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a b per

Understanding
Unit Rates

Master the concept of unit rate \(\frac{a}{b}\) associated with a ratio \(a : b\), and learn to use rate language to describe real-world relationships.

6th Grade Math

What Is a Ratio?

A ratio compares two quantities. If you have 3 apples and 4 oranges, the ratio of apples to oranges is \(3 : 4\).

A A A : O O O O Ratio = 3 : 4

What Is a Rate?

A rate is a special ratio that compares two quantities with different units. For example, miles per hour compares distance to time.

Rate
Compares quantities with different units: dollars per pound, miles per hour
Ratio
Compares quantities with the same or different units: 3 boys to 4 girls

What Is a Unit Rate?

A unit rate is a rate where the second quantity is exactly 1. It tells you "how much per one."

Rate $75 15 burgers → ÷ 15 Unit Rate $5 1 burger "$5 per burger"

Why Do Unit Rates Matter?

Unit rates make it easy to compare options and make smart decisions in everyday life.

Shopping
Which cereal box is the better deal? Compare price per ounce.
Travel
Which route is faster? Compare miles per hour.
Cooking
Need to scale a recipe? Use cups per serving.

The Unit Rate Formula

For a ratio \(a : b\) where \(b \neq 0\)
Unit Rate = \(a \div b\)
Result: how much "a" for every 1 of "b"

Given a ratio \(a : b\) with \(b \neq 0\), the unit rate is \(\frac{a}{b}\). This tells you the amount of \(a\) for each single unit of \(b\).

Visualizing Unit Rate on a Number Line

If you drive 150 miles in 3 hours, the unit rate is \(\frac{150}{3} = 50\) miles per hour. Each hour, you travel the same distance.

Miles 0 50 100 150 +50 +50 +50 Hours 0 1 2 3 Unit Rate = 50 miles per hour

Ratio vs. Rate vs. Unit Rate

Ratio Rate Unit Rate
Definition Compares two quantities Ratio with different units Rate per 1 unit
Units Same or different Always different Always different
Example \(3 : 4\) \(\$75\) for 15 burgers \(\$5\) per burger
Denominator Any number Any number Always 1

How to Find a Unit Rate

1
Identify the two quantities and their units (e.g., dollars and pounds).
2
Write them as a fraction \(\frac{a}{b}\), placing the quantity you want "per one" in the denominator.
3
Divide the numerator by the denominator to get the value per 1 unit.
4
Label your answer with the correct units using "per" language.

Example 1: Price Per Pound

A bag of apples costs \(\$6\) for \(4\) pounds. What is the unit price?

Step-by-Step $6 4 lb = $1.50 1 lb

Example 2: Typing Speed

Marcus types 210 words in 3 minutes. What is his typing rate in words per minute?

1
Identify: \(a = 210\) words, \(b = 3\) minutes
2
Write the fraction: \(\frac{210 \text{ words}}{3 \text{ min}}\)
3
Divide: \(210 \div 3 = 70\)
4
Label: 70 words per minute

Example 3: Comparing Unit Rates

Which is the better deal: Store A sells 8 muffins for \(\$12\), or Store B sells 6 muffins for \(\$8.40\)?

Store A Store B
Total Price \(\$12\) \(\$8.40\)
Quantity 8 muffins 6 muffins
Unit Rate \(\$12 \div 8 = \$1.50\) \(\$8.40 \div 6 = \$1.40\)
Winner? ✓ Better deal

Using Rate Language

Rate language uses the words "per," "for each," and "for every" to describe how two quantities relate.

"per"
\(\$5\) per hamburger
"for each"
\(\frac{3}{4}\) cup for each cup of sugar
"for every"
50 miles for every hour

Real-World: Recipe Ratios

A recipe uses 3 cups of flour for 4 cups of sugar. Express this as a unit rate.

F F F 3 cups flour : S S S S 4 cups sugar
Unit rate = \(\frac{3}{4}\) cup of flour per cup of sugar

Common Mistakes to Avoid

✕ Flipping the Fraction
Writing \(\frac{4}{3}\) when asked for flour per sugar instead of \(\frac{3}{4}\). Always check which quantity goes on top.
✕ Forgetting Units
Saying "the unit rate is 5" instead of "\(\$5\) per hamburger." Without units, the answer is incomplete.
✕ Not Dividing Fully
Stopping at \(\frac{75}{15}\) instead of computing \(75 \div 15 = 5\). A unit rate must have 1 in the denominator.

Summary: Key Concepts

Concept Definition Example
Ratio Compares two quantities \(3 : 4\)
Rate Ratio with different units 150 miles in 3 hours
Unit Rate Rate where denominator = 1 50 miles per hour
Rate Language "per," "for each," "for every" \(\$5\) per burger
Finding It Divide: \(\frac{a}{b}\) \(\frac{210}{3} = 70\) wpm

Question 1 of 5

A car travels 240 miles in 4 hours. What is the unit rate?

Question 2 of 5

You pay \(\$9.60\) for 8 notebooks. What is the price per notebook?

Question 3 of 5

A recipe uses 5 cups of water for every 2 cups of rice. How many cups of water per cup of rice?

Question 4 of 5

Store X sells 10 oranges for \(\$4.00\). Store Y sells 6 oranges for \(\$2.10\). Which store has the better price per orange?

Question 5 of 5

Which of the following is NOT an example of a unit rate?