Unit Rates

A unit rate is a ratio that compares a quantity to exactly one unit of something else. You encounter unit rates constantly in daily life: miles per hour, price per pound, calories per serving. The word “per” is a signal that you are dealing with a rate, and when the second quantity is one, you have a unit rate.

Understanding unit rates gives you a powerful tool for comparing options and making decisions, whether you are shopping for groceries, planning a road trip, or tracking your productivity at work.

What Is a Unit Rate?

A rate compares two quantities measured in different units. For example, driving 150 miles in 3 hours is a rate:

$\frac{150 \text{ miles}}{3 \text{ hours}}$

A unit rate simplifies that rate so the denominator is 1:

$\frac{150 \text{ miles}}{3 \text{ hours}} = \frac{50 \text{ miles}}{1 \text{ hour}} = 50 \text{ miles per hour}$

To find a unit rate, divide the numerator by the denominator.

$\text{Unit Rate} = \frac{\text{Total Quantity}}{\text{Number of Units}}$

Finding Unit Rates

The process is straightforward: divide to make the denominator equal to 1.

Example 1: Earnings Per Hour

Suppose you earn $168 for 12 hours of work. What is your hourly rate?

$\text{Unit Rate} = \frac{168}{12} = 14$

Your unit rate is $14 per hour.

Example 2: Miles Per Gallon

A car travels 372 miles on 12 gallons of gas. What is the fuel efficiency?

$\text{Unit Rate} = \frac{372 \text{ miles}}{12 \text{ gallons}} = 31 \text{ miles per gallon}$

The car gets 31 miles per gallon.

Example 3: Cost Per Ounce

A 32-ounce bottle of juice costs $4.80. What is the cost per ounce?

$\text{Unit Rate} = \frac{4.80}{32} = 0.15$

The juice costs $0.15 per ounce.

Using Unit Rates for Comparison

One of the most practical uses of unit rates is comparing two options to find the better deal. When package sizes differ, you cannot compare total prices directly. Instead, find the unit rate for each option and compare.

Example 4: Best Buy at the Store

You are choosing between two brands of rice:

• Brand A: 5-pound bag for $6.25
• Brand B: 8-pound bag for $9.20

Find the unit price for each:

Brand A:

$\frac{6.25}{5} = 1.25 \text{ per pound}$

Brand B:

$\frac{9.20}{8} = 1.15 \text{ per pound}$

Brand B is the better deal at $1.15 per pound compared to $1.25 per pound.

Example 5: Speed Comparison

Two runners complete a training run:

• Runner 1: 6 miles in 48 minutes
• Runner 2: 8 miles in 68 minutes

Find the unit rate (minutes per mile) for each:

Runner 1:

$\frac{48 \text{ min}}{6 \text{ miles}} = 8 \text{ min per mile}$

Runner 2:

$\frac{68 \text{ min}}{8 \text{ miles}} = 8.5 \text{ min per mile}$

Runner 1 is faster with a pace of 8 minutes per mile, compared to Runner 2’s pace of 8.5 minutes per mile.

When the Unit Rate Is Not a Whole Number

Unit rates do not always come out to neat whole numbers. Round to a reasonable number of decimal places based on the context.

Example: A package of 6 light bulbs costs $8.50. What is the price per bulb?

$\frac{8.50}{6} \approx 1.4167$

Rounding to the nearest cent: $1.42 per bulb.

Practice Problems

Test your understanding with these problems. Work each one out before checking the answer.

Problem 1: A factory produces 840 widgets in 7 hours. What is the production rate per hour?

Problem 2: A 24-pack of water bottles costs $5.76. A 36-pack costs $7.92. Which is the better deal?

Problem 3: A car travels 256 miles on 8 gallons of gas. What is the fuel efficiency in miles per gallon?

Problem 4: A nurse administers 750 mL of IV fluid over 5 hours. What is the flow rate in mL per hour?

Problem 5: Two internet plans are available. Plan A offers 200 Mbps for $60 per month. Plan B offers 500 Mbps for $90 per month. Which plan gives more speed per dollar?

Key Takeaways

• A unit rate expresses a ratio with a denominator of 1, found by dividing the numerator by the denominator.
• Unit rates let you compare options that have different sizes, quantities, or timeframes on equal footing.
• The word “per” signals a rate. Common unit rates include miles per hour, cost per item, and calories per serving.
• When comparing prices, always calculate the unit price for each option before deciding.
• Round unit rates appropriately for the context (usually to the nearest cent for prices).

Return to Arithmetic for more foundational math topics.