Ratio and Proportion

Grade 10 Math - Ratio and Proportion

Lesson Objectives

Define and understand the concepts of ratio and proportion.
Simplify ratios and convert them into proportions.
Apply ratios and proportions to solve real-life problems.
Interpret and solve past examination questions involving these concepts.

Lesson Introduction

Ratios and proportions are widely used in everyday life — from sharing profits, cooking recipes, to mixing chemicals. A ratio compares two quantities, while a proportion shows that two ratios are equal. This lesson teaches you how to work with these concepts effectively in mathematical and real-world contexts.

Core Lesson Content

What is a Ratio?

A ratio is a comparison of two numbers by division. It can be written in three forms: as a fraction, using a colon, or using the word “to”.

For example, the ratio of 2 to 3 can be written as:

$\frac{2}{3},\ 2:3,\ \text{or}\ 2\ \text{to}\ 3$

What is a Proportion?

A proportion is an equation that states that two ratios are equal.

$\frac{a}{b} = \frac{c}{d}$

Worked Example

Example 1: Simplify the ratio 24:36.
Find the HCF of 24 and 36 which is 12.

\frac{24}{36} = \frac{2}{3}

So, the simplified ratio is 2:3.

Example 2: If

\frac{4}{x} = \frac{2}{3}

, find the value of

x

.
Cross-multiply:

4 \times 3 = 2 \times x

12 = 2x

x = \frac{12}{2} = 6

Example 3: A recipe uses ingredients in the ratio 2:5. If you have 10 cups of flour, how many cups of sugar are needed?
Let the sugar be

x

. Then:

\frac{2}{5} = \frac{10}{x}

Cross-multiplying:

2x = 50

→

x = 25

Example 4: The ages of two brothers are in the ratio 3:5. If the younger is 12 years old, how old is the elder?

\frac{3}{5} = \frac{12}{x}

Cross-multiply:

3x = 60

→

x = 20

Example 5: Find the fourth proportion of 8, 12, and 20.
Let the fourth term be

x

\frac{8}{12} = \frac{20}{x}

Cross-multiply:

8x = 240

→

x = 30

Example 6: [WAEC] Divide ₦600 in the ratio 2:3.
Total parts = 2 + 3 = 5
One part = ₦600 ÷ 5 = ₦120
Share 1 = ₦120 × 2 = ₦240
Share 2 = ₦120 × 3 = ₦360

Example 7: A tank is filled in the ratio 5:8. If 20 litres is used, how much is the total capacity?
Let total =

x

, then:

\frac{5}{8} = \frac{20}{x}

Cross-multiply:

5x = 160

→

x = 32

Example 8: If

\frac{x}{6} = \frac{10}{15}

, find the value of

x

.
Cross-multiply:

15x = 60

→

x = 4

Example 9: [NECO] Two workers share a bonus of ₦4500 in the ratio 4:5. How much does each get?
Total parts = 4 + 5 = 9
One part = ₦4500 ÷ 9 = ₦500
Worker 1: ₦500 × 4 = ₦2000
Worker 2: ₦500 × 5 = ₦2500

Example 10: [JAMB] Solve

\frac{7}{x} = \frac{21}{63}

.
Cross-multiply:

21x = 441

→

x = \frac{441}{21} = 21

Exercises

Simplify the ratio 35:49.
If $\frac{5}{x} = \frac{15}{21}$ , find $x$ .
A box contains red and blue balls in the ratio 3:4. If there are 12 red balls, how many blue balls are there?
[WAEC] Divide ₦7200 in the ratio 5:7. (Past Question)
Two numbers are in the ratio 6:11. If the smaller is 48, find the larger number.
[NECO] Find the third proportion to 6 and 9. (Past Question)
If $\frac{2}{5} = \frac{10}{x}$ , find $x$ .
Three partners share a profit of ₦15000 in the ratio 2:3:5. What is each person’s share?
[JAMB] Solve for $x$ if $\frac{x}{18} = \frac{4}{9}$ (Past Question)
[WAEC] A recipe requires sugar and flour in the ratio 2:7. How much flour is needed if 10 cups of sugar are used? (Past Question)

Conclusion/Recap

In this lesson, we learned how to simplify ratios and solve proportions. We also applied these concepts in real-life problems like sharing money, interpreting recipes, and solving examination questions. Understanding these skills is essential in both academic and everyday contexts.