ICSE Class 7 Maths Chapter 06 Unitary Method

Unit 2: Commercial Arithmetic

Chapter 6: Unitary Method

Including Time and Work

6.1 Basic Concept

Consider the following examples:

1. If the cost of 15 m cloth = ₹ 300

The cost of 1 m cloth = \(\frac{₹ 300}{15} = ₹ 20\)

and, the cost of 10 m cloth = 10 × ₹ 20 = ₹ 200

2. If 15 men can do some work in 300 days

1 man will do the same work in 300 × 15 days = 4500 days

and, 10 men will do it in \(\frac{4500}{10}\) days = 450 days

In part 1, given above, the cost of 1 m cloth is first obtained from the given cost of 15 m and then the cost of 10 m is found.

In the same way; in part 2, given above, the number of days taken by 1 man is first calculated from the number of days by 15 men and then the number of days taken by 10 men is obtained.

The method in which the value of a unit quantity is first calculated to get the value of any quantity is called the unitary method.

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

Teacher's Note

Understanding the unitary method helps in everyday situations like calculating the price per unit when shopping or determining how long a task will take with more workers.

6.2 Examples of Direct Variation

Example 1:

A man earns ₹ 400 in 10 days. How much will he earn in 28 days?

Solution:

In 10 days, the man earns = ₹ 400

In 1 day, he will earn = \(\frac{₹ 400}{10} = ₹ 40\)

Less money is earned in 1 day, so divide

In 28 days, he will earn = 28 × ₹ 40 = ₹ 1,120 (Ans.)

Note: For solving problems (using unitary method), the sentences (statements) should be framed in such a way that, whatever is to be found is written at the end of the statement.

Arrow method:

Steps:

1. Form two columns, one heading earnings and the other heading no. of days (as shown alongside). [The quantity to be obtained must be written at the extreme right column. Here, quantity to be obtained is earnings of 28 days].

2. Let earnings of 28 days be ₹ x. Write the no. of days and corresponding earnings as shown alongside.

3. For the column on the extreme right, mark an arrow in the downward direction.

4. If it is the case of direct variation, the arrow for the column (headed: no. of days) must be in the same direction as that for earnings.

In the case of inverse variation, the arrow for this column must be in the direction opposite to the direction of the first arrow.

Since, here we have the case of direct variation, therefore, for both the columns arrows must be in the same direction.

5. Now according to the arrows marked, take:

\(\frac{\text{value on the head}}{\text{value on the tail}}\) for one arrow = \(\frac{\text{value on the head}}{\text{value on the tail}}\) for the other arrow.

Thus, we get:

\(\frac{₹ x}{₹ 400} = \frac{28}{10}\) => x = \(\frac{28}{10} × 400 = 1,120\)

28 men will earn = ₹ 1,120 (Ans.)

Teacher's Note

The arrow method provides a visual way to organize information, making it easier to see the relationship between quantities and avoid calculation errors.

Example 2:

0-75 metre cloth costs ₹ 45. What will be the cost of 0-6 metre of same cloth?

Solution:

Given: the cost of 0-75 m cloth = ₹ 45

The cost of 1 m cloth = \(\frac{₹ 45}{0.75} = ₹ 60\)

And, the cost of 0-6 m cloth = 0-6 × ₹ 60 = ₹ 36 (Ans.)

Arrow method:

=> \(\frac{x}{45} = \frac{0-6}{0.75}\)

=> x = \(\frac{0-6}{0.75} × 45 = 36\)

Cost of 0-6 m cloth = ₹ 36 (Ans.)

6.3 Examples of Inverse Variation

Example 3:

4 men can do a piece of work in 5 days. How many men will do it in 4 days?

Solution:

In 5 days, the work is done by 4 men

In 1 day, the work will be done by 4 × 5 = 20 men [More number of men are required to do the work in 1 day, so multiply]

In 4 days, the work will be done by \(\frac{20}{4}\) men = 5 men (Ans.)

Arrow method:

=> \(\frac{x}{4} = \frac{5}{4}\) and x = 5

5 men will do the work in 4 days (Ans.)

Example 4:

With a speed of 60 km/h, it takes 4 hours to cover a certain distance. What should be the speed, if the same journey is to be completed in 3 hours?

Solution:

To cover a certain distance in 4 hours; speed required = 60 km/h

To cover the same distance in 1 hour; speed required = 60 km/h × 4 = 240 km/h

And, to cover the same distance in 3 hours; speed required = \(\frac{240}{3}\) km/h = 80 km/h (Ans.)

Arrow method:

=> \(\frac{x}{60} = \frac{4}{3}\)

=> x = \(\frac{4}{3} × 60 = 80\)

Required speed = 80 km h-1 (Ans.)

Remember: (i) For getting more, multiply. (ii) For getting less, divide.

Teacher's Note

Inverse variation appears in real life when faster speeds require less time, or when more workers complete a job in fewer days.

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