### Key Concepts

• Comparing two different quantities using ratio.

• Using Bar Graph to solve a problem in the ratio.

• Using double number line to solve a problem in the ratio.

**Introduction:**

**5.1 Understand Ratios**

**Ratio:**

Ratio can be defined as a comparison of two or more numerical quantities. Ratios are a helpful tool that helps in comparing things to each other in mathematics and real life, so it is important to know what they mean and how to use them.

**Notation of ratio:**

The relationship between two quantities can be represented by a ratio using appropriate notations, such as *a* to *b*, a:b, and **a/b**.

**Example:**

What is the ratio of the number of boys to the number of girls in the class?

Let us say, number of boys in the class = 25

Number of girls in the class = 15

Then the ratio of the number of boys to the number of girls will be written as 25 to15 or 25:15 or 25 /15.

But we mostly use 25:15 notation.

**5.1 Write ratios to compare quantities**

**Example 1:**

Identify different quantities from the given tabular form and answer the following questions:

**Questions:**

- Find the ratio of the number of fans to the number of switches.

- What is the ratio of the number of chairs to the number of windows?

- What is the ratio of the number of doors to the number of objects in total?

- What is the ratio of total objects to the number of fans?

**Answers:**

- 1:15

- 4:3

- 2:25

- 25:1

**Example 2:**

Identify different quantities from the given tabular form and answer the following questions:

**Questions:**

- What is the ratio of terrestrial to aquatic animals?

- What is the ratio of birds to non-birds?

- What is the ratio of lions to pigeons?

- What is the ratio of carnivores to omnivores?

**Answers:**

- 7:24

- 10:21

- 5:0

- 29:2

**5.1.2 Using a bar diagram to solve ratio problem**

**Example 1:**

In a high school, the ratio of number of juniors to seniors is found to be 7:5. How many seniors are there in the school if juniors are 56?

**Solution:** Use 7 boxes to show juniors and 5 boxes to show seniors

Juniors

Seniors

**Step 1:** Again, make use of the same diagram to represent 56 juniors. But, before that, divide 56 by 7 to get 8.

**Step 2:** Write the value of 8 in all the boxes of juniors and seniors.

Juniors

Seniors

**Step 3:** Add all the 8’s in senior boxes (8+8+8+8+8 = 40)

Hence, we can conclude that seniors in the school were 40.

**Example 2:**

The ratio of pens to pencils is 2:5. If there are 8 pens, then find the number of pencils.

**Solution:** Use 2 boxes to show pens and 5 boxes to show pencils.

Pens:

Pencils:

**Step 1:** Again, make use of the same diagram to represent 8 pens. But, before that, divide 8 by 2 to get 4.

**Step 2:** Write the value of 4 in all the boxes of pens and pencils.

Pens:

Pencils:

**Step 3:** Add all the 4’s in pencils (4+4+4+4+4 = 20). Hence, we conclude that there are 20 pencils.

**5.1.2 Using a double number line to solve ratio problem**

**Example 1:**

Rodrigo receives $20 every month from his dad as pocket money. Rodrigo plans to donate $200 for a foster home that looks after animals. How many months does Rodrigo wait before he donates?

**Solution:** We can use double number line to see some of the possible money to month scenarios.

**Step 1:** To make a double number line, we must start with two number lines by marking from 0.

**Step 2:** Increase by 20 on the first line until we reach 200, which represents money, increase by 1 on the second line representing months.

**Step 3:** Make sure that both the lines are lined up in an order by making comparison.

Analyze the diagram, which says that $0 is collected for $0 months, $20 for first, $40 for the second and so on until you reach $200 for the tenth month.

Hence, Rodrigo will have to wait for 10 months to collect $200 and donate to the foster home.

**Example 2:**

A company can enlarge a 2-inch length by 3-inch width photograph to any size you want. If the width of the photograph is enlarged to 15 inches, what will be the length?

**Solution:** We can use double number line to see some of the possible money to month scenarios.

**Step 1:** To make a double number line, we must start with two number lines by marking from 0.

** Step 2:** Increase by 2 on the first line until we reach 10, which represents the length. Increase by 3 on the second line representing width until we reach 15.

**Step 3:** Make sure that both the lines are lined up in an order by making comparison.

Analyse the diagram which says that if the length is o inches, then the width is also o inches; if the length is 4 inches, then the width is 6 inches.

Hence, the length will be 10 inches when the width is 15 inches, as observed from the figure.

## Exercise Problems:

1. If Samuel takes 3 days to complete 5 diagrams, how many diagrams can he finish 15 days?

2. The teacher said that the ratio of the number of fish to that of the tortoise is 7:2 in the lab. Find how many fishes the lab has if there are 6 tortoises.

3. Alice and Ella were watching birds and trying to find the ratio between the number of beaks to that of legs. After observing for some time, they found it to be 1:2. Can you find how many legs do 25 birds have in total?

4. Audrey can run 2 miles in 15 min. Find how many miles can she run in 75 min.

5. A sixth-grade basketball team has 3 centers, 8 forwards and 5 guards. Write a few possible ratios.

6. The ratio of boys to girls in a swimming club was 2:4. There were 14 girls. How many total members were there in the club?

7. A herd of 56 horses has 20 white and rest black horses. What is the ratio of white to black horses?

8. A classroom has 35 glue sticks. If the ratio of glue sticks to glue bottles was 5:2, how many glue bottles did the classroom have?

9. On Sunday, a library checked out 52 books. 11 22 of the books were fiction, what is the ratio of non-fiction books to the fiction books checked out?

10. The ratio of red cars to blue cars in a parking lot is 5:3.1f there were 35 red cars, how many blue cars were there?

### What have we learned:

• Define and write ratios. Also, explain why and how ratios are used mathematically.

• Demonstrate the concept of ratio by using ratio language to describe relationship between quantities.

• How a bar graph can be used to solve a ratio problem.

• Use double number line that can be also used to solve a ratio problem.

• Analyze the importance of making use of the concepts discussed today

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