2 men and 3 women together or 4 men together can complete a piece of work in 20 days.3 men and 3 women will complete the same work in :

Options

A. 12 days
B. 16 days
C. 18 days
D. 19 days

Answer

To solve the problem step by step, we will first determine the work done by men and women, then find out how long it will take for 3 men and 3 women to complete the same work.

### Step 1: Determine the total work done by 2 men and 3 women or 4 men in 20 days.
- Let the work done by 1 man in 1 day be \( M \) and the work done by 1 woman in 1 day be \( W \).
- According to the problem, \( 2M + 3W \) (work done by 2 men and 3 women) can complete the work in 20 days. Therefore, the total work \( W_{total} \) can be calculated as:
\[ W_{total} = (2M + 3W) \times 20 \]

- Similarly, \( 4M \) (work done by 4 men) can also complete the work in 20 days:
\[ W_{total} = 4M \times 20 \]

### Step 2: Set the equations equal to each other.
Since both expressions equal the total work, we can set them equal:
\[ (2M + 3W) \times 20 = 4M \times 20 \]
Dividing both sides by 20, we get:
\[ 2M + 3W = 4M \]

### Step 3: Rearranging the equation.
Rearranging the equation gives:
\[ 3W = 4M - 2M \]
\[ 3W = 2M \]
From this, we can express the relationship between men and women:
\[ \frac{M}{W} = \frac{3}{2} \]

### Step 4: Calculate the total work in units.
We can express the total work in terms of \( M \):
- From \( 4M \times 20 \):
\[ W_{total} = 4M \times 20 = 80M \]

### Step 5: Calculate the work done by 3 men and 3 women in one day.
Now, we need to find out how much work 3 men and 3 women can do in one day:
\[ \text{Work done by 3 men} = 3M \]
\[ \text{Work done by 3 women} = 3W \]
Substituting \( W \) in terms of \( M \):
\[ 3W = 3 \times \frac{2M}{3} = 2M \]
Thus, the total work done by 3 men and 3 women in one day is:
\[ 3M + 2M = 5M \]

### Step 6: Find the number of days to complete the total work.
Now, we need to find out how many days it will take for 3 men and 3 women to complete the total work:
\[ \text{Number of days} = \frac{W_{total}}{\text{Work done in one day}} = \frac{80M}{5M} = 16 \text{ days} \]

### Final Answer:
Thus, 3 men and 3 women will complete the work in **16 days**.

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