If X and Y can complete a piece of work in 15 days, and X alone can do the same work in 30 days, find how many days will Y alone take to complete the same work?

Options

A. 60 days
B. 45 days
C. 30 days
D. 15 days

Answer

To solve the problem, we need to determine how many days Y alone will take to complete the work, given that X and Y together can complete it in 15 days and X alone can complete it in 30 days.

### Step-by-Step Solution:

1. **Calculate the work done by X and Y together in one day:**
If X and Y together can complete the work in 15 days, then the work done by both in one day is:
\[ \text{Work done by (X + Y) in 1 day} = \frac{1}{15} \text{ of the work} \]

2. **Calculate the work done by X alone in one day:**
If X alone can complete the work in 30 days, then the work done by X in one day is:
\[ \text{Work done by X in 1 day} = \frac{1}{30} \text{ of the work} \]

3. **Calculate the work done by Y alone in one day:**
To find the work done by Y alone in one day, we can use the equation:
\[ \text{Work done by Y in 1 day} = \text{Work done by (X + Y) in 1 day} - \text{Work done by X in 1 day} \]
Substituting the values we calculated:
\[ \text{Work done by Y in 1 day} = \frac{1}{15} - \frac{1}{30} \]

4. **Finding a common denominator:**
The common denominator for 15 and 30 is 30. We can rewrite \(\frac{1}{15}\) as \(\frac{2}{30}\):
\[ \text{Work done by Y in 1 day} = \frac{2}{30} - \frac{1}{30} = \frac{1}{30} \]

5. **Calculate the number of days Y will take to complete the work alone:**
If Y does \(\frac{1}{30}\) of the work in one day, then Y will take:
\[ \text{Days taken by Y} = \frac{1}{\text{Work done by Y in 1 day}} = \frac{1}{\frac{1}{30}} = 30 \text{ days} \]

### Conclusion:
Y alone will take **30 days** to complete the work.