The time taken by A alone to finish a piece of work is `60%` more than that taken by A and B together to finish the same piece of work. C is twice as efficient as B. If B and C together can complete the same piece of work in `13'1/3` days, in how many days can A alone finish the same piece of work?

Options

A. 32
B. 24
C. 16
D. 28

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let the time taken by A and B together to complete