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.

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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.

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Frequently Asked Questions

How is finding equivalent ratios like finding equivalent fractions?

They use the exact same process. A ratio \(\color{blue}{a : b}\) is equivalent to \(\color{blue}{c : d}\) when \(\color{blue}{\frac{a}{b} = \frac{c}{d}}\). Multiplying or dividing both terms by the same number keeps the value of the fraction — and the ratio — unchanged.

Can you add or subtract to find an equivalent ratio?

No. You must multiply or divide both terms by the same number. Adding or subtracting would change the relationship between the quantities and produce a different ratio.

What does it mean if a ratio has no equivalent with smaller whole numbers?

It means the ratio is already in its simplest (lowest) form — the GCF of the two terms is 1. For example, \(\color{blue}{5 : 7}\) is already in simplest form.

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