ICSE Class 7 Maths Chapter 08 Ratio Proportion

Chapter 8: Ratio And Proportion

Including Sharing In A Ratio

8.1 Definition

A ratio is a relationship between two quantities of the same kind with same unit and is obtained on dividing first quantity by the second.

The symbol for ratio is ":" and it is put in between the two quantities to be compared.

Thus, the ratio between 15 kg and 20 kg = 15 kg : 20 kg = \(\frac{15}{20} = \frac{3}{4}\) = 3 : 4.

1. The two quantities must be of the same kind. Thus, there can be a ratio between ₹ 50 and ₹ 80, but there can be no ratio between ₹ 50 and 80 kg.

2. The ratio between 3 and 4 is written as 3 : 4 (read as 3 is to 4) or \(\frac{3}{4}\).

3. In the ratio 3 : 4, the first term (i.e., 3) is called antecedent and the second term (i.e., 4) is called consequent.

4. A ratio is a pure number.

5. In order to find the ratio between two quantities, both the quantities must be in the same unit, e.g., ratio between 30 cm and 2 metre = 30 cm : 200 cm [As, 2 metre = 200 cm] = \(\frac{30}{200} = \frac{3}{20}\) = 3 : 20

6. A ratio must always be expressed in its lowest terms in simplest form.

7. A ratio has no unit because it is simply a number.

8.2 To Convert A Fractional Ratio Into A Whole Number Ratio

Example 1:

Convert \(\frac{1}{3} : \frac{1}{4}\) into whole number ratio, i.e., to ratio in simple form.

Solution:

\(\frac{1}{3} : \frac{1}{4} = \frac{1}{3} \times \frac{4}{1}\) [Dividing 1st quantity by the 2nd]

= \(\frac{4}{3}\) = 4 : 3 (Ans.)

Alternative method:

Steps: 1. Find the L.C.M. of denominators 3 and 4 of the given ratio \(\frac{1}{3} : \frac{1}{4}\).

2. Multiply each term of the ratio by this L.C.M. and simplify.

If each term of a ratio is multiplied or divided by the same number (quantity), the ratio remains the same.

Thus, \(\frac{1}{3} : \frac{1}{4} = \frac{1}{3} \times 12 : \frac{1}{4} \times 12\) [Since, L.C.M. of 3 and 4 = 12]

= 4 : 3 (Ans.)

Teacher's Note

Ratios help us compare quantities in real life, like mixing paint colors or adjusting recipe ingredients to serve different numbers of people.

8.3 To Divide A Given Quantity Into A Given Ratio

Example 2:

20 sweets are distributed between A and B in the ratio 2 : 3. Find, how many does each get?

Solution:

A and B get sweets in the ratio 2 : 3 => If A gets 2 parts, then B gets 3 parts.

In other words, if we make (2 + 3) = 5 equal parts, then A should get 2 parts out of these 5 equal part

=> A gets = \(\frac{2}{5}\) of the total number of sweets = \(\frac{2}{5}\) of 20 = \(\frac{2}{5} \times 20\) = 8 sweets

Similarly, B gets 3 parts out of 5 equal parts

=> B gets = \(\frac{3}{5}\) of the total number of sweets = \(\frac{3}{5}\) of 20 = \(\frac{3}{5} \times 20\) = 12 sweets

Thus, A gets 8 sweets and B gets 12 sweets. (Ans.)

Direct method:

Since, the given ratio = 2 : 3 and 2 + 3 = 5

∴ A gets = \(\frac{2}{5}\) of the total no. of sweets = \(\frac{2}{5} \times 20\) sweets = 8 sweets (Ans.)

and, B gets = \(\frac{3}{5} \times 20\) sweets = 12 sweets (Ans.)

Example 3:

Divide ₹ 260 among A, B and C in the ratio \(\frac{1}{2} : \frac{1}{3} : \frac{1}{4}\).

Solution:

First of all convert the given ratio into its simple form.

Since, L.C.M. of denominators 2, 3 and 4 is 12.

∴ \(\frac{1}{2} : \frac{1}{3} : \frac{1}{4}\) = \(\frac{1}{2} \times 12 : \frac{1}{3} \times 12 : \frac{1}{4} \times 12\) = 6 : 4 : 3

And, 6 + 4 + 3 = 13.

