ICSE Class 8 Maths Chapter 07 Unitary Method

Unit 2: Commercial Arithmetic

Chapter 7: Unitary Method

(Including Time and Work)

7.1 Review

Teacher's Note

The unitary method is used daily when shopping - if 5 apples cost 50 rupees, we find the cost of 1 apple to determine the price of any other quantity.

7.2 Types of Variations

In unitary method, we come across two types of variations:

1. Direct variation - 2. Inverse variation

1. Direct Variation: Two quantities are said to have direct variation, if the increase in one quantity causes the increase in the other and, the decrease in one quantity causes the decrease in the other.

2. Inverse Variation: Two quantities are said to vary inversely, if by increasing one quantity, the second quantity decreases. And, if by decreasing one quantity, the second quantity increases.

Example 1

A fort had provisions for 300 men for 90 days. After 20 days, 50 men left the fort. How long would the food last at the same rate?

Solution

After 20 days:

For 300 men, provisions will last (90 - 20) days = 70 days

- For 1 man, the provisions will last 300 - 70 days

And, for (300 - 50) = 250 men, the provisions will last \(\frac{300 \times 70}{250}\) days

= 84 days (Ans.)

Teacher's Note

This inverse relationship appears in real logistics - when fewer soldiers need to be fed with the same food supply, it lasts longer.

Example 2

A hostel had provisions for 75 students for 30 days. After 6 days, 15 more students come to hostel. How long would the remaining provisions last at the same rate?

Solution

After 6 days:

For 75 students, provisions are sufficient for (30 - 6) days = 24 days

- For 1 student, the provisions are sufficient for (75 - 24) days

And, for 90 students, the provisions are sufficient for \(\left(\frac{75 \times 24}{90}\right)\) days

= 20 days (Ans.)

Teacher's Note

Hostel management uses this calculation when unexpected additional students arrive - the existing food supply gets stretched over more people and lasts fewer days.

Example 3

6 men or 8 women earn - 960 in one day.

Find: (i) one day's earning of a man.

(ii) one day's earning of a woman.

(iii) one day's earning of 4 men and 5 women.

Solution

(i) Since, one day's earning of 6 men = - 960

- One day's earning of a man = \(\frac{960}{6}\) = - 160 (Ans.)

(ii) Since, one day's earning of 8 women = - 960

- One day's earning of a woman = \(\frac{960}{8}\) = - 120 (Ans.)

(iii) One day's earning of 4 men and 5 women

= 4 - - 160 + 5 - - 120 = - 1,240 (Ans.)

Teacher's Note

Wage calculations in factories and farms often compare earnings of different types of workers to ensure fair compensation based on individual productivity rates.

While applying unitary method, arrange a statement in such a way that, whatever is asked to find in the question, is written at the end of the statement.

Example 4

2 men or 3 women can do a piece of work in 45 days. Find, in how many days will 6 men and 1 woman be able to complete the same work?

Solution

According to the amount of work done, in the same time, 2 men are equivalent to 3 women

i.e. 2 men - 3 women

1 man = \(\frac{3}{2}\) women

and, 6 men - \(\frac{3}{2}\) - 6 women - 9 women.

. 6 men + 1 woman = 9 women + 1 woman = 10 women

Since, 3 women can do the work in 45 days

- 1 woman will do the work in 45 - 3 days = 135 days

- 10 women will do the work in \(\frac{135}{10}\) days = \(13\frac{1}{2}\) days.

. 6 men and 1 woman will complete the work in \(13\frac{1}{2}\) days (Ans.)

Example 5

3 men and 4 boys can complete a certain amount of work in 28 days, whereas 4 men and 6 boys can complete the same work in 20 days.

Find: (i) according to the amount of work done, one man is equivalent to how many boys.

(ii) the number of days required by 7 men and 6 boys to complete the same work.

Solution

(i) In 28 days, the work can be completed by 3 men and 4 boys

. In 1 day, the work can be completed by 28(3 men + 4 boys)

i.e. by 84 men + 112 boys.

. In 20 days, the same work can be completed by 4 men + 6 boys

. In 1 day, the same work can be completed by 20(4 men + 6 boys)

i.e. by 80 men + 120 boys.

. According to the amount of work done,

84 men + 112 boys - 80 men + 120 boys

- 4 men - 8 boys and 1 man - 2 boys (Ans.)

(ii) Since, 3 men + 4 boys - 3 - 2 boys + 4 boys [1 man = 2 boys]

- 10 boys

And, 7 men + 6 boys - 7 - 2 boys + 6 boys

- 20 boys

Given, 3 men + 4 boys can complete the work in 28 days

i.e., 10 boys can complete the work in 28 days

- 1 boy will complete the same work in 28 - 10 days = 280 days

- 20 boys will complete the same work in \(\frac{280}{20}\) days = 14 days

- 7 men and 6 boys require 14 days to complete the same work (Ans.)

Teacher's Note

Construction projects use this method to determine if adding more workers of different efficiency levels can complete work faster than originally planned.

Test Yourself

1. If 18 identical articles cost - 1,530

- Cost of 1 article = ................... .

And, cost of 13 articles = ................... = ..........

2. If for - 80, the number of pencils bought = 50

- for - 1, the number of pencils bought = ............... = ..............

And, for - 96, the number of pencils bought = ................... = ........... .

3. If 4 men can do a piece of work in 30 days

- 1 man will do the same work in ............... days = ........... days

And, 10 men will do the same work in ........... days = ........... days.

4. If a certain quantity of food is sufficient for 50 men for 20 days

- the same quantity of food is sufficient for 1 man for ........... days = ........... days

And, the same quantity of food is sufficient for 40 men for ........... days = ........... days

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