A and B together can do a piece of work in 15 days. If one day's work of A be `1 (1)/(2)` times one day's work of B, find how many days will each take to finish the work alone?

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given.

### Step 1: Define Variables
Let:
- \( x \) = number of days A takes to complete the work alone.
- \( y \) = number of days B takes to complete the work alone.

### Step 2: Express One Day's Work
The work done by A in one day is \( \frac{1}{x} \) and the work done by B in one day is \( \frac{1}{y} \).

### Step 3: Set Up the First Equation
According to the problem, one day's work of A is \( \frac{3}{2} \) times one day's work of B. Therefore, we can write:
\[ \frac{1}{x} = \frac{3}{2} \cdot \frac{1}{y} \]
This can be rearranged to:
\[ y = \frac{3}{2}x \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Equation
A and B together can complete the work in 15 days. Therefore, their combined one day's work is:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{15} \]
Substituting \( y \) from Equation 1 into this equation gives:
\[ \frac{1}{x} + \frac{1}{\frac{3}{2}x} = \frac{1}{15} \]
Simplifying \( \frac{1}{\frac{3}{2}x} \) gives \( \frac{2}{3x} \), so we have:
\[ \frac{1}{x} + \frac{2}{3x} = \frac{1}{15} \]
Combining the fractions on the left side:
\[ \frac{3}{3x} + \frac{2}{3x} = \frac{1}{15} \]
This simplifies to:
\[ \frac{5}{3x} = \frac{1}{15} \]

### Step 5: Solve for \( x \)
Cross-multiplying gives:
\[ 5 \cdot 15 = 3x \]
\[ 75 = 3x \]
Dividing both sides by 3:
\[ x = 25 \]

### Step 6: Find \( y \)
Now substitute \( x \) back into Equation 1 to find \( y \):
\[ y = \frac{3}{2} \cdot 25 = \frac{75}{2} = 37.5 \]

### Conclusion
Thus, A takes 25 days to finish the work alone, and B takes 37.5 days to finish the work alone.

### Final Answers:
- A's time to complete the work alone: **25 days**
- B's time to complete the work alone: **37.5 days**