12 men and 16 women together can complete a work in 4 days . It takes 80 days for one man alone to complete the same work , then how many days would be required for one woman alone to complete the same work ?

Options

A. 160
B. 150
C. 130
D. 175

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Work Done by Men and Women
We know that 12 men and 16 women can complete the work in 4 days.

### Step 2: Calculate Total Work in Man-Days
First, we need to convert the work done into man-days. Since it takes 80 days for one man to complete the work alone, the total work can be expressed in man-days as:
\[ \text{Total Work} = 1 \text{ man} \times 80 \text{ days} = 80 \text{ man-days} \]

### Step 3: Work Done by 12 Men in 4 Days
Now, let’s find out how much work 12 men can do in 4 days:
\[ \text{Work done by 12 men in 4 days} = 12 \text{ men} \times 4 \text{ days} = 48 \text{ man-days} \]

### Step 4: Work Done by 16 Women in 4 Days
Let’s denote the work done by one woman in one day as \( W \). Therefore, the work done by 16 women in 4 days is:
\[ \text{Work done by 16 women in 4 days} = 16 \text{ women} \times 4 \text{ days} = 64W \text{ woman-days} \]

### Step 5: Total Work Equation
According to the problem, the total work done by both men and women together in 4 days is equal to the total work in man-days:
\[ 48 \text{ man-days} + 64W \text{ woman-days} = 80 \text{ man-days} \]

### Step 6: Solve for Woman's Work
Now, we can solve for \( W \):
\[ 64W = 80 - 48 \]
\[ 64W = 32 \]
\[ W = \frac{32}{64} = \frac{1}{2} \text{ man-days} \]

### Step 7: Calculate Days Required for One Woman
If one woman does \( \frac{1}{2} \) of a man-day of work in one day, then the number of days required for one woman to complete the entire work (80 man-days) is:
\[ \text{Days required for one woman} = \frac{80 \text{ man-days}}{\frac{1}{2} \text{ man-days per day}} = 80 \times 2 = 160 \text{ days} \]

### Final Answer
Thus, one woman alone would require **160 days** to complete the same work.

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