If 9 men or 15 women can do a piece of work in 18 days working 9 hours a day. How many days will it take to complete a work twice as large with 6 men and 8 women working together 6 hours a day?

Options

A. 22 days
B. 27.5 days
C. 45 days
D. 22.5 days

Answer

To solve the problem step by step, we will first determine the work done by men and women, then calculate the total work required for the larger task, and finally find out how long it will take for 6 men and 8 women to complete that work.

### Step 1: Determine the efficiency of men and women
Given:
- 9 men can complete the work in 18 days working 9 hours a day.
- 15 women can complete the same work in the same time.

**Total work done by men:**
\[ \text{Total Work} = \text{Number of Men} \times \text{Days} \times \text{Hours per Day} \times \text{Efficiency of Men} \]
Let the efficiency of men be \( m \).
\[ \text{Total Work} = 9 \times 18 \times 9 \times m = 1458m \]

**Total work done by women:**
Let the efficiency of women be \( w \).
\[ \text{Total Work} = 15 \times 18 \times 9 \times w = 2430w \]

Since both expressions represent the same total work:
\[ 1458m = 2430w \]
Dividing both sides by 486 gives:
\[ 3m = 5w \quad \Rightarrow \quad \frac{m}{w} = \frac{5}{3} \]

### Step 2: Calculate total work for twice the amount
Since the work is now twice as large:
\[ \text{Total Work} = 2 \times 1458m = 2916m \]

### Step 3: Calculate the combined efficiency of 6 men and 8 women
Using the ratio of efficiencies:
- Efficiency of 6 men: \( 6m \)
- Efficiency of 8 women: \( 8w \)

Now substituting \( w \) in terms of \( m \):
\[ w = \frac{3m}{5} \quad \Rightarrow \quad 8w = 8 \times \frac{3m}{5} = \frac{24m}{5} \]

**Combined efficiency of 6 men and 8 women:**
\[ \text{Combined Efficiency} = 6m + \frac{24m}{5} = \frac{30m + 24m}{5} = \frac{54m}{5} \]

### Step 4: Calculate the number of days required to complete the work
Let \( D \) be the number of days required to complete the work with 6 men and 8 women working 6 hours a day.
\[ \text{Total Work} = \text{Combined Efficiency} \times \text{Days} \times \text{Hours per Day} \]
\[ 2916m = \left(\frac{54m}{5}\right) \times D \times 6 \]

Now simplifying:
\[ 2916m = \frac{324mD}{5} \]
Cancelling \( m \) from both sides (assuming \( m \neq 0 \)):
\[ 2916 = \frac{324D}{5} \]
Multiplying both sides by 5:
\[ 14580 = 324D \]
Dividing both sides by 324:
\[ D = \frac{14580}{324} = 45 \]

### Final Answer
The number of days required to complete the work is **45 days**.

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