**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:

- 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?
- 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?
- 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?
- The month of June has seen 960 mm of rainfall. Month of July has seen 1085mm. Which month has seen more rainfall per day?
- Which has better value, 2 books for $15 or 6 books for $45? Explain.
- 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?
- Store A takes $27 for 4 large pizzas. Store B charges $32 for 5 large pizzas. Which store has better price being offered?
- Dave runs 1000 meters in 5 minutes. Rachel runs 1500 meters in 12 minutes. Who runs faster?
- A photographer charges $8 for 36 pictures or $5 for 24 pictures. Which offer must be taken to gain profit?
- 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.

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