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1 unit

Solve Unit Rate
Problems

Master the art of finding how much per one: unit pricing, constant speed, and real-world ratios that make comparison easy.

6th Grade Math

What Is a Rate?

A rate is a ratio that compares two quantities measured in different units. Whenever you see words like "per," "each," or "for every," you are looking at a rate.

Speed
\( 60 \) miles per \( 1 \) hour
Price
\( \$2.50 \) per \( 1 \) pound
Work
\( 4 \) lawns per \( 7 \) hours

What Makes It a Unit Rate?

A unit rate is a special rate where the second quantity is exactly \( 1 \). It answers the question: "How much for just one?"

Regular Rate 120 miles 2 hours →
\( \div 2 \)
Unit Rate 60 miles 1 hour

Why Do Unit Rates Matter?

Unit rates let you compare things fairly, even when the original numbers are different sizes. They also let you scale up or down to find any amount you need.

Compare
Which store has the better deal? Find the price per one item and compare.
Predict
If you know the rate, you can predict how far, how many, or how much for any amount.

The Unit Rate Formula

UNIT RATE Quantity A Quantity B
\( = 1 \)

Visualizing with a Bar Model

Imagine \( \$12 \) buys \( 4 \) notebooks. We can split both the money and the notebooks into \( 4 \) equal groups to find the price of one notebook.

$12 total $3 $3 $3 $3
\( \div 4 \)
4 notebooks 1 1 1 1 Unit Rate: $3 per notebook

Rate vs. Unit Rate

Rate Unit Rate
Definition Compares two quantities in different units Rate where the second quantity is \( 1 \)
Example \( 150 \) miles in \( 3 \) hours \( 50 \) miles per \( 1 \) hour
Best For Describing a specific situation Comparing and predicting
Form \( \frac{A}{B} \) (B can be anything) \( \frac{A}{B} \) (B must equal \( 1 \))

How to Solve Unit Rate Problems

1
Identify the two quantities and their units. Ask: "What is being compared?"
2
Decide which quantity should be "per one." This goes in the denominator.
3
Divide the first quantity by the second. The result is your unit rate.
4
Use the unit rate to answer the question: multiply to scale up, or compare unit rates to find the better deal.

Example 1: Constant Speed

A car travels \( 210 \) miles in \( 3.5 \) hours. What is the car's speed as a unit rate?

1
Quantities: \( 210 \) miles and \( 3.5 \) hours
2
We want miles per 1 hour, so hours go in the denominator.
3
Divide: \( \frac{210}{3.5} = 60 \)

Example 2: Unit Pricing

Store A sells \( 6 \) apples for \( \$4.50 \). Store B sells \( 8 \) apples for \( \$5.60 \). Which store has the better deal?

Store A
\( \frac{\$4.50}{6} = \$0.75 \) per apple
Store B
\( \frac{\$5.60}{8} = \$0.70 \) per apple

Example 3: Scaling Up

It took \( 7 \) hours to mow \( 4 \) lawns. At that rate, how many lawns could be mowed in \( 35 \) hours?

1
Find the unit rate: \( \frac{4 \text{ lawns}}{7 \text{ hours}} = \frac{4}{7} \) lawns per hour
2
Multiply: \( \frac{4}{7} \times 35 = \frac{140}{7} = 20 \)
0 hr 7 hr 14 hr 21 hr 28 hr 35 hr
\( +4 \)
\( +4 \)
\( +4 \)
\( +4 \)
\( +4 \)
 Total: 20 lawns

Two Sides of the Same Rate

Every rate problem can give you two different unit rates, depending on which quantity goes in the denominator.

Lawns per Hour
\( \frac{4}{7} \approx 0.57 \) lawns per hour: "How fast am I mowing?"
Hours per Lawn
\( \frac{7}{4} = 1.75 \) hours per lawn: "How long does each lawn take?"

Example 4: Decimal Division

A printer prints \( 45 \) pages in \( 2.5 \) minutes. How many pages per minute?

1
Set up: \( \frac{45 \text{ pages}}{2.5 \text{ minutes}} \)
2
To divide by \( 2.5 \), multiply both by \( 10 \): \( \frac{450}{25} \)
3
Divide: \( 450 \div 25 = 18 \)

Example 5: Fraction Rates

A recipe uses \( \frac{3}{4} \) cup of sugar for every \( \frac{1}{2} \) batch of cookies. How much sugar per \( 1 \) batch?

1
Set up: \( \frac{\frac{3}{4} \text{ cup}}{\frac{1}{2} \text{ batch}} \)
2
Dividing fractions: multiply by the reciprocal: \( \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} \)
3
Simplify: \( \frac{3}{2} = 1\frac{1}{2} \) cups per batch

Common Mistakes

✕ Dividing the Wrong Way
Putting the wrong quantity on top. If you need "price per item," divide \( \frac{\text{price}}{\text{items}} \), not \( \frac{\text{items}}{\text{price}} \).
✕ Forgetting Units
Writing just "\( 60 \)" without saying "\( 60 \) miles per hour." Units make the answer meaningful.
✕ Not Simplifying
Giving \( \frac{12}{4} \) as the answer instead of simplifying to \( 3 \). Always reduce to the simplest form.

Summary: Types of Unit Rate Problems

Problem Type You Divide Example Answer
Unit Pricing \( \frac{\text{Total cost}}{\text{number of items}} \) \( \$2.50 \) per lb
Constant Speed \( \frac{\text{Distance}}{\text{time}} \) \( 60 \) mph
Work Rate \( \frac{\text{Tasks}}{\text{time}} \) \( 5 \) pages/min
Scaling Up Find unit rate, then multiply \( 20 \) lawns in \( 35 \) hrs
Better Deal Compare two unit prices Store B is cheaper

Question 1 of 5

A runner covers \( 18 \) miles in \( 3 \) hours. What is her unit rate?

Question 2 of 5

Brand A: \( 5 \) bottles for \( \$8.75 \). Brand B: \( 3 \) bottles for \( \$5.10 \). Which is the better deal?

Question 3 of 5

It took \( 7 \) hours to mow \( 4 \) lawns. At that rate, how many lawns could be mowed in \( 35 \) hours?

Question 4 of 5

A factory makes \( 240 \) toys in \( 4.5 \) hours. About how many toys per hour is that? (Round to the nearest whole number.)

Question 5 of 5

A recipe calls for \( \frac{2}{3} \) cup of flour for every \( \frac{1}{3} \) batch. How many cups of flour are needed for \( 1 \) full batch?