Question

2 men and 5 boys together can do a piece of work in 4 days while 1 man and 2 boys together can do the same work 9 days. Find the time taken by : (a) one man alone, (b) one boy alone to complete the work.

Answer 1

Given:

• 2 men and 5 boys working together can finish the work in 4 days.
• 1 man and 2 boys working together can finish the same work in 9 days.

Let:

• Time taken by 1 man alone to complete the work = (x) days
• Time taken by 1 boy alone to complete the work = (y) days

Step-wise calculation:

Step 1: Express the one day work of man and boy.

One man’s one day work = `1/x`

One boy’s one day work = `1/y`

Step 2: Write equations based on the given conditions.

Work done by 2 men and 5 boys in one day = `1/4`

⇒ `2 xx 1/x + 5 xx 1/y = 1/4`

Work done by 1 man and 2 boys in one day = `1/9`

⇒ `1/x + 2 xx 1/y = 1/9`

Step 3: Set the equations for ease of solving.

`2/x + 5/y = 1/4`   ...(1)

`1/x + 2/y = 1/9`   ...(2)

Step 4: Multiply equation (2) by 2 to align with equation (1).

`2/x + 4/y = 2/9`   ...(3)

Step 5: Subtract equation (3) from equation (1) to eliminate `2/x`. 

`(2/x + 5/y) - (2/x + 4/y) = 1/4 - 2/9`

⇒ `1/y = 1/4 - 2/9`

⇒ `1/y = 9/36 - 8/36`

⇒ `1/y = 1/36`

⇒ y = 36

Step 6: Substitute (y = 36) into equation (2). 

`1/x + 2/36 = 1/9`

⇒ `1/x + 1/18 = 1/9`

⇒ `1/x = 1/9 - 1/18`

⇒ `1/x = 2/18 - 1/18`

⇒ `1/x = 1/18`

⇒ x = 18

(a) One man alone can complete the work in 18 days.

(b) One boy alone can complete the work in 36 days.

Hence, the time taken by one man alone is 18 days and by one boy alone is 36 days to complete the work.