Lesson 7: Equivalent Ratios Have the Same Unit Rates

Let’s revisit equivalent ratios.

7.1: Which One Doesn’t Belong: Comparing Speeds

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

  1. 5 miles in 15 minutes
  2. 3 minutes per mile
  1. 20 miles per hour
  2. 32 kilometers per hour

7.2: Price of Burritos

  1. Two burritos cost \$14.00. 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)
    row 1 2 14.00  
    row 2 4    
    row 3 5    
    row 4 10    
    row 5 $b$    
     

    2. What do you notice about the values in this table?

  1. 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)
    row 1 Noah $b$ $c$ $\frac{c}{b}$
    row 2 Lin $2 \boldcdot b$ $2 \boldcdot c$  

  2. Explain why, if you can buy $b$ burritos for $c$ dollars, or buy $2 \boldcdot b$ burritos for $2 \boldcdot c$ dollars, the cost per item is the same in either case.

7.3: Making Bracelets

  1. Complete the table. Then, explain the strategy you used to do so.
      time in hours number of bracelets speed (bracelets per hour)
    row 1 2   6
    row 2 5   6
    row 3 7   6
    row 4   66 6
    row 5   100 6
    Leather Lanyard Bracelet Copyright Owner: Maegan Tintari License: CC BY 2.0 Via: Flickr
  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)
    row 1   14.00 7.00
    row 2   28.00 7.00
    row 3 5   7.00
    row 4 10   7.00
  3. Next, compare your results with those in the first table in the previous activity. Do they match? Explain why or why not.

7.4: How Much Applesauce?

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

  1. At this rate, how much applesauce can you make with:

    1. 7 pounds of apples?
    2. 10 pounds of apples?

  2. How many pounds of apples would you need to make:

    1. 9 cups of applesauce?
    2. 20 cups of applesauce?
  pounds of apples cups of applesauce
row 1 4 6
row 2 7  
row 3 10  
row 4   9
row 5   20

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)
row 1 4 10 $10 \div 4 = 2.50$
row 2 8 20 $20 \div 8 = 2.50$
row 3 20 50 $50 \div 20 = 2.50$
 

The unit price is always the same. Whether we buy 10 pounds of apples for 4 dollars or 20 pounds of apples for 8 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
row 1 4 10 $4 \div 10 = 0.4$
row 2 8 20 $8 \div 20 = 0.4$
row 3 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 10 pounds of apples for 4 dollars or 20 pounds of apples for 8 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
row 1 $a$ $b$ $\frac{a}{b}$ $\frac{b}{a}$
row 2 $s \boldcdot a$ $s \boldcdot b$ $\frac{s \boldcdot a}{s \boldcdot b} = \frac{a}{b}$ $\frac{s \boldcdot b}{s \boldcdot a} = \frac{b}{a}$

Practice Problems ▶