12 men can complete a work in 16 days and 16 women can complete the same work in 12 days In how many days will 4 men and 8 women complete that work ?

Options

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

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the work done by men and women
We know that:
- 12 men can complete the work in 16 days.
- 16 women can complete the same work in 12 days.

First, we calculate the total work done in terms of "man-days" and "woman-days".

**Work done by men:**
Total work = Number of men × Number of days
= 12 men × 16 days = 192 man-days

**Work done by women:**
Total work = Number of women × Number of days
= 16 women × 12 days = 192 woman-days

### Step 2: Find the work rate of one man and one woman
Now, we can find the work done by one man and one woman in one day.

**Work done by one man in one day:**
1 man can complete = 1/192 of the work in one day.

**Work done by one woman in one day:**
1 woman can complete = 1/192 of the work in one day.

### Step 3: Calculate the combined work rate of 4 men and 8 women
Now, we need to find out how much work 4 men and 8 women can do in one day.

**Work done by 4 men in one day:**
= 4 × (1/192) = 4/192 = 1/48 of the work.

**Work done by 8 women in one day:**
= 8 × (1/192) = 8/192 = 1/24 of the work.

### Step 4: Combine the work rates
Now, we add the work done by 4 men and 8 women in one day:

Total work done in one day by 4 men and 8 women:
= (1/48) + (1/24)

To add these fractions, we need a common denominator. The least common multiple of 48 and 24 is 48.

So, we convert (1/24) to have a denominator of 48:
1/24 = 2/48

Now, we can add:
= (1/48) + (2/48) = 3/48 = 1/16 of the work.

### Step 5: Calculate the total time taken to complete the work
If 4 men and 8 women can complete 1/16 of the work in one day, then the total time taken to complete the entire work is the reciprocal of the work done in one day.

Total days required = 1 / (1/16) = 16 days.

### Final Answer
Therefore, 4 men and 8 women will complete the work in **16 days**.

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