How to Find Unit Rates and Rates?

To find a unit rate, divide the amount by the number of units. For example, 240 miles in 4 hours is 240 ÷ 4 = 60 miles per hour. Keep the units in that order: dollars divided by items gives dollars per item; miles divided by hours gives miles per hour. Rates help you compare prices, speeds, and other quantities, and they are an important middle-school skill used again in Algebra 1.

What Are Rates and Unit Rates?

A rate is a ratio that compares two quantities measured in different units. For example, driving 240 miles in 4 hours is a rate: 240 miles per 4 hours.

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A unit rate is a rate with a denominator of 1. It tells you how much of one quantity corresponds to exactly one unit of the other. To find a unit rate, divide both parts of the rate so the denominator equals 1.

Unit \(\color{blue}{\text{ rate } = \text{ quantity }}\) \(\color{blue}{A \div \text{ quantity }}\) B

How to Find a Unit Rate

Step 1, Set up the rate as a fraction

Write the rate with the quantity you want “per one” in the denominator.

Example: $15 for 3 items ⇒ \(\color{blue}{$\frac{15}{3} \text{ items }}\)

Step 2, Divide

Divide both numerator and denominator by the denominator’s value:

\(\color{blue}{$\frac{15}{3} = $5 \text{ per } 1 \text{ item }}\)

Comparing rates

Convert both rates to unit rates, then compare. The better deal has the lower unit price.

Example: Brand A: $8.40 for 6 oz ⇒ $\(\color{blue}{\frac{1.40}{\text{ oz }}}\). Brand B: $9.75 for 6.5 oz ⇒ $\(\color{blue}{\frac{1.50}{\text{ oz }}}\). Brand A is cheaper per ounce.

Step-by-Step Summary

1. Identify the two quantities being compared and their units.
2. Write the rate as a fraction: numerator / denominator.
3. Divide to make the denominator equal to 1.
4. Label the unit rate with both units (e.g., miles per hour, dollars per item).

Watch: Rates and Unit Rates (Video Lesson)

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

Unit Rates and Rates – Worked Examples

Example 1: Find the unit price: $15 for 3 items.

\(\color{blue}{$15 \div 3 \text{ items } = $5 \text{ per item }}\)

Example 2: A car travels 240 miles in 4 hours. Find the unit rate (speed).

\(\color{blue}{240 \text{ miles } \div 4 \text{ hours } = 60 \text{ miles per hour }}\)

Example 3: A typist types 180 words in 3 minutes. Find the unit rate.

\(\color{blue}{180 \text{ words } \div 3 \text{ minutes } = 60 \text{ words per minute }}\)

Example 4: Which is a better deal: $8.40 for 6 oz or $9.75 for 6.5 oz?

Brand A: \(\color{blue}{$8.40 \div 6 = $1.40 \text{ per oz }}\)
Brand B: \(\color{blue}{$9.75 \div 6.5 = $1.50 \text{ per oz }}\)
Brand A is the better deal at $1.40 per oz.

Example 5: A unit rate with a fractional denominator

A walker travels 3/4 mile in 1/2 hour. Miles per hour means distance divided by time: (3/4) ÷ (1/2) = (3/4) × 2 = 3/2, or 1.5 miles per hour. Dividing by half an hour tells you the distance for one full hour at the same pace. Reversing the order would give hours per mile instead.

More Practice: Solving Unit Rate Problems (Video)

Khan Academy walks through unit rate problems including those with fractions and multi-step calculations:

Exercises for Unit Rates and Rates

1. A runner completes 12 miles in 2 hours. What is the unit rate in miles per hour?
2. A store sells 5 notebooks for $7.50. What is the cost per notebook?
3. A faucet drips 90 drops in 3 minutes. What is the unit rate in drops per minute?
4. Which is a better buy: 8 oz for $3.20 or 12 oz for $4.80?
5. A factory makes 500 parts in 4 hours. How many parts does it make per hour?

Answers

1. \(\color{blue}{6 \text{ miles per hour }}\)
2. \(\color{blue}{$1.50 \text{ per notebook }}\)
3. \(\color{blue}{30 \text{ drops per minute }}\)
4. Both are $\(\color{blue}{\frac{0.40}{\text{ oz }}}\), the same unit price.
5. \(\color{blue}{125 \text{ parts per hour }}\)

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Want More Practice?

Practice this skill with our free Unit Rates worksheet with answers. Start with its worked example, solve the problems, then check the answer explanations. For related lessons and printable practice, use the Pre-Algebra learning center. The two Algebra 1 worksheets below extend the idea to direct variation and slope; use them after per-one comparisons feel comfortable.

• Download Direct Variation Worksheet
• Download Slope and Rate of Change Worksheet

Frequently Asked Questions

What is the difference between a rate and a ratio?

A ratio compares two quantities, such as 3 red marbles to 5 blue marbles. A rate is a ratio often expressed with different units, such as miles and hours. A unit rate expresses the comparison per one unit: 60 miles in 1 hour is 60 miles per hour. Ratios are not limited to quantities with the same units.

How do I find a unit rate from a table?

First read the column labels so you know which quantity goes on top. For each row with a nonzero x-value, divide y by x and attach the units. If those quotients agree and the table represents a proportional relationship, the common value is the unit rate. Do not divide by zero: a row with x = 0 is checked separately; proportional quantities have y = 0 there.

Why are unit rates useful?

Unit rates make it easy to compare quantities. For example, comparing $\(\color{blue}{\frac{1.40}{\text{ oz }}}\) to $\(\color{blue}{\frac{1.50}{\text{ oz }}}\) is much simpler than comparing $\(\color{blue}{\frac{8.40}{6}}\) oz to $\(\color{blue}{\frac{9.75}{6.5}}\) oz directly.

Related Topics

• Convert Units of Measurement
• Metric Units
• Arithmetic Sequences
• Finding Slope
• Writing Linear Equations

Curriculum reference: Common Core Grade 6 ratio and unit-rate guidance connects per-one comparisons with tables, unit prices, and speed.