ICSE Class 6 Maths Chapter 08 Ratio

Unit 2: Arithmetic Problems

Chapter 8: Ratio (Including Proportions)

8.1 Introduction

Most of the time, we compare things, numbers, etc. (say, x and y) by saying:

x is greater than y

x is less than y

x is double of y

x is one-third of y

\(\frac{x}{y} = \frac{4}{5}\)

\(\frac{y}{x} = \frac{3}{2}\), etc.

The method of comparing two quantities (numbers, things, etc.) by dividing one quantity by the other, is called ratio.

Thus: \(\frac{x}{y} = \frac{4}{5}\) represents the ratio of x to y.

and, \(\frac{y}{x} = \frac{3}{2}\) represents the ratio of y to x.

8.2 Ratio

The relation of two quantities (both of the same kind and in the same unit) obtained on dividing one quantity by the other is called their ratio.

The ratio of two quantities x to y, both of the same kind and in the same unit, is \(\frac{x}{y}\), and is often written as x : y (read as x to y or x is to y).

Meaning of the two quantities of the same kind and in the same unit:

Both the quantities must of the same kind, means: If one quantity is length, the other quantity must also be length; if quantity represents mass the other quantity must also be representing mass and so on.

The ratio between unlike quantities has no meaning.

For example, the ratio of length to mass has no meaning.

Both the quantities must be in the same unit, means: The two quantities must have the same unit of measurement.

For example, if the lengths of two objects are given to be 60 cm and 1-5 m; then before finding the ratio of one length to that of other, both of these lengths must either be converted into cm or into m.

Examples:

The ratio of 5 kg to 15 kg = \(\frac{5 \text{ kg}}{15 \text{ kg}} = \frac{1}{3} = 1 : 3\)

The ratio of 800 gm to 1.2 kg

Since 1.2 kg = 1.2 \(\times\) 1000 gm = 1200 gm

\(= \frac{800 \text{ gm}}{1200 \text{ gm}} = \frac{2}{3} = 2 : 3\)

The ratio of 2 m to 80 cm

Since 2 m = 2 \(\times\) 100 cm = 200 cm

\(= \frac{200 \text{ cm}}{80 \text{ cm}} = \frac{5}{2} = 5 : 2\)

The ratio of 1\(\frac{1}{2}\) years to 10 months

Since 1\(\frac{1}{2}\) years = \(\frac{3}{2} \times 12\) months = 18 months

\(= \frac{18 \text{ months}}{10 \text{ months}} = \frac{9}{5} = 9 : 5\)

A ratio is a pure number and so has no unit.

The ratio of two numbers or quantities is denoted by the colon mark ":". Thus, the ratio of two quantities p and q = p : q

The ratio of two quantities of same kind and in the same unit is obtained on dividing one quantity by the other.

Thus, the ratio of 20 kg to 80 kg = \(\frac{20 \text{ kg}}{80 \text{ kg}} = \frac{1}{4} = 1 : 4\)

The first term of a ratio is called the antecedent and the second term is called the consequent.

In the ratio 1 : 4, antecedent = 1 and consequent = 4.

A ratio must always by expressed in its lowest terms.

Whatever be the units of the terms of a ratio, the ratio has no unit. The ratio of 15 km and 20 km = \(\frac{15 \text{ km}}{20 \text{ km}} = \frac{3}{4} = 3:4\). Here, the two quantities 15 km and 20 km have unit km, but their ratio 3:4 has no unit. On dividing, units cancel out.

The terms of a ratio are written in a definite order:

The ratio of 5 kg and 8 kg = \(\frac{5 \text{ kg}}{8 \text{ kg}} = \frac{5}{8} = 5 : 8\) and

the ratio of 8 kg and 5 kg = \(\frac{8 \text{ kg}}{5 \text{ kg}} = \frac{8}{5} = 8 : 5\)

Remember: 5 : 8 and 8 : 5 are not equal to each other.

Example 1:

Find the ratio of: (i) 60 to 48 (ii) 3.75 kg to 750 gm

Solution:

Required ratio = \(\frac{60}{48} = \frac{5}{4}\)

= 5 : 4

Since 3.75 kg = 3.75 \(\times\) 1000 gm = 3750 gm

Required ratio = \(\frac{3.75 \text{ kg}}{750 \text{ gm}}\)

\(= \frac{3750 \text{ kg}}{750 \text{ gm}} = \frac{5}{1} = 5 : 1\)

8.3 Converting Into Simple Ratio

Example 2:

Express as simple ratio: (i) 3\(\frac{1}{2}\) : 2\(\frac{1}{3}\) (ii) \(\frac{2}{3}\) : \(\frac{4}{5}\) : \(\frac{1}{2}\)

Solution:

Divide the first term of the ratio by its second term and then simplify.

Given ratio = 3\(\frac{1}{2}\) : 2\(\frac{1}{3}\) = \(\frac{7}{2}\) : \(\frac{7}{3}\)

\(= \frac{7}{2} \times \frac{3}{7} = \frac{3}{2} = 3 : 2\)

Alternative method:

Multiply each terms of the ratio by the L.C.M. of their denominators and then simplify.

Given ratio = \(\frac{7}{2}\) : \(\frac{7}{3}\) = \(\frac{7}{2} \times 6 : \frac{7}{3} \times 6\)

L.C.M. of 2 and 3 = 6

= 21 : 14 = \(\frac{21}{14} = \frac{3}{2} = 3 : 2\)

Given ratio = \(\frac{2}{3}\) : \(\frac{4}{5}\) : \(\frac{1}{2}\) = \(\frac{2}{3} \times 30 : \frac{4}{5} \times 30 : \frac{1}{2} \times 30\)

L.C.M. of 3,5 and 2 = 30

= 20 : 24 : 15

Example 3:

The strength of a class is 50 with 30 boys and the remaining girls. Find the ratio of the number of boys to the number of girls in the class.

Solution:

Since the strength of the class = 50

and the number of boys in the class = 30

The number of girls in the class = 50 - 30 = 20

Required ratio = \(\frac{\text{No. of boys in the class}}{\text{No. of girls in the class}}\)

\(= \frac{30}{20} = \frac{3}{2} = 3 : 2\)

Example 4:

A man's monthly income is ₹ 15,000, out of which he spends ₹ 12,500 every month. Find the ratio of his:

(i) savings to expenditure (ii) expenditure to income

(iii) income to savings

Solution:

Since the monthly income of the man = ₹ 15,000

And his monthly expenditure = ₹ 12,500

His savings per month = ₹ 15,000 - ₹ 12,500 = ₹ 2,500

Ratio of savings to expenditure = \(\frac{₹ 2,500}{₹ 12,500} = \frac{1}{5} = 1 : 5\)

Ratio of expenditure to income = \(\frac{₹ 12,500}{₹ 15,000} = \frac{5}{6} = 5 : 6\)

Ratio of income to savings = \(\frac{₹ 15,000}{₹ 2,500} = \frac{6}{1} = 6 : 1\)

Teacher's Note

Understanding ratios helps us compare prices when shopping, such as determining which product offers better value per unit. This practical application makes ratio a fundamental skill in everyday financial decisions.

This is a preview of the first 3 pages. To get the complete book, click below.