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?

- 16 days
- 32 days
- 64 days
- 30 days

32 days

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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.

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?"**

Choice B is the correct answer.

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