A piece of work can be completed by 10 men and 6 women in 18 days. Men works 9 hours per day while women works 7.5 hours per day. Per hour efficiency of a woman is `(2)/(3)rd` of a man's efficiency. In how many days 10 men and 9 women complete the work?

Options

A. 16 days
B. 20 days
C. 30 days
D. 25 days

Answer

To solve the problem, we need to find out how many days 10 men and 9 women will take to complete the same piece of work. Let's break down the solution step by step.

### Step 1: Determine the Work Done by 10 Men and 6 Women in One Day

Given that 10 men and 6 women can complete the work in 18 days, we first calculate the total work done in terms of man-hours.

1. **Daily Work Calculation**:
- Men work for 9 hours a day.
- Women work for 7.5 hours a day.
- Let the efficiency of one man be \( M \).
- Then the efficiency of one woman is \( \frac{2}{3}M \).

2. **Total Work Done in One Day**:
- Work done by 10 men in one day:
\[ \text{Work by men} = 10 \times M \times 9 = 90M \]
- Work done by 6 women in one day:
\[ \text{Work by women} = 6 \times \left(\frac{2}{3}M\right) \times 7.5 = 6 \times \frac{2}{3}M \times 7.5 = 30M \]
- Total work done in one day by 10 men and 6 women:
\[ \text{Total Work per Day} = 90M + 30M = 120M \]

### Step 2: Calculate Total Work

Since they complete the work in 18 days, the total work \( W \) can be calculated as:
\[ W = \text{Total Work per Day} \times \text{Number of Days} = 120M \times 18 = 2160M \]

### Step 3: Determine the Work Done by 10 Men and 9 Women in One Day

Next, we calculate how much work 10 men and 9 women can do in one day.

1. **Work Done by 10 Men**:
- As calculated earlier, the work done by 10 men in one day is:
\[ 90M \]

2. **Work Done by 9 Women**:
- Work done by 9 women in one day:
\[ \text{Work by women} = 9 \times \left(\frac{2}{3}M\right) \times 7.5 = 9 \times \frac{2}{3}M \times 7.5 = 45M \]

3. **Total Work Done in One Day by 10 Men and 9 Women**:
\[ \text{Total Work per Day} = 90M + 45M = 135M \]

### Step 4: Calculate the Number of Days Required

Now, we can find out how many days \( D \) it will take for 10 men and 9 women to complete the total work \( W \):
\[ D = \frac{W}{\text{Total Work per Day}} = \frac{2160M}{135M} \]

### Step 5: Simplify the Calculation

Now, we simplify:
\[ D = \frac{2160}{135} = 16 \]

Thus, 10 men and 9 women will complete the work in **16 days**.

### Final Answer
The answer is **16 days**.

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