Another common set of situations that can be represented by proportions are rate problems. Rates are ratios that compare two related quantities. The table below shows three examples of commonly used rates.
| Rate | Ratio | Units |
|---|---|---|
| Rate of Speed | distance over time distance time | Miles per hour or meters per second |
| Rate of Pay | pay over time pay time | Dollars per week or per hour |
| Prices | cost over amount cost amount | Dollars per pound or per item |
What word do you notice in the units for each of these examples?
Interactive popup. Assistance may be required. Per
When a rate is given in a problem, the units often contain or are described by the word “per.”
| Rate | Ratio | Units |
|---|---|---|
| Margie can drive 95 miles in 1.5 hours. | 95 miles over 1.5 hours 95 miles 1.5 hours | Miles per hour |
| Steve built 15 birdhouses in 3 days. | 15 birdhouses over 3 days 15 birdhouses 3 days | Birdhouses per day |
| A 3-pound bag of onions cost $2.59. | $2.59 over 3 pounds $2.59 3 pounds | Dollars per pound |
| Admission is $50 for 7 students. | $50 over 7 students $50 7 students | Price per student |
How can you use the units to help you set up the correct ratio for your proportion?
Interactive popup. Assistance may be required. The unit that is listed before the word "per" is the numerator (the top) of the ratio. The unit that is listed after the word "per" is the denominator (the bottom) of the ratio.
Read the following two examples and answer the questions below.
Maria is reading a book for her English class. Last night, she read 84 pages in 2 hours. Tonight she plans to read for 5 hours. If she reads at the same rate tonight as she read last night, how many pages will she read tonight?
What are the two related quantities that are being compared?
Interactive popup. Assistance may be required. Number of pages and time.
What information is given in the problem that relates the two quantities?
Interactive popup. Assistance may be required. She reads 84 pages in 2 hours.
Does it make more sense to use this information in pages per hour or in hours per page?
Interactive popup. Assistance may be required. It makes more sense to use pages per hour.
Write the information from the problem as a ratio.
Interactive popup. Assistance may be required. 84 pages over 2 hours 84 pages 2 hours
Use this ratio to set up a proportion that you could use to solve for p, the number of pages she reads in 5 hours.
Interactive popup. Assistance may be required. 84 over 2 84 2 = p over 5 p 5
Solve your proportion for p.
Interactive popup. Assistance may be required.The factor that converts 2 → 5 is 2.5.
The same factor will convert 84 → p.
p = 84(2.5)
p = 210
She can read 210 pages in 5 hours.
Interactive popup. Assistance may be required.
The factor that converts 2 → 84 is 42.
The same factor will convert 5 → p.
p = 5(42)
p = 210
She can read 210 pages in 5 hours.
Interactive popup. Assistance may be required.
84 over 2 84 2 = p over 5 p 5
5(84) = 2p
p = 5 times 84 over 2 5(84) 2
p = 210
She can read 210 pages in 5 hours.
Tommy is helping his mom at the grocery store. He notices that potatoes are on sale, and you can buy a 5-pound bag for $2.29. Using the same unit rate, find the cost of a 7-pound bag of potatoes.
What is the price rate written as a ratio?
Interactive popup. Assistance may be required. $2.29 over 5 pounds $2.29 5 pounds
Write a proportion that could be used to calculate c, the cost of a 7-pound bag.
Interactive popup. Assistance may be required. c over 7 c 7 = 2.29 over 5 2.29 5
Solve your proportion for c.
Interactive popup. Assistance may be required.c = 3.206
A 7-pound bag would cost $3.21.