3 men or 4 women can do a piece of work in 43 days. In how many days 7 men & 5 women can do the same work ?

Answer

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

### Step 1: Understand the given information
We know that 3 men or 4 women can complete a piece of work in 43 days.

### Step 2: Find the work done by men and women
Let's denote the work done by 1 man in 1 day as \( M \) and the work done by 1 woman in 1 day as \( W \).

From the information given:
- The total work done by 3 men in 43 days is:
\[ \text{Work} = 3M \times 43 \]
- The total work done by 4 women in 43 days is:
\[ \text{Work} = 4W \times 43 \]

Since both expressions represent the same total work, we can equate them:
\[ 3M \times 43 = 4W \times 43 \]
Dividing both sides by 43:
\[ 3M = 4W \]

### Step 3: Express men in terms of women
From the equation \( 3M = 4W \), we can express \( M \) in terms of \( W \):
\[ M = \frac{4}{3}W \]

### Step 4: Calculate the total work done by 7 men and 5 women
Now, we need to find out how much work 7 men and 5 women can do in 1 day:
- Work done by 7 men:
\[ \text{Work by 7 men} = 7M = 7 \times \frac{4}{3}W = \frac{28}{3}W \]
- Work done by 5 women:
\[ \text{Work by 5 women} = 5W \]

Now, adding the work done by 7 men and 5 women:
\[ \text{Total work in 1 day} = \frac{28}{3}W + 5W = \frac{28}{3}W + \frac{15}{3}W = \frac{43}{3}W \]

### Step 5: Calculate the total work in terms of women
We know that 4 women can complete the work in 43 days. Therefore, the total work \( T \) can be calculated as:
\[ T = 4W \times 43 \]
So,
\[ T = 172W \]

### Step 6: Find the number of days taken by 7 men and 5 women to complete the work
Let \( x \) be the number of days taken by 7 men and 5 women to complete the work. The equation can be set up as:
\[ \text{Total work} = \text{Work done in 1 day} \times \text{Number of days} \]
Thus,
\[ 172W = \frac{43}{3}W \times x \]

### Step 7: Solve for \( x \)
Dividing both sides by \( W \) (assuming \( W \neq 0 \)):
\[ 172 = \frac{43}{3} \times x \]
Multiplying both sides by 3:
\[ 516 = 43x \]
Now, dividing both sides by 43:
\[ x = \frac{516}{43} = 12 \]

### Final Answer
Thus, 7 men and 5 women can complete the work in **12 days**.

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