5 men can do a piece of work in 6 days while 10 women can do it in 5 days. In how many days can 5 women and 3 men do it?

Options

A. 4 days
B. 5 days
C. 6 days
D. 8 days

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Determine the total work done by men and women
We know that:
- 5 men can complete the work in 6 days.
- 10 women can complete the same work in 5 days.

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

**Total work in man-days**:
\[ \text{Total Work} = \text{Number of Men} \times \text{Days} = 5 \text{ men} \times 6 \text{ days} = 30 \text{ man-days} \]

**Total work in woman-days**:
\[ \text{Total Work} = \text{Number of Women} \times \text{Days} = 10 \text{ women} \times 5 \text{ days} = 50 \text{ woman-days} \]

### Step 2: Find the efficiency of men and women
We can equate the total work done by men and women since they both complete the same task.

From the above calculations:
- 30 man-days = 50 woman-days

Now, we can find the relationship between the efficiencies of men and women.

Let the efficiency of a man be \( m \) and the efficiency of a woman be \( w \).

\[ 5 \text{ men} \times 6 \text{ days} \times m = 10 \text{ women} \times 5 \text{ days} \times w \]
This simplifies to:
\[ 30m = 50w \]
Dividing both sides by 10 gives:
\[ 3m = 5w \quad \Rightarrow \quad \frac{m}{w} = \frac{5}{3} \]

### Step 3: Assign values to efficiencies
Let’s assign a value to the efficiency of a woman:
- Let \( w = 3 \) units/day (as derived from the ratio).

Then, the efficiency of a man will be:
\[ m = \frac{5}{3} \times 3 = 5 \text{ units/day} \]

### Step 4: Calculate the total work in units
Now, we can calculate the total work in units:
\[ \text{Total Work} = 30 \text{ man-days} \times 5 \text{ units/man-day} = 150 \text{ units} \]

### Step 5: Calculate the combined efficiency of 5 women and 3 men
Now, we need to find out how much work 5 women and 3 men can do in one day.

**Efficiency of 5 women**:
\[ \text{Efficiency of 5 women} = 5 \times 3 = 15 \text{ units/day} \]

**Efficiency of 3 men**:
\[ \text{Efficiency of 3 men} = 3 \times 5 = 15 \text{ units/day} \]

**Total efficiency of 5 women and 3 men**:
\[ \text{Total Efficiency} = 15 + 15 = 30 \text{ units/day} \]

### Step 6: Calculate the number of days to complete the work
Now, we can find out how many days it will take for 5 women and 3 men to complete the total work of 150 units.

\[ \text{Number of Days} = \frac{\text{Total Work}}{\text{Total Efficiency}} = \frac{150 \text{ units}}{30 \text{ units/day}} = 5 \text{ days} \]

### Final Answer:
Thus, 5 women and 3 men can complete the work in **5 days**.

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