∴ A's share = \(\frac{6}{13}\) of ₹ 260 = ₹ \(\frac{6}{13} \times 260\) = ₹ 120

B's share = \(\frac{4}{13}\) of ₹ 260 = ₹ \(\frac{4}{13} \times 260\) = ₹ 80

and C's share = \(\frac{3}{13}\) of ₹ 260 = ₹ \(\frac{3}{13} \times 260\) = ₹ 60

Thus, A gets ₹ 120, B gets ₹ 80 and C gets ₹ 60. (Ans.)

Teacher's Note

Dividing quantities in specific ratios is essential in business partnerships, where profits are shared based on investment amounts.

Example 4:

Two numbers are in the ratio 10 : 13. If the difference between the numbers is 48, find the numbers.

Solution:

Let the two numbers be 10 and 13.

∴ The difference between these numbers = 13 - 10 = 3

Applying unitary method:

When difference between the numbers = 3, 1st number = 10

=> " " " " = 1, 1st number = \(\frac{10}{3}\)

=> " " " " = 48, 1st number = \(\frac{10}{3} \times 48\) = 160

In the same way,

when difference between the numbers = 3, 2nd number = 13

=> " " " " = 48, 2nd number = \(\frac{13}{3} \times 48\) = 208

∴ Required numbers are 160 and 208 (Ans.)

Alternative method (algebraic method):

Since, the ratio between the required numbers = 10 : 13

Let the numbers be 10x and 13x.

∴ 13x - 10x = 48 => 3x = 48 and x = \(\frac{48}{3}\) = 16

∴ Required numbers = 10x and 13x = 10 × 16 and 13 × 16 = 160 and 208 (Ans.)

Teacher's Note

Understanding ratios helps in comparing ages of people or time durations in real-world scenarios like calculating the age difference between siblings.

Exercise 8(A)

1. Express each of the given ratio in its simplest form:

(i) 22 : 66 (ii) 1.5 : 2.5 (iii) 6\(\frac{1}{4}\) : 12\(\frac{1}{2}\)

(iv) 40 kg : 1 quintal (v) 10 paise : ₹ 1 (vi) 200 m : 5 km

(vii) 3 hours : 1 day (viii) 6 months : 1\(\frac{1}{3}\) years (ix) 1\(\frac{1}{4}\) : 2\(\frac{1}{4}\) : 2\(\frac{1}{2}\)

2. Divide 64 cm long string into two parts in the ratio 5 : 3.

3. ₹ 720 is divided between x and y in the ratio 4 : 5. How many rupees will each get?

4. The angles of a triangle are in the ratio 3 : 2 : 7. Find each angle.

5. A rectangular field is 100 m by 80 m. Find the ratio of:

(i) length to its breadth (ii) breadth to its perimeter.

6. The sum of three numbers, whose ratios are 3\(\frac{1}{3}\) : 4\(\frac{1}{5}\) : 6\(\frac{1}{8}\), is 4917. Find the numbers.

7. The ratio between two quantities is 3 : 4. If the first is ₹ 810, find the second.

8. Two numbers are in the ratio 5 : 7. Their difference is 10. Find the numbers.

9. Two numbers are in the ratio 10 : 11. Their sum is 168. Find the numbers.

10. A line is divided in two parts in the ratio 2.5 : 1.3. If the smaller one is 35.1 cm, find the length of the line.

11. In a class, the ratio of boys to the girls is 7 : 8. What part of the whole class are girls?

12. The population of a town is 180,000, out of which males are \(\frac{1}{3}\) of the whole population. Find the number of females. Also, find the ratio of the number of females to the whole population.

13. Ten gram of an alloy of metals A and B contains 7.5 gm of metal A and the rest is metal B. Find the ratio between:

(i) the weights of metals A and B in the alloy.

(ii) the weight of metal B and the weight of the alloy.

14. The ages of two boys A and B are 6 years 8 months and 7 years 4 months respectively. Divide ₹ 3,150 in the ratio of their ages.

15. Three persons start a business and spend ₹ 25,000; ₹ 15,000 and ₹ 40,000 respectively. Find the share of each out of a profit of ₹ 14,400 in a year.

16. A plot of land, 600 sq m in area, is divided between two persons such that the first person gets three-fifth of what the second gets. Find the share of each.

17. Two poles of different heights are standing vertically on a horizontal field. At a particular time, the ratio between the lengths of their shadows is 2 : 3. If the height of the smaller pole is 7.5 m, find the height of the other pole.

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