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.

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?

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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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Three Ways to Solve the Same Problem

Problem: 4 pencils cost $3. How much do 12 pencils cost?

1. Ratio table

Keep adding 4 pencils and $3 until you reach 12 pencils. The cost is $9.

2. Scale factor

12 ÷ 4 = 3, so multiply both parts of 4 : 3 by 3. That gives 12 : 9, so the cost is $9.

3. Unit rate

One pencil costs $3 ÷ 4 = $0.75. Then 12 × $0.75 = $9.

All three give the same answer. Tables are great for seeing the pattern, scale factors are fastest with whole numbers, and unit rates work for any amount.

Part-to-Total Problems and a Common Mistake

Some problems give the total instead of one part. Example: The ratio of boys to girls in a club is 2 : 3, and there are 30 members. How many are girls?

1. Add the parts: 2 + 3 = 5 parts in all.
2. Find one part: 30 ÷ 5 = 6 members.
3. Girls are 3 parts: 3 × 6 = 18 girls. Boys are 2 × 6 = 12.
4. Check: 12 + 18 = 30, and 12 : 18 simplifies to 2 : 3.

Watch out: equivalent ratios come from multiplying or dividing, never from adding the same number. Adding 2 to both parts of 2 : 3 gives 4 : 5, which is not equivalent. Multiplying by 2 gives 4 : 6, which is.

Frequently Asked Questions

How do you solve equivalent ratio word problems?

Write the ratio you know, find the number that turns one part into the new amount, and multiply the other part by that same number. Example: a paint mix uses 3 parts blue to 4 parts white. With 20 cups of white, 4 × 5 = 20, so use 3 × 5 = 15 cups of blue. You can also set up 3/4 = b/20 and cross-multiply to get 4b = 60.

What is an example of an equivalent ratio question?

Here is a typical one: a school has 2 teachers for every 25 students. How many teachers are needed for 175 students? Since 25 × 7 = 175, multiply the teachers by 7 too: 2 × 7 = 14 teachers. The ratios 2 : 25 and 14 : 175 are equivalent because both parts were multiplied by the same number.

How do you find the missing number in an equivalent ratio?

Compare the two parts you know to find the scale factor, then apply it to the missing part. For 5 : 8 = ? : 32, ask what 8 is multiplied by to get 32. The answer is 4, so the missing number is 5 × 4 = 20. When the factor is not a whole number, cross-multiply: for 6 : 9 = 10 : x, 6x = 90, so x = 15.

Is there an equivalent ratios word problems worksheet with answers?

Yes. The exercises that follow the worked examples work like a short worksheet, with recipe, map and mixture problems and step-by-step solutions. Try solving each one two ways, by scaling and by cross-multiplying, and confirm both give the same answer. If the two answers differ, the ratio was probably written in the wrong order, such as boys : girls instead of girls : boys.

Related Topics

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