When 6 men and 8 women can do a piece of work in 15 days, then 11 men and 16 women can do the same work in 8 days. In how many days can 4 men and 4 women do the work ?

Options

A. 28
B. 24
C. 26
D. 30

Explanation

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

### Step 1: Understand the given information
We have two scenarios:
1. 6 men and 8 women can complete the work in 15 days.
2. 11 men and 16 women can complete the same work in 8 days.

### Step 2: Set up the equations based on work done
Let the work done by 1 man in a day be \( M \) and the work done by 1 woman in a day be \( W \).

From the first scenario:
- Work done by 6 men and 8 women in 1 day = \( 6M + 8W \)
- Total work done in 15 days = \( 15(6M + 8W) \)

From the second scenario:
- Work done by 11 men and 16 women in 1 day = \( 11M + 16W \)
- Total work done in 8 days = \( 8(11M + 16W) \)

Since both scenarios complete the same work, we can set the equations equal to each other:
\[ 15(6M + 8W) = 8(11M + 16W) \]

### Step 3: Simplify the equation
Expanding both sides:
\[ 90M + 120W = 88M + 128W \]

Now, rearranging the equation:
\[ 90M - 88M = 128W - 120W \]
\[ 2M = 8W \]

### Step 4: Find the relationship between men and women
Dividing both sides by 2:
\[ M = 4W \]

This means that 1 man is equivalent to 4 women in terms of work done.

### Step 5: Calculate the work done by 4 men and 4 women
Now we need to find out how many days 4 men and 4 women will take to complete the work.

Substituting \( M = 4W \):
- Work done by 4 men = \( 4M = 4(4W) = 16W \)
- Work done by 4 women = \( 4W \)

Total work done by 4 men and 4 women in 1 day:
\[ 16W + 4W = 20W \]

### Step 6: Calculate total work in terms of W
We can express the total work \( W \) from either of the previous scenarios. Let's use the first scenario:
Total work = \( 15(6M + 8W) = 15(6(4W) + 8W) \)
\[ = 15(24W + 8W) = 15(32W) = 480W \]

### Step 7: Find the number of days taken by 4 men and 4 women
Let \( x \) be the number of days taken by 4 men and 4 women to complete the work:
\[ 20W \cdot x = 480W \]

Dividing both sides by \( W \):
\[ 20x = 480 \]
\[ x = \frac{480}{20} = 24 \]

Thus, **4 men and 4 women can complete the work in 24 days.**

### Summary of Steps:
1. Set up equations based on the work done.
2. Simplify the equations to find the relationship between men and women.
3. Calculate the total work done by the new group (4 men and 4 women).
4. Use the total work to find how many days they will take.