Class 6 Ratio and Proportion

Introduction to Ratio

Properties of Ratio

Difference between fraction and ratio

Introduction to Proportion

Unitary Method

Percentage

Ratio and Proportion Test

Ratio and Proportion Worksheet

Answer Sheet

Introduction to Ratio

It is a comparison of the sizes of two or more quantities of same kind by division.

If A and B are two quantities of the same kind, then the fraction ^A⁄_B is called the ratio of A and B. It is written as A : B and read as A is to B. Here A is known as first term or antecedent and B is known as second term or consequent. Let's have a look at some examples.

Example 1. In a class there are 15 boys and 25 girls. Find the ratio of boys to girls.

Solution. Ratio of boys to girls = ¹⁵⁄₂₅ = ³⁄₅

Example 2. In a class there are 15 boys and 25 girls. Find the ratio of boys to total no. of students.

Solution. Number of boys = 15
Total number of students = 15 + 25 = 40
Ratio of boys to total students = ¹⁵⁄₄₀ = ³⁄₈

Example 3. In a class there are 15 boys and 25 girls. Find the ratio of girls to total no. of students.

Solutions. Number of girls = 25
Total number of students = 15 + 25 = 40
Ratio of girls to total students = ²⁵⁄₄₀ = ⁵⁄₈

Properties of Ratio

1. Ratio has no units, because the units in the numerator and denominator cancel out.

2. Two ratios are called equivalent if the fractions corresponding to them are equivalent. So, the ratio 2 : 3 is equivalent to 4 : 6.

Difference between fraction and ratio

A fraction is a number which represents a part of a whole, but ratio is a comparison of sizes of quantities of same kind by division. Let's have a look at some examples.

Fraction ³⁄₄ means a quantity is divided into 4 equal parts and 3 parts are taken.

Ratio 3 : 4 means a quantity is divided into 7 equal parts and it is the comparison of 3 parts to 4 parts by division.

In fraction, denominator shows the number of parts.
In a ratio sum of numerator and denominator will give the number of parts.

Example 1. Find the ratio of 50 paisa to Rs. 3.

Solution. First, we have to convert rupees into paise.
Rs 3 = 3 x 100 = 300 paise
Ratio of 50 paise to Rs. 3 = ⁵⁰⁄₃₀₀ = ¹⁄₆ or 1 : 6.

Example 2. Find the ratio of 2 kg to 500 g.

Solution. 2 kg = 2 x 1000 = 2000 g
Ratio of 2 kg to 500 g = ²⁰⁰⁰⁄₅₀₀ = ⁴⁄₁ or 4 : 1.

Example 3. Find the ratio of 6 months to 3 years.

Solution. 3 years = 3 x 12 = 36 months
Ratio of 6 months to 3 years = ⁶⁄₃₆ = ¹⁄₆ = 1 : 6

Introduction to Proportion

An equality of two ratios is known as proportion. In other words four quantities A, B, C and D are said to be in proportion if A : B = C : D.

^A⁄_B = ^C⁄_D

=> AD = BC

Above method is known as cross product rule.
If AD ≠ BC, then A, B, C and D are not in proportion.

Example 1. Check if 6 : 9 and 24 : 36 form a proportion or not.

Solution. Let's express both the ratios in simplest form.
Simplest form of 6 : 9 = ⁶⁄₉ = ²⁄₃
Simplest form of 24 : 36 = ²⁴⁄₃₆ = ^{(2 x 12)}⁄_{(3 x 12)} = ²⁄₃
2/3 = 2/3
So, the given ratios form a proportion.

Example 2. Are the 25 cm : 30 cm and 40 cm : 54 cm in proportion?

Solution. Let's express both the ratio in their simplest form.
25 cm : 30 cm = ²⁵⁄₃₀ = ⁵⁄₆
40 cm : 54 cm = ⁴⁰⁄₅₄ = ²⁰⁄₂₇
⁵⁄₆ ≠ ²⁰⁄₂₇
So, the given ratios do not form a proportion.

Example 3. Are 6, 10, 60 and 100 in proportion.

Solution. 6, 10, 60 and 100 are in proportion if the product of extremes is equal to product of means.
Product of extremes = 6 x 100 = 600
Product of means = 10 x 60 = 600
Hence, are 6, 10, 60 and 100 in proportion.

Unitary Method

It is a method by which the value of a unit quantity is first obtained to find the value of any required quantity is called unitary method. Let's have a look at some examples.

Example 1. If the cost of 10 chocolates is Rs. 80, find the cost of 4 chocolates.

Solution. Cost of 10 chocolates = Rs. 80
Cost of one chocolate = ⁸⁰⁄₁₀ = Rs. 8
Cost of 4 chocolates = 8 x 4 = Rs. 32

Examples 2. If cost of 9 pens is Rs. 90, find the cost of 25 pens.

Solution. Cost of 9 pens = Rs. 90
Cost of one pen = 90/9 = Rs. 10
Cost of 25 pens = 10 x 25 = Rs. 250

Percentage

It is the numerator of a fraction having denominator 100. ⁸⁵⁄₁₀₀ is a fraction with denominator 100. It's numerator is 85, so percentage is 85%.

If the denominator of a fraction is not 100, then convert it into an equivalent fraction with denominator 100. Let's see some example.

Example 1. Convert 8/25 into percentage.

Solution. ⁸⁄₂₅ can be written as ^{(8 X 4)}⁄_{(25 X 4)} = ³²⁄₁₀₀
Fraction having numerator 32 and denominator 100, can be called as 32%

Example 2. What is the 40% of Rs. 1000?

Solution. 40% of Rs. 1000 = ⁴⁰⁄₁₀₀ x 1000 = Rs. 400

Example 3. What is the 75% of 600 marks?

Solution. 75% of 600 marks = ⁷⁵⁄₁₀₀ x 600 = 75 x 6 = 450
So, 75% of 600 marks is equal to 450 marks.