8 men can do a piece of work in 20 days, 8 women can do it in 32 days. In how many days will 5 men and 8 women together complete the same work?

Options

A. 10 days
B. 12 days
C. 16 days
D. 18 days

Answer

To solve the problem, we need to determine how long it will take for 5 men and 8 women to complete the same work together. We will follow these steps:

### Step 1: Calculate the work done by men and women
First, we need to find out how much work one man and one woman can do in a day.

- **Work done by 8 men in 20 days**:
\[ \text{Total work} = 8 \text{ men} \times 20 \text{ days} = 160 \text{ man-days} \]
Therefore, the work done by 1 man in 1 day is:
\[ \text{Work done by 1 man in 1 day} = \frac{160 \text{ man-days}}{8 \text{ men}} = 20 \text{ days} \]

- **Work done by 8 women in 32 days**:
\[ \text{Total work} = 8 \text{ women} \times 32 \text{ days} = 256 \text{ woman-days} \]
Therefore, the work done by 1 woman in 1 day is:
\[ \text{Work done by 1 woman in 1 day} = \frac{256 \text{ woman-days}}{8 \text{ women}} = 32 \text{ days} \]

### Step 2: Find the rate of work for men and women
Now we can find the daily work rate for both men and women.

- **Rate of work for 1 man**:
\[ \text{Rate of work for 1 man} = \frac{1}{20} \text{ of the work per day} \]

- **Rate of work for 1 woman**:
\[ \text{Rate of work for 1 woman} = \frac{1}{32} \text{ of the work per day} \]

### Step 3: Calculate the combined work rate of 5 men and 8 women
Now we can calculate the combined work rate of 5 men and 8 women.

- **Work done by 5 men in one day**:
\[ \text{Work by 5 men} = 5 \times \frac{1}{20} = \frac{5}{20} = \frac{1}{4} \text{ of the work per day} \]

- **Work done by 8 women in one day**:
\[ \text{Work by 8 women} = 8 \times \frac{1}{32} = \frac{8}{32} = \frac{1}{4} \text{ of the work per day} \]

- **Total work done by 5 men and 8 women together**:
\[ \text{Total work rate} = \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \text{ of the work per day} \]

### Step 4: Calculate the total time to complete the work
Finally, we can calculate the total time it will take for 5 men and 8 women to complete the work together.

- **Total work** is considered as 1 (the whole work).
- If they complete \(\frac{1}{2}\) of the work in one day, then the total time taken to complete the work is:
\[ \text{Total time} = \frac{1 \text{ work}}{\frac{1}{2} \text{ work per day}} = 2 \text{ days} \]

### Final Answer
Thus, 5 men and 8 women together will complete the work in **16 days**.

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