The question is from Time and work. Details about the efficiency of two people is given and we need to find out the time required to complete the task by one prson. Another one from CAT Time and Work for more practice.

Question 9: A and B together can finish a task in 12 days. If A worked half as efficiently as he usually does and B works thrice as efficiently as he usually does, the task gets completed in 9 days. How long would A take to finish the task if he worked independently?

- 12 days
- 24 days
- 27 days
- 18 days

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Let A take ‘a’ days to complete the task and B take ‘b’ days to complete the task.

Thus in one day, A will complete (1/a)th of the task.

Similarly in one day, B will complete (1/b)th of the task.

So in one day, if A and B work together they will complete (1/a + 1/b)^{th} of the task.

Given that A and B together take 12 days to complete the task, then in one day A and B together complete (1/12)th of the task.

Thus, 1/a + 1/b = 1/12 ……Eqn. 1

If A worked half as efficiently as he usually does, then A will take twice the time as he usually takes, i.e., 2a days. Thus in one day, A completes (1/2a)^{th} of the task.

Similarly if B worked thrice as efficiently as he usually does, then B will take one-third the time he usually takes, i.e., b/3 days. Thus in one day, B completed (1/(b⁄3))^{th} or (3/b)^{th} of the task.

Thus when both of them work together, they will complete (1/2a + 3/b)^{th} of the task, given that A and B take 9 days to complete the task.

Thus, 1/a + 1/b = 1/12 ……Eqn. 1

1/2a + 3/b = 1/9 …… Eqn. 2

1/2a + 3/b = 1/9 …… Eqn. 2

Solving Equations 1 and 2 for ‘a’ we should get the answer,

From equation (1) we get 12(a + b) = ab

From equation (2), we get 9(b + 6a) = 2ab

Substituting ab as 12(a + b) in equation (2) we get 9b + 54a = 2 x 12 x ( a + b)

9b + 54a = 24a + 24b;

Or, 30a = 15b,

Or, b = 2a

Now, 12(a + b) = ab, or 12 x 3a = 2a^{2}

a = 18 days.

The question is **"How long would A take to finish the task if he worked independently?"**

Choice D is the correct answer.

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