Lesson 18Equivalent Ratios Have the Same Unit Rates

Learning Goal

Let’s revisit equivalent ratios.

Learning Targets

I can give an example of two equivalent ratios and show that they have the same unit rates.
I can multiply or divide by the unit rate to calculate missing values in a table of equivalent ratios.

Lesson Terms

unit price
unit rate

Warm Up: Which One Doesn’t Belong: Comparing Speeds

Which one doesn’t belong? Be prepared to explain your reasoning.

5 miles in 15 minutes
3 minutes per mile
20 miles per hour
32 kilometers per hour

Activity 1: Price of Burritos

Two burritos cost $14. Complete the table to show the cost for 4, 5, and 10 burritos at that rate. Next, find the cost for a single burrito in each case.
number of burritos
cost in dollars
unit price
(dollars per burrito)
$2$
$14$
$4$
$5$
$10$
$b$
What do you notice about the values in this table?
Noah bought $b$ burritos and paid $c$ dollars. Lin bought twice as many burritos as Noah and paid twice the cost he did. How much did Lin pay per burrito?
number of burritos
cost in dollars
unit price
(dollars per burrito)
Noah
$b$
$c$
$\frac{c}{b}$
Lin
$2 \cdot b$
$2 \cdot c$
Explain why, if you can buy $b$ burritos for $c$ dollars, or buy $2 \cdot b$ burritos for $2 \cdot c$ dollars, the cost per item is the same in either case.

	number of burritos	cost in dollars	unit price (dollars per burrito)
Noah	$b$	$c$	$\frac{c}{b}$
Lin	$2 \cdot b$	$2 \cdot c$

Activity 2: Making Bracelets

Problem 1

Complete the table. Then, explain the strategy you used to do so.

time in hours	number of bracelets	speed (bracelets per hour)
$2$		$6$
$5$		$6$
$7$		$6$
	$66$	$6$
	$100$	$6$

Problem 2

Here is a partially filled table from an earlier activity. Use the same strategy you used for the bracelet problem to complete this table.

number of burritos	cost in dollars	unit price (dollars per burrito)
	$14.00$	$7.00$
	$28.00$	$7.00$
$5$		$7.00$
$10$		$7.00$

Problem 3

Next, compare your results with those in the first table in the previous activity. Do they match? Explain why or why not.

Activity 3: How Much Applesauce?

It takes 4 pounds of apples to make 6 cups of applesauce.

pounds of apples	cups of applesauce
$4$	$6$
$7$
$10$
	$9$
	$20$

At this rate, how much applesauce can you make with 7 pounds of apples?
How much applesauce can you make with 10 pounds of apples?
How many pounds of apples would you need to make 9 cups of applesauce?
How many pounds of apples would you need to make 20 cups of applesauce?

Are you ready for more?

Problem 1

Jada eats 2 scoops of ice cream in 5 minutes. Noah eats 3 scoops of ice cream in 5 minutes. How long does it take them to eat 1 scoop of ice cream working together (if they continue eating ice cream at the same rate they do individually)?

Problem 2

The garden hose at Andre’s house can fill a 5-gallon bucket in 2 minutes. The hose at his next-door neighbor’s house can fill a 10-gallon bucket in 8 minutes. If they use both their garden hoses at the same time, and the hoses continue working at the same rate they did when filling a bucket, how long will it take to fill a 750-gallon pool?

Lesson Summary

The table shows different amounts of apples selling at the same rate, which means all of the ratios in the table are equivalent. In each case, we can find the unit price in dollars per pound by dividing the price by the number of pounds.

apples (pounds)	price (dollars)	unit price (dollars per pound)
4	10	$10 \div 4 = 2.50$
8	20	$20 \div 8 = 2.50$
20	50	$50 \div 20 = 2.50$

The unit price is always the same. Whether we buy 4 pounds of apples for 10 dollars or 8 pounds of apples for 20 dollars, the apples cost 2.50 dollars per pound.

We can also find the number of pounds of apples we can buy per dollar by dividing the number of pounds by the price.

apples (pounds)	price (dollars)	pounds per dollar
4	10	$4 \div 10 = 0.4$
8	20	$8 \div 20 = 0.4$
20	50	$20 \div 50 = 0.4$

The number of pounds we can buy for a dollar is the same as well! Whether we buy 4 pounds of apples for 10 dollars or 8 pounds of apples for 20 dollars, we are getting 0.4 pounds per dollar.

This is true in all contexts: when two ratios are equivalent, the two unit rates will always be equal.

quantity $x$	quantity $y$	unit rate 1	unit rate 2
$a$	$b$	$\frac{a}{b}$	$\frac{b}{a}$
$s \cdot a$	$s \cdot b$	$\frac{s \cdot a}{s \cdot b} = \frac{a}{b}$	$\frac{s \cdot b}{s \cdot a} = \frac{b}{a}$