X and Y together can finish a piece of work in 15 days, while Y alone can finish it in 40 days. X alone can finish the work in:

Options

A. 24 days
B. 26 days
C. 25 days
D. 23 days

Answer

To solve the problem step by step, we will use the information given about the work done by X and Y.

### Step 1: Understand the work done by X and Y together
X and Y together can finish the work in 15 days. This means that their combined work rate (efficiency) is:
\[ \text{Efficiency of (X + Y)} = \frac{1}{15} \text{ work/day} \]

### Step 2: Determine Y's efficiency
Y alone can finish the work in 40 days. Therefore, Y's efficiency is:
\[ \text{Efficiency of Y} = \frac{1}{40} \text{ work/day} \]

### Step 3: Calculate X's efficiency
Using the combined efficiency of X and Y, we can find X's efficiency. We know:
\[ \text{Efficiency of (X + Y)} = \text{Efficiency of X} + \text{Efficiency of Y} \]
Let the efficiency of X be \( E_X \). Then:
\[ \frac{1}{15} = E_X + \frac{1}{40} \]

### Step 4: Solve for X's efficiency
To solve for \( E_X \), we first need a common denominator for the fractions. The least common multiple of 15 and 40 is 120. Thus, we can rewrite the equation:
\[ \frac{1}{15} = \frac{8}{120}, \quad \frac{1}{40} = \frac{3}{120} \]
Substituting these values into the equation gives us:
\[ \frac{8}{120} = E_X + \frac{3}{120} \]
Now, subtract \( \frac{3}{120} \) from both sides:
\[ E_X = \frac{8}{120} - \frac{3}{120} = \frac{5}{120} \]
Simplifying gives:
\[ E_X = \frac{1}{24} \text{ work/day} \]

### Step 5: Calculate the time taken by X to finish the work alone
Now that we have X's efficiency, we can find out how long it takes for X to finish the work alone. The time taken by X to complete the work is the reciprocal of X's efficiency:
\[ \text{Time taken by X} = \frac{1}{E_X} = \frac{1}{\frac{1}{24}} = 24 \text{ days} \]

### Conclusion
Thus, X alone can finish the work in **24 days**.

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