A takes thrice as long to do a piece of work as B takes. A and B together can finish a piece of work in 15 days. A alone can do it in-

Options

A. 30 days
B. 45 days
C. 60 days
D. 120 days

Answer

To solve the problem, we will define the time taken by B to complete the work and then express the time taken by A in terms of B. We will then use the information about their combined work rate to find the time taken by A alone.

### Step-by-Step Solution:

1. **Define the time taken by B**:
Let the time taken by B to complete the work be \( x \) days.

2. **Define the time taken by A**:
Since A takes thrice as long as B, the time taken by A will be \( 3x \) days.

3. **Calculate the work done by A and B**:
The work done by A in one day is \( \frac{1}{3x} \) (since A takes \( 3x \) days to complete the work).
The work done by B in one day is \( \frac{1}{x} \).

4. **Combine their work rates**:
Together, A and B can complete the work in 15 days. Therefore, their combined work rate is:
\[ \frac{1}{3x} + \frac{1}{x} = \frac{1}{15} \]

5. **Find a common denominator and simplify**:
The common denominator for \( 3x \) and \( x \) is \( 3x \):
\[ \frac{1}{3x} + \frac{3}{3x} = \frac{1 + 3}{3x} = \frac{4}{3x} \]
Thus, we have:
\[ \frac{4}{3x} = \frac{1}{15} \]

6. **Cross-multiply to solve for \( x \)**:
Cross-multiplying gives:
\[ 4 \cdot 15 = 3x \implies 60 = 3x \implies x = 20 \]

7. **Calculate the time taken by A alone**:
Since \( x = 20 \), the time taken by A is:
\[ 3x = 3 \cdot 20 = 60 \text{ days} \]

### Final Answer:
A alone can complete the work in **60 days**.