Finding Equivalent Ratio

Two ratios are equivalent if they represent the same relationship between two quantities, just expressed using different numbers. Recognizing and generating equivalent ratios is a core skill for solving proportion problems on the GED Math test.

What Are Equivalent Ratios?

Just as equivalent fractions name the same value (e.g., \(\color{blue}{\frac{1}{2} = \frac{2}{4} = \frac{3}{6}}\)), equivalent ratios represent the same comparison. The ratio \(\color{blue}{2 : 3}\) is equivalent to \(\color{blue}{4 : 6}\), \(\color{blue}{6 : 9}\), \(\color{blue}{10 : 15}\), and so on, because each pair has the same simplified form.

GED Math for Beginners 2026 The Ultimate Step by Step Guide to Preparing for the GED Math Test

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

Rated 4.34 out of 5 based on 106 customer ratings

Satisfied 92 Students

Two Methods for Finding Equivalent Ratios

Method 1: Multiply both terms by the same number

To scale a ratio up, multiply both numbers by any whole number (the scale factor).

Example: Find three ratios equivalent to \(\color{blue}{3 : 5}\).

• × 2: \(\color{blue}{6 : 10}\)
• × 3: \(\color{blue}{9 : 15}\)
• × 4: \(\color{blue}{12 : 20}\)

Method 2: Divide both terms by the same number (simplify)

To scale a ratio down, divide both numbers by a common factor.

Example: Simplify \(\color{blue}{18 : 24}\) to find an equivalent ratio in lower terms.

\(\color{blue}{\text{ GCF }(18, 24) = 6}\). Divide both: \(\color{blue}{18 \div 6 = 3}\), \(\color{blue}{24 \div 6 = 4}\). Equivalent ratio: \(\color{blue}{3 : 4}\).

Checking equivalence

Two ratios \(\color{blue}{a : b}\) and \(\color{blue}{c : d}\) are equivalent if and only if \(\color{blue}{a \times d = b \times c}\) (cross-multiply and check).

Example: Are \(\color{blue}{4 : 6}\) and \(\color{blue}{6 : 9}\) equivalent?

\(\color{blue}{4 \times 9 = 36}\) and \(\color{blue}{6 \times 6 = 36}\). Yes, they are equivalent.

Step-by-Step Summary

1. To find an equivalent ratio, multiply (or divide) both terms by the same nonzero number.
2. To check equivalence, cross-multiply: \(\color{blue}{a \times d = b \times c}\).
3. The simplest form of a ratio is found by dividing both terms by their GCF.

Watch: Understanding Equivalent Ratios (Video Lesson)

Khan Academy builds intuition for why some ratios are equivalent and others are not:

Worked Examples

Example 1: Find two equivalent ratios for \(\color{blue}{5 : 8}\).

× 2: \(\color{blue}{10 : 16}\)\(\color{blue}{. \times 3}\): \(\color{blue}{15 : 24}\).

Example 2: Are \(\color{blue}{3 : 7}\) and \(\color{blue}{12 : 28}\) equivalent?

Cross-multiply: \(\color{blue}{3 \times 28 = 84}\) and \(\color{blue}{7 \times 12 = 84}\). Equal, so yes, they are equivalent.

Example 3: Find the missing value: \(\color{blue}{4 : 7 = ? : 35}\)

Scale factor from 7 to 35: \(\color{blue}{35 \div 7 = 5}\). Multiply numerator: \(\color{blue}{4 \times 5 = 20}\). Answer: \(\color{blue}{20 : 35}\).

Example 4: Find the missing value: \(\color{blue}{? : 6 = 15 : 18}\)

Cross-multiply: \(\color{blue}{? \times 18 = 6 \times 15 = 90}\). Divide: \(\color{blue}{90 \div 18 = 5}\). Answer: \(\color{blue}{5 : 6}\).

More Practice: Ratio Problems with Tables

This Khan Academy video solves equivalent ratio problems using a table approach:

Exercises

1. Write two equivalent ratios for \(\color{blue}{2 : 9}\).
2. Are \(\color{blue}{6 : 10}\) and \(\color{blue}{9 : 15}\) equivalent? Show your work.
3. Find the missing value: \(\color{blue}{3 : 5 = 12 : ?}\)
4. Find the missing value: \(\color{blue}{? : 8 = 6 : 16}\)
5. Simplify \(\color{blue}{20 : 36}\) to an equivalent ratio in simplest form.
6. Are \(\color{blue}{5 : 9}\) and \(\color{blue}{20 : 36}\) equivalent?

Answers

1. \(\color{blue}{4 : 18}\) and \(\color{blue}{6 : 27}\) (multiply by 2 and 3)
2. \(\color{blue}{6 \times 15 = 90}\) and \(\color{blue}{10 \times 9 = 90}\). Equal, yes, equivalent.
3. Scale factor: \(\color{blue}{12 \div 3 = 4}\); \(\color{blue}{5 \times 4 = 20}\). Answer: \(\color{blue}{20}\).
4. Cross-multiply: \(\color{blue}{? \times 16 = 8 \times 6 = 48}\); \(\color{blue}{48 \div 16 = 3}\). Answer: \(\color{blue}{3}\).
5. \(\color{blue}{\text{ GCF }(20, 36) = 4}\); \(\color{blue}{5 : 9}\).
6. \(\color{blue}{5 \times 36 = 180}\) and \(\color{blue}{9 \times 20 = 180}\). Equal, yes, equivalent.

Frequently Asked Questions

What is an equivalent ratio?

Equivalent ratios compare quantities in the same way even though they use different numbers. The ratio 2 : 3 is equivalent to 4 : 6, 6 : 9 and 10 : 15, because each one simplifies to 2 : 3. It works just like equivalent fractions: 1/2, 2/4 and 3/6 all name the same amount.

How do you find equivalent ratios?

Multiply or divide both terms of the ratio by the same nonzero number. To scale 3 : 5 up, multiply by 2, 3 or 4 to get 6 : 10, 9 : 15 and 12 : 20. To scale down, divide both terms by a common factor. Changing only one term breaks the ratio, so always do the same thing to both numbers.

How do you find an equivalent ratio in simplest form?

Divide both terms by their greatest common factor (GCF). For 18 : 24, the GCF is 6, so 18 ÷ 6 = 3 and 24 ÷ 6 = 4, giving 3 : 4. The ratio is in simplest form when the two numbers share no common factor other than 1.

How do you find equivalent ratios with a missing number?

Find the scale factor from the pair of terms you know, then use it on the other term. If 4 : 7 = 12 : ?, notice that 4 × 3 = 12, so multiply 7 by 3 to get 21. You can also cross-multiply: 4 × ? = 7 × 12 = 84, and 84 ÷ 4 = 21.

How do you find equivalent ratios in a table?

In a ratio table, every row (or column) is an equivalent ratio, so you build it by multiplying or dividing both values by the same number. If a recipe uses 2 cups of flour for 3 eggs, the table can continue 4 : 6, 6 : 9 and 8 : 12. To check a filled table, divide the two numbers in each pair; the result should be the same every time.

How can you tell if two ratios are equivalent?

Simplify both ratios or cross-multiply. For 6 : 8 and 15 : 20, both simplify to 3 : 4, so they are equivalent. Cross-multiplying gives 6 × 20 = 120 and 8 × 15 = 120, which also confirms it. If the cross products are different, the ratios are not equivalent.

Related Topics

• Write a Ratio
• Ratio Tables
• Word Problems Involving Equivalent Ratio
• Word Problems Involving Comparing Ratio
• Using a Fraction to Write Down a Ratio