C is `50%` more efficient than A. A and B together can finish a piece of work in 18 days. B and C together can do it in `13 1/2`days. In how many days A alone can finish the. same piece of work?

Options

A. 27
B. 33
C. 18
D. 25

Answer

To solve the problem step by step, we will denote the efficiencies and work done by A, B, and C, and then derive the time taken by A to complete the work alone.

### Step 1: Define the efficiencies
Let the efficiency of A be \( x \). Since C is 50% more efficient than A, the efficiency of C will be:
\[ \text{Efficiency of C} = x + 0.5x = \frac{3x}{2} \]

### Step 2: Work done by A and B together
A and B together can finish the work in 18 days. Therefore, their combined work done in one day is:
\[ \text{Work done by A and B in 1 day} = \frac{1}{18} \]
The work done by A in one day is:
\[ \text{Work done by A in 1 day} = \frac{x}{x} = \frac{1}{x} \]
Let the efficiency of B be \( y \). Thus, the work done by B in one day is:
\[ \text{Work done by B in 1 day} = \frac{1}{y} \]
So we have:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{18} \quad \text{(1)} \]

### Step 3: Work done by B and C together
B and C together can finish the work in \( 13 \frac{1}{2} \) days, which is \( \frac{27}{2} \) days. Therefore, their combined work done in one day is:
\[ \text{Work done by B and C in 1 day} = \frac{2}{27} \]
Thus, we have:
\[ \frac{1}{y} + \frac{3}{2x} = \frac{2}{27} \quad \text{(2)} \]

### Step 4: Solve the equations
From equation (1):
\[ \frac{1}{y} = \frac{1}{18} - \frac{1}{x} \]
Substituting this into equation (2):
\[ \left(\frac{1}{18} - \frac{1}{x}\right) + \frac{3}{2x} = \frac{2}{27} \]
Combining the terms gives:
\[ \frac{1}{18} + \frac{3}{2x} - \frac{1}{x} = \frac{2}{27} \]
This simplifies to:
\[ \frac{1}{18} + \frac{3 - 2}{2x} = \frac{2}{27} \]
\[ \frac{1}{18} + \frac{1}{2x} = \frac{2}{27} \]

### Step 5: Find a common denominator
The common denominator for \( 18 \) and \( 27 \) is \( 54 \):
\[ \frac{3}{54} + \frac{27}{54x} = \frac{4}{54} \]
This simplifies to:
\[ 3 + \frac{27}{x} = 4 \]
\[ \frac{27}{x} = 1 \quad \Rightarrow \quad x = 27 \]

### Step 6: Calculate the time taken by A
Now we can find the time taken by A to finish the work alone:
\[ \text{Time taken by A} = \frac{Total \, Work}{Efficiency \, of \, A} = \frac{1}{\frac{1}{27}} = 27 \, days \]

### Final Answer
A alone can finish the work in **27 days**.