Unit Rate – Concept and Its Uses

5.6 Compare unit rates  

Unit rate: 

Unit rate is a rate in which the comparison is to 1 unit. 

Rates can be converted to unit rates. Below is an example depicting how to convert rates to unit rates. 

Note: A unit can also be defined as a rate having 1 in the denominator. 

5.6.1 Comparing to find the greater unit rate 

Example 1: 

Sophia is comparing the cost of two packages of socks. Pack A has 8 pairs of socks for $32. Pack B has 3 pairs of socks for $6. Which pack is expensive? 

Solution: 

Cost of 1 pair of Pack A socks = $𝟑𝟐/𝟖

Cost of 1 pair of Pack B socks = $6/2

Make the denominator 1 to find the unit rate 

Unit rate in cost per pair of Pack A socks =

$32 ÷8 / 𝟖 ÷𝟖 = $𝟒/𝟏 = $4

Unit rate in cost per pair of Pack B socks = 

$6 ÷3 / 3÷3 = $2 / 1 = $2 

After analyzing, we can tell that pack A is expensive as one pair of pack A socks costs $4 

Example 2:  

Marcia and Jane decided to donate certain amount to a charity group by washing cars. Marcia raised $168 by washing 42 cars. Jane raised $152 by washing 38 cars. Who charges more per car? 

Solution:  

Step 1: First, write fractions for both Marcia and Jane. 

Marcia –

168 / 42

Jane – 

152 / 38

Step 2: Recognize which term needs to be reduced to 1. In this case divide by the terms in denominator. So, divide with 42 for Marcia and 38 for Jane. 

Marcia –

168 ÷42/42 ÷42 = 4/1

Jane – 

152 ÷ 38/ 38 ÷ 38 = 4/1

After analyzing we find that Marcia and Jane charged equally per car, i.e., is $4. 

Example 3: 

Benjamin ’s car travels 600 feet in 20 seconds.  James’s motorcycle travels 300 feet in 12 seconds. Which is faster car or bike? Explain. 

Solution: 

Step 1: First, write fractions for car and bike. 

Car –

600 / 20

Bike – 

300 / 12

Step 2: Recognize which term needs to be reduced to 1. In this time helps us to analyze. Hence, divide with 20 for car and 12 for bike. 

Car –

600 ÷ 20 / 20 ÷ 20 = 30 feet / 1 second

Bike – 

300 ÷ 12 / 12 ÷ 12 = 25 feet 1 second

After analyzing we can tell that car travels faster than motorbike for the same time, i.e., 1 second. 

5.6.2 Comparing to find the lesser unit rate 

Example 1: 

Explain how to decide which is the better value, 4 greeting cards for $10 or 6 greeting cards for $14. 

Solution: 

Step 1: First, write fractions for both the offers. 

Offer 1 – 4/10

Offer 2 – 6/14

Step 2: Recognize which term needs to be reduced to 1. In this case cost helps us to analyze. Hence, divide with 10 for offer 1 and 14 for offer 2. 

Offer 1 – 4 ÷10 / 10 ÷ 10 = 0.40 /1

Offer 2 –  6 ÷ 14 / 14 ÷ 14 = 0.42 / 1

After analyzing we can tell that 0.40 < 0.42 is lesser. Hence, offer 1 is better compared to offer 2. 

Example 2:  

Nick and Jonas’ orchids had a harvest of 450 lbs. and 1215 lbs. of oranges, respectively. If Nick has 6 orange trees and Jonas has 15 orange trees, who made less profit per tree? 

Solution: 

Step 1: First, write fractions for both the packs. 

Nick – 450/6

Jonas – 1215/15

Step 2: Recognize, which term needs to be reduced to 1. In this case tree helps us to analyze. Hence divide with 6 for Nick and 15 for Jonas. 

Nick –

450 ÷ 6 / 6 ÷6 = 75/1

Pack B – 

1215 ÷ 15/15 ÷ 15 =81/1

After analyzing we can tell that Nick earned less profit compared to Jonas.  

Example 3: 

Henry bought pack of 108 oz. of dried figs for $36 from a wholesale store. Karen picked 21 oz. pack of dried figs for $7. Who clinched the better buy? 

Solution: 

Step 1: First, write fractions for both the packs. 

Henry – 108/36

Karen – 21/7

Step 2: Recognize which term needs to be reduced to 1. In this case cost helps us to analyze. Hence divide with 36 for Henry’s pack and 7 for Karen’s pack. 

Henry –

108 ÷ 36 / 36 ÷ 36 = 3/1

Karen – 

21 ÷ 7 / 7 ÷ 7 = 3/1

After analyzing we can tell that both Henry and Karen clinched better buy, since cost per pack is same, i.e., $3. 

Example 4: 

A total of 348 people attended a medical camp over a period of 6 hours on day one. The team of doctors attended to 228 people in 4 hours. Determine the day which recorded lesser number of people per hour. 

Solution: 

Step 1: First, write fractions for both days. 

Day one – 348/6

Day two –  228/4

Step 2: Recognize which term needs to be reduced to 1. In this case time helps us to analyze. Hence, divide with 6 for day one and 4 for day two. 

Day one – 348 ÷ 6 / 6 ÷ 6 = 58/1

Day 2 –  228 ÷ 4 / 4 ÷ 4 =57/1

After analyzing we can tell that day two has seen less number of people per hour.  

Exercise:

1. Flight A travelled a distance of 420 miles in 60 minutes. Flight B reached its destination in 75 minutes by travelling a distance of 600 miles. Which flight travelled faster?
2.  Steve creates a flyer for a Christmas yard sale. Printer A took 48 seconds to make 8 copies. Printer B reproduced 7 copies in 49 seconds. Which printer took lesser time to print a copy?
3. A pet rescue center provides new homes for abandoned dogs. Fort – two Labradors were given away for adoption in 6 days and 56 Beagles were given new homes in 8 days. Which breed of dog found more homes per day?
4. The month of June has seen 960 mm of rainfall. Month of July has seen 1085mm. Which month has seen more rainfall per day?
5. Which has better value, 2 books for $15 or 6 books for $45? Explain.
6. John’s team score 35 points in 20 minutes. Where Bret’s team scores 49 points in 35 minutes. Whose team has better points per minute?
7. Store A takes $27 for 4 large pizzas. Store B charges $32 for 5 large pizzas. Which store has better price being offered?
8. Dave runs 1000 meters in 5 minutes. Rachel runs 1500 meters in 12 minutes. Who runs faster?
9. A photographer charges $8 for 36 pictures or $5 for 24 pictures. Which offer must be taken to gain profit?
10. Compare the rate 24 laps in 11 minutes or 26 laps in 13 minutes to find which has greater unit rate.

What have we learned?

■ Compare to find the greater unit rate.

■ Compare to find the lesser unit rate.