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The question is from Time and Work: Making toys. Some details are given about A, B and C's efficiency. We need to find out A's efficiency to determine how many days he would take to complete 256 toys? Interesting Time and Work Question.

Question 23: A, B and C are to make 100 toys. In a day, they can together make 25 toys. A starts to work alone and makes 32 toys in some days. A then leaves and B and C works to make the remaining toys. It takes 8 days overall to make the 100 toys. How many days will it take for A to make 256 toys alone?

  1. 16 days
  2. 32 days
  3. 64 days
  4. 30 days

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Explanatory Answer

Method of solving this CAT Question from Pipes Cisterns and Work Time: How many days A took to complete 32 toys?

A, B and C together could make 100 toys in \\frac{100}{25}\\) = 4 days
A and (B+C)as a sub team totally worked for 8 days to finish the 100 toys.
Let's say that A worked for 'n' out of the total 8 days. That means, (B+C) worked for the remaining '8-n' days.

Let's say that A makes 'x' toys per day, and (B+C) together make 'y' toys per day.

4(x + y) = 100.
A, B and C complete 100 toys in 4 days
n * x + (8 - n) * y = 100.
A and (B+C)as a sub team totally worked for 8 days to finish the 100 toys.

n * x + (8 - n) * y = 4(x + y)
(4 - n) * y = (4 - n) * x

There are two scenarios where this equation is valid:
1) x = y (n can be anything)
That is A does the same number of toys in a day as (B+C) do in a day
This way no matter what the value of n is, the job gets completed in exactly 8(2 * 4) days.
Because now we are employing either A or (B+C) who are both equally efficient on any given day. while previously, A and (B+C), with equal efficiencies would have done the job in 4 days (half time of 8 days).

2) x ≠ y (n has to be 4)
A and (B+C) are not equally efficient, but if we make each of them work for exactly 4 days, it is the same as making all of them work together for 4 days. In both the cases we get equal amount of job done.

Instead of relying on the equations, please try to argue and convince yourself that these are the only two cases.

If it was the first case, That is A and (B+C) are of the same efficiency, then A would have made 50 toys in the span of 8 days.(working on some of the 8 days)
But we are told that A made 32 toys in the span of those 8 days.(working on some of the 8 days)
So, it has to be the second case, that is, A worked exactly for 4 days(of the total 8).
We know that he made 32 toys in this span.
Hence A made 32 toys in 4 days.

So, A can make 32 toys in 4 days => In a day, A can make = \\frac{32}{4}\\) = 8 toys/day
To make 256 toys, A will take = \\frac{256}{8}\\) = 32 days

The question is "How many days will it take for A to make 256 toys alone?"

Hence, the answer is "32 days".

Choice B is the correct answer.

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