Equivalent Rates

How to equivalent rates, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

A rate compares two quantities with different units, like miles per hour or dollars per pound. Equivalent rates express the same relationship in different terms, just as equivalent fractions name the same value. Mastering equivalent rates is key to solving unit-price problems, speed calculations, and many real-world GED math questions.

What Are Equivalent Rates?

Two rates are equivalent when they describe the same comparison, even though the numbers look different. For example:

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• \(\color{blue}{60 \text{ miles per hour }}\) = \(\color{blue}{1 \text{ mile per minute }}\)
• \(\color{blue}{$15 \text{ for } 3 \text{ shirts }}\) = \(\color{blue}{$5 \text{ per shirt }}\)
• \(\color{blue}{120 \text{ words in } 2 \text{ minutes }}\) = \(\color{blue}{60 \text{ words per minute }}\)

You create an equivalent rate by multiplying or dividing both parts of the rate by the same nonzero number.

How to Find Equivalent Rates

Method 1, Scale both parts

Multiply or divide both the numerator and denominator by the same factor.

Example: \(\color{blue}{$4.50 \text{ for } 6 \text{ servings }}\) → divide both by 6 → \(\color{blue}{$0.75 \text{ per serving }}\).

Method 2, Find the unit rate first

Divide the first quantity by the second to get the unit rate (rate per 1 unit). Then scale up or down from there.

Example: 180 km in 3 hours. Unit rate: \(\color{blue}{180 \div 3 = 60 \frac{\text{ km }}{h}}\). For 5 hours: \(\color{blue}{60 \times 5 = 300 \text{ km }}\).

Method 3, Cross-multiply to check equivalence

To verify that two rates are equivalent, set them as a proportion and cross-multiply. If both products are equal, the rates are equivalent.

Step-by-Step Summary

1. Write the given rate as a fraction: quantity A / quantity B.
2. Identify whether you need to scale up or down.
3. Multiply or divide both parts by the same factor.
4. Check: does the simplified form equal the original unit rate?

Watch: Rates and Unit Rates (Video Lesson)

Math with Mr. J explains rates and unit rates with clear, step-by-step examples:

Worked Examples

Example 1: A car travels 60 miles in 1 hour. Express this as miles per minute.

1 \(\color{blue}{\text{ hour } = 60}\) minutes. Rate: \(\color{blue}{60 \frac{\text{ miles }}{60} \text{ minutes } = 1 \text{ mile per minute }}\).
Answer: 1 mile per minute

Example 2: $15 for 3 shirts. What is the cost per shirt?

Divide: \(\color{blue}{$15 \div 3 = $5 \text{ per shirt }}\).
Answer: $5 per shirt

Example 3: A typist types 120 words in 2 minutes. What is the equivalent rate per minute?

Divide both by 2: \(\color{blue}{120 \div 2 = 60}\) and \(\color{blue}{2 \div 2 = 1}\). Rate: \(\color{blue}{60 \text{ words per minute }}\).
Answer: 60 words per minute

Example 4: A recipe uses 4.5 cups of broth for 6 servings. How much broth is needed per serving?

Divide: \(\color{blue}{4.5 \div 6 = 0.75 \text{ cups per serving }}\).
Answer: 0.75 cups per serving

More Practice: Ratios and Rates Video Review

Math Antics covers both ratios and rates, perfect preparation for GED rate problems:

Exercises

1. A cyclist rides 45 miles in 3 hours. What is the equivalent rate in miles per hour?
2. A store sells 5 pounds of apples for $3.75. What is the price per pound?
3. A printer prints 80 pages in 4 minutes. How many pages per minute does it print?
4. A worker earns $210 for 14 hours. What is the hourly rate?
5. A car uses 8 gallons of gas for 240 miles. Write an equivalent rate for miles per gallon.
6. A faucet drips 12 ounces of water every 3 minutes. How many ounces per minute does it drip?

Answers

1. \(\color{blue}{15 \text{ miles per hour }}\)
2. \(\color{blue}{$0.75 \text{ per pound }}\)
3. \(\color{blue}{20 \text{ pages per minute }}\)
4. \(\color{blue}{$15 \text{ per hour }}\)
5. \(\color{blue}{30 \text{ miles per gallon }}\)
6. \(\color{blue}{4 \text{ ounces per minute }}\)

Frequently Asked Questions

What is an equivalent rate in math?

An equivalent rate describes the same comparison between two quantities with different units, using different numbers. For example, $15 for 3 shirts is equivalent to $5 per shirt, and 120 words in 2 minutes is equivalent to 60 words per minute. Equivalent rates are the rate version of equivalent fractions.

How do you find equivalent rates?

Multiply or divide both parts of the rate by the same nonzero number. For $4.50 for 6 servings, divide both by 6 to get $0.75 per serving. Or multiply both by 2 to get $9.00 for 12 servings. As long as both parts change by the same factor, the rate stays the same.

How do you write equivalent rates?

Write the rate as a fraction with units, such as 180 km / 3 h, then scale the top and bottom together. Dividing both by 3 gives 60 km / 1 h, or 60 km per hour. Keep the units on both parts so you do not mix up which quantity is on top.

Is there a formula for equivalent rates?

There is no single special formula, but the idea is a/b = (a × k)/(b × k) for any nonzero number k. A useful shortcut is to find the unit rate first by dividing: 180 km in 3 hours is 180 ÷ 3 = 60 km per hour. Then multiply by any amount of time, so 5 hours gives 60 × 5 = 300 km.

What are some examples of equivalent rates?

60 miles per hour is equivalent to 1 mile per minute, because an hour has 60 minutes. $2 for 4 apples equals $0.50 per apple and $6 for 12 apples. Typing 300 words in 5 minutes equals 60 words per minute. In each case, dividing the first quantity by the second gives the same unit rate.

Related Topics

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• Write a Ratio
• Finding Equivalent Ratio
• Word Problems Involving Equivalent Ratio
• Ratio Tables