In how much time can A and B together complete the entire work?

Given:

1. A and B work alternately starting with A and finish the work in 16 days.
2. The ratio of efficiencies of A and B is 2 : 1.
3. C alone can complete the work in 36 days, and A and C together can complete it in 9 days.

Which of the following statements or combinations of statements are sufficient to find the time taken by A and B together to complete the work?

Question

In how much time can A and B together complete the entire work?

Given:

1. A and B work alternately starting with A and finish the work in 16 days.
2. The ratio of efficiencies of A and B is 2 : 1.
3. C alone can complete the work in 36 days, and A and C together can complete it in 9 days.

Which of the following statements or combinations of statements are sufficient to find the time taken by A and B together to complete the work?

Accepted Answer

**Solution:**

1. From statement 1, A and B work alternately starting with A and complete the work in 16 days. This gives a relation involving their work rates but does not directly give the combined rate if they work together simultaneously.

2. Statement 2 states the ratio of efficiencies of A and B is 2 : 1. This means if A's efficiency is 2x, B's efficiency is x.

3. Using statements 1 and 2 together:
   - Let A's efficiency be 2 units/day and B's efficiency be 1 unit/day.
   - When working alternately starting with A, the work done in 2 days = 2 (A's work) + 1 (B's work) = 3 units.
   - Total work = 3 units per 2 days.
   - Given total time = 16 days, so total work = (3/2) × 16 = 24 units.
   - Therefore, combined efficiency of A and B working together = 2 + 1 = 3 units/day.
   - Time taken by A and B together = Total work / Combined efficiency = 24 / 3 = 8 days.

4. Statement 3 involves C's work and A and C together but does not directly help find A and B's combined time without statement 2.

5. Statements 2 and 3 together do not provide enough information about B's efficiency or combined work of A and B.

6. Hence, only statements 1 and 2 together are sufficient to find the time taken by A and B working together.

**Answer:** (a) Statements 1 and 2 together

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