Ratio and Proportion
Arithmetic Notes with Word Problems & Exercise
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
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
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
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
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
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
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
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.
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
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
A car travels 180 km in 3 hours.
Distance in 5 hours = 300 km
11. Homework / Exercise
Exercise 1: Simplify the Ratios
- 8 : 16
- 12 : 20
- 18 : 27
- 25 : 75
- 14 : 35
Exercise 2: Find the Missing Value
- 5 : x = 10 : 20
- x : 7 = 21 : 35
- 6 : 12 = x : 24
- 9 : x = 27 : 81
Exercise 3: Word Problems
- The ratio of boys to girls is 4 : 5. If there are 20 boys, find girls.
- Divide Rs. 900 between A and B in the ratio 2 : 7.
- If 10 workers can complete a job in 12 days, how many days will 5 workers take?
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