Word Problems Involving Equivalent Ratio

Equivalent ratios appear everywhere in daily life — from scaling recipes to reading maps to mixing paint. When a word problem asks you to find a missing quantity using a known ratio, you are working with equivalent ratios. Once you know how to set up the proportion correctly, these problems become straightforward and quick to solve.

What Are Equivalent Ratios?

Two ratios are equivalent when they represent the same relationship between two quantities. Just as equivalent fractions name the same number, equivalent ratios name the same rate or comparison. You create an equivalent ratio by multiplying or dividing both terms of a ratio by the same nonzero number.

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Example: The ratio \(\color{blue}{2 : 3}\) is equivalent to \(\color{blue}{4 : 6}\), \(\color{blue}{6 : 9}\), and \(\color{blue}{10 : 15}\) because each is formed by multiplying both terms by the same factor.

How to Solve Equivalent Ratio Word Problems

1. Identify the known ratio and the missing quantity

Read the problem carefully. Find the ratio that is given (e.g., 3 apples for every 2 oranges) and the value you know for one quantity.

2. Write a proportion

Set up two equivalent fractions (a proportion) with the missing value as a variable:

known part / known \(\color{blue}{\text{ total } = \text{ unknown }}\) part / unknown total

3. Solve by cross-multiplying or by scaling

Either cross-multiply and divide, or find the scale factor that links the two ratios.

Step-by-Step Summary

1. Read the problem and identify both quantities in the original ratio.
2. Decide which quantity you are given and which you need to find.
3. Write the proportion: \(\color{blue}{\frac{a}{b} = x / ?}\) or \(\color{blue}{\frac{a}{b} = ? / n}\).
4. Find the scale factor (divide or multiply) or cross-multiply.
5. Check: confirm the two ratios simplify to the same fraction.

Watch: Finding the Missing Number in an Equivalent Ratio

Math with Mr. J walks through setting up equivalent ratios step by step:

Worked Examples

Example 1: If 3 apples cost $6, how much do 9 apples cost?

Set up the proportion: \(\color{blue}{\frac{3}{6} = \frac{9}{x}}\). Scale factor: \(\color{blue}{9 \div 3 = 3}\). Multiply cost: \(\color{blue}{6 \times 3 = 18}\).
Answer: $18

Example 2: A recipe uses 2 cups of flour for 3 cookies. How much flour is needed for 12 cookies?

Ratio: \(\color{blue}{2 \text{ cups } : 3 \text{ cookies }}\). Scale factor: \(\color{blue}{12 \div 3 = 4}\). Flour: \(\color{blue}{2 \times 4 = 8}\) cups.
Answer: 8 cups

Example 3: In a class, the ratio of boys to girls is 4 : 5. If there are 20 boys, how many girls are there?

Proportion: \(\color{blue}{\frac{4}{5} = \frac{20}{x}}\). Scale factor: \(\color{blue}{20 \div 4 = 5}\). Girls: \(\color{blue}{5 \times 5 = 25}\).
Answer: 25 girls

Example 4: A map uses a scale of 1 cm : 50 km. Two cities are 3 cm apart on the map. What is the actual distance?

Ratio: \(\color{blue}{1 \text{ cm } : 50 \text{ km }}\). Scale factor: \(\color{blue}{3 \times 1 = 3}\) cm, so distance: \(\color{blue}{3 \times 50 = 150 \text{ km }}\).
Answer: 150 km

More Practice: Proportions Video Review

Math Antics covers proportions — including how to verify equivalent ratios — in this clear lesson:

Exercises

1. If 5 notebooks cost $10, how much do 8 notebooks cost?
2. A car travels 120 miles in 2 hours. How far does it travel in 5 hours?
3. The ratio of red to blue beads is 3 : 7. If there are 21 blue beads, how many red beads are there?
4. A recipe calls for 4 cups of sugar for every 6 cups of flour. How many cups of sugar are needed for 18 cups of flour?
5. A map scale is 2 cm : 30 km. Two towns are 7 cm apart on the map. What is the actual distance?

Answers

1. \(\color{blue}{$16}\)
2. \(\color{blue}{300 \text{ miles }}\)
3. \(\color{blue}{9 \text{ red beads }}\)
4. \(\color{blue}{12 \text{ cups of sugar }}\)
5. \(\color{blue}{105 \text{ km }}\)

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Frequently Asked Questions

What is the difference between a ratio and a proportion?

A ratio compares two quantities (e.g., 3 : 4). A proportion is a statement that two ratios are equal (e.g., \(\color{blue}{\frac{3}{4} = \frac{6}{8}}\)). Solving equivalent ratio word problems means setting up and solving a proportion.

How do I know which quantity goes on top in the proportion?

Be consistent. If the first ratio is apples/cost, write the second ratio as apples/cost too. Mixing up the order will give you the wrong answer.

Can I solve equivalent ratio problems without cross-multiplying?

Yes. Find the scale factor by dividing the known value in one ratio by the matching value in the other ratio, then multiply the remaining term by that factor. Both methods give the same result.

Related Topics

• Finding Equivalent Ratio
• Ratio Tables
• Equivalent Rates
• Word Problems Involving Rates and Ratios
• Word Problems Involving Comparing Ratio