A man and a woman working together can do a certain work in 18 days. Their skills in doing the work are in the ratio 3:2. How many days will the woman take to finish the work alone?

Options

A. 45 days
B. 36 days
C. 27 days
D. 30 days

Answer

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

### Step 1: Understand the Work Done Together
The man and woman can complete the work together in 18 days. This means that their combined work rate is such that they complete 1/18 of the work in one day.

### Step 2: Determine the Ratio of Their Efficiencies
The skills of the man and woman are in the ratio of 3:2. Let's denote the efficiency of the man as 3x and the efficiency of the woman as 2x, where x is a constant.

### Step 3: Calculate Their Combined Efficiency
The combined efficiency of the man and woman can be calculated as:
\[ \text{Combined Efficiency} = \text{Efficiency of Man} + \text{Efficiency of Woman} = 3x + 2x = 5x \]

### Step 4: Relate Combined Efficiency to Work Done
Since they can complete the work in 18 days, their combined efficiency can also be expressed in terms of work done:
\[ \text{Total Work} = \text{Combined Efficiency} \times \text{Time} = 5x \times 18 \]
Thus, the total work done is:
\[ \text{Total Work} = 90x \]

### Step 5: Calculate the Woman's Efficiency
The efficiency of the woman is given as 2x.

### Step 6: Calculate the Time Taken by the Woman Alone
To find out how many days the woman will take to complete the work alone, we can use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of Woman}} = \frac{90x}{2x} \]
Simplifying this gives:
\[ \text{Time} = \frac{90}{2} = 45 \text{ days} \]

### Conclusion
The woman will take **45 days** to finish the work alone.

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