answers Checking your work

Question 1 of 5

Which fraction is equivalent to \(\frac{3}{5}\)?

Question 2 of 5

Simplify \(\frac{14}{21}\) to lowest terms.

Question 3 of 5

Solve for the missing value: \(\frac{4}{9} = \frac{?}{36}\)

Question 4 of 5

Which pair of fractions is NOT equivalent?

Question 5 of 5

A student says \(\frac{3}{4} = \frac{6}{7}\) because they added 3 to both the numerator and denominator. What is wrong?