### Key Concepts

■ Finding equivalent rates.

■ Comparison of quantities in two ways. • Using unit rate in solving problems.

**Introduction**:

## 5.5 Understand Rates and Unit Rates

To understand, what is a rate? Firstly, we have to know about ratio.

**Ratio: **A ratio can be defined as the comparison of two quantities. The ratio can be represented in three forms.

**Example 1: **In a class of 30 students, 12 are girls.** **Write the ratio of number of boys to the total number of students in the class.** **

**Solution****:**

Number of boys = Total number of students – Number of girls

Number of boys = 30 – 12

Number of boys = 18

The ratio will be 18: 12. The three forms of representation of the ratio will be as follows.

**Example2****:**

**Rate: **A rate is also a ratio comparing two different quantities having different units.

**Example: **A bike travels 60 miles in 4 hours.

**Explanation: Here**, we observe that there is a comparison being done between two different quantities, i.e., distance and time, the units of which are unlike.

**5.5.1 Find equivalent rates**

**Example 1: **The price of pears in a shop A is listed as 5 pears for $12. At the same rate, what would be the cost for 25 pears and how many pears can we buy using $84?

**Method 1: Ratio table**

**Solution: **

**Step 1:** Form a table to write ratios that are equivalent to

5 pears $ 12

**Step 2:** Multiply both the terms of the ratio by a non-zero positive integer until we get 25 in the pears column and 84 in the dollars column.

**Step 3: **On analyzing the table, we observe that 25 pears cost $60. Whereas $84 can buy us 35 pears.

**Method 2**

**Solution: **

- To find the cost of 25 pears.

**Step 1: **Write the rate as a fraction:

5 12

**Step 2: **Multiply both terms of the rate by the same non-zero number to find an equivalent rate.

b) To find the number of pears that can be bought using $84.

**Step 1: **Write the rate as a fraction:

5 12

**Step 2: **Multiply both terms of the rate by the same non-zero number to find an equivalent rate.

Hence, we conclude that 25 pears cost $60. Whereas $84 can buy us 35 pears.

**5.5.2 Compare quantities in two ways**

**Example 1: **A recipe for scrambled eggs uses 2 tablespoons of milk for every 3 eggs. What are the two unit rates that could represent the recipe?

**Solution: **

- Find the unit rate in eggs per spoons.

The unit rate in eggs per spoons is

1.5 eggs1 spoons 1.5 eggs1 spoons

- Find the unit rate in spoons per eggs.

The unit rate in eggs per spoons is

0.6 table spoons1 egg 0.6 table spoons1 egg

**5.5.2 Use unit rates to solve problems**

**What is a unit rate?**

Unit rate is an equivalent rate with a denominator of 1.

or

A unit rate is a rate in which the comparison is made to 1 unit.

**Examples of unit rates:**

- The rate of heart beats in 72 times per minute.

- There are 24 hours per day.

- There are seven days in a week.

- There are 12 months in a year.

- There are 365 days in a year.

**Ratio and Unit Rate**

A unit is a ratio that compares two measurements(units) and the second measurement is 1(per).

**Note:**

Remember that the word ‘’per’’ is always associated with rates, mostly unit rates.

**Example 1: **A canoeing club travels 80 miles in 5 days. How far could they travel in 8 days, if they maintain the same speed?

**Solution: **

**Step 1: **Find the unit rate in miles per hour.

The unit rate is 16/1 or 16 miles per hour.

**Step 2: **Use the unit rate to find how far the club travel in 8 hours. Multiply the unit rate by 8.

Therefore, the canoeing club can travel 128 miles in 8 hours.

**Concept Map**

## Exercise:

1. A car uses 8 gallons of gas to travel SO miles. How far can the car travel with 40 gallons of gas?

2. Sierra can make 5 friendship bands in 2 hours. How long will she take to make 15 such bands?

3. Mary can type 60 words in 3 minutes. How many words can she type in 12 minutes?

4. Hannah read 40 pages in 60 minutes. What are two different unit rates that could represent the situation?

5. It took jasmine 25 minutes to run 10 laps. What are the two different unit rates that could represent the situation?

6. Eight pairs of gloves are required for 16 patients. What are two different unit rates that could represent the situation?

7. John spent $40 to buy 8 basketballs. What is the cost of 5 basketballs?

8. Dave runs 1000 meters in S minutes. How much can he run in 20 minutes?

9. Isabella flew 600 miles in 120 minutes. How many miles did she fly in 60 minutes?

10. Lenin paid $30 dollars for 6 pitas, what would he pay for 9 pizzas of the same order?

### What have we learned?

• Understand how to find equivalent rates.

• Understand how to compare quantities in two ways.

• Understand how to use unit rates in solving problems.

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