Ratio and Proportion – Easy Explanation with Word Problems and Homework

Ratio and Proportion

Arithmetic Notes with Word Problems & Exercise

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1. Introduction

Ratio and Proportion is an important topic of Arithmetic. It is widely used in daily life, exams, money problems, age problems, speed and work questions. A clear understanding of this chapter makes many mathematical problems easy.

2. Ratio

A ratio is a comparison of two quantities of the same kind by division.

How to Write a Ratio

• Using colon: a : b
• As a fraction: a / b

Example:
If there are 4 pens and 8 pencils, the ratio of pens to pencils is:
4 : 8 = 1 : 2

3. Simplifying a Ratio

To simplify a ratio, divide both terms by the same common number.

Example:
20 : 30
Divide both terms by 10
= 2 : 3

4. Types of Ratio

(a) Part to Part Ratio

Comparison between two parts.

Example:
Boys = 12, Girls = 18
Ratio = 12 : 18 = 2 : 3

(b) Part to Whole Ratio

Comparison of one part with the total.

Example:
Total students = 12 + 18 = 30
Boys to total = 12 : 30 = 2 : 5
Girls to total = 18 : 30 = 3 : 5

5. Properties of Ratio

• Both quantities must be in the same units.
• Ratio has no unit.
• Multiplying or dividing both terms by the same number does not change the ratio.

6. Proportion

When two ratios are equal, they form a proportion.

Form: a : b = c : d

7. Cross Multiplication Rule

If a : b = c : d, then:

a × d = b × c

8. Arithmetic Problems

Example 1:
4 : x = 8 : 16
4 × 16 = 8 × x
64 = 8x
x = 8

Example 2:
x : 6 = 10 : 15
x × 15 = 6 × 10
15x = 60
x = 4

9. Word Problems on Ratio

Age Problem:
Ratio of Ali’s age to Ahmed’s age is 3 : 5.
Ali’s age = 12 years
Ahmed’s age = 20 years

Sum Problem:
Two numbers are in the ratio 2 : 3.
Their sum is 25.
Numbers are 10 and 15.

10. Word Problems on Proportion

Money Problem:
5 notebooks cost Rs. 150.
Cost of 8 notebooks = Rs. 240

Speed Problem:
A car travels 180 km in 3 hours.
Distance in 5 hours = 300 km

11. Homework / Exercise

Exercise 1: Simplify the Ratios

1. 8 : 16
2. 12 : 20
3. 18 : 27
4. 25 : 75
5. 14 : 35

Exercise 2: Find the Missing Value

1. 5 : x = 10 : 20
2. x : 7 = 21 : 35
3. 6 : 12 = x : 24
4. 9 : x = 27 : 81

Exercise 3: Word Problems

1. The ratio of boys to girls is 4 : 5. If there are 20 boys, find girls.
2. Divide Rs. 900 between A and B in the ratio 2 : 7.
3. If 10 workers can complete a job in 12 days, how many days will 5 workers take?

Note: Practice these exercises carefully. Ratio and Proportion is a base chapter for many advanced arithmetic topics.