3 men and 5 women can do a work in 14 days while 5 men can do it in 14 days. 5 men and 5 women can complete the work in

Options

A. 13 days
B. 11 days
C. 10 days
D. 12 days

Answer

To solve the problem step by step, we need to determine how long it will take for 5 men and 5 women to complete the work. Let's break it down:

### Step 1: Determine the work done by 3 men and 5 women in 14 days.
Given that 3 men and 5 women can complete the work in 14 days, we can express their combined work rate.

Let the work done by 1 man in 1 day be \( M \) and the work done by 1 woman in 1 day be \( W \).

The total work done by 3 men and 5 women in 1 day is:
\[ 3M + 5W \]

Since they complete the work in 14 days, the total work \( W_{total} \) can be expressed as:
\[ W_{total} = (3M + 5W) \times 14 \]

### Step 2: Determine the work done by 5 men in 14 days.
It is given that 5 men can also complete the work in 14 days. Thus, the total work done by 5 men in 1 day is:
\[ 5M \]

So, the total work can also be expressed as:
\[ W_{total} = 5M \times 14 \]

### Step 3: Set the equations equal to each other.
Since both expressions represent the same total work, we can set them equal:
\[ (3M + 5W) \times 14 = 5M \times 14 \]

Dividing both sides by 14:
\[ 3M + 5W = 5M \]

### Step 4: Rearranging the equation.
Rearranging gives us:
\[ 5W = 5M - 3M \]
\[ 5W = 2M \]

### Step 5: Find the ratio of the efficiencies of men and women.
From \( 5W = 2M \), we can express the ratio of men to women:
\[ \frac{M}{W} = \frac{5}{2} \]

This means the efficiency of 1 man is equivalent to 2.5 times that of 1 woman.

### Step 6: Calculate the total work in units.
Now, we can calculate the total work done in terms of units. From the work done by 5 men in 14 days:
\[ W_{total} = 5M \times 14 = 70M \]

### Step 7: Substitute the value of \( M \) in terms of \( W \).
Using the ratio \( \frac{M}{W} = \frac{5}{2} \):
Let \( W = 2 \) (for simplicity), then \( M = 5 \).

Thus, substituting:
\[ W_{total} = 70M = 70 \times 5 = 350 \text{ units} \]

### Step 8: Calculate the efficiency of 5 men and 5 women.
Now, we find the efficiency of 5 men and 5 women:
- The efficiency of 5 men is \( 5 \times M = 5 \times 5 = 25 \)
- The efficiency of 5 women is \( 5 \times W = 5 \times 2 = 10 \)

So, total efficiency of 5 men and 5 women together:
\[ \text{Total Efficiency} = 25 + 10 = 35 \text{ units/day} \]

### Step 9: Calculate the time taken by 5 men and 5 women to complete the work.
Now, we can find the time taken to complete the total work of 350 units:
\[ \text{Time} = \frac{W_{total}}{\text{Total Efficiency}} = \frac{350}{35} = 10 \text{ days} \]

### Final Answer:
5 men and 5 women can complete the work in **10 days**.

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