In how many days can ten men complete a work?

Given:

I. Four men and six women can complete the same work in 20 days.

II. The number of days taken by two men to complete the work is the same as the number of days taken by three women to complete the same work.

Determine which of the statements is/are sufficient to answer the question.

Question

In how many days can ten men complete a work?

Given:

I. Four men and six women can complete the same work in 20 days.

II. The number of days taken by two men to complete the work is the same as the number of days taken by three women to complete the same work.

Determine which of the statements is/are sufficient to answer the question.

Accepted Answer

**Solution:**

1. Let the work rates of one man and one woman be *M* and *W* units per day respectively.

2. From statement I:
   - Four men and six women complete the work in 20 days.
   - So, total work = (4M + 6W) × 20.
   - This gives the total work in terms of *M* and *W* but does not provide their individual rates.

3. From statement II:
   - Two men take the same time as three women to complete the work.
   - This implies their rates relate as: 2M = 3W or M = $\frac{3}{2}W$.

4. Using both statements together:
   - Substitute M =$ \frac{3}{2}$W into total work from statement I:
     Total work = (4 × $\frac{3}{2}$W + 6W) × 20 = (6W + 6W) × 20 = 12W × 20 = 240W.

5. Now, to find the time taken by 10 men:
   - Work rate of 10 men = 10M = 10 × $\frac{3}{2}W$ = 15W.
   - Time taken = Total work / rate = $\frac{240W}{15W}$ = 16 days.

6. Hence, neither statement alone is sufficient, but together they allow us to find the exact time.

**Answer:** Both statements together are necessary, but neither alone is sufficient.

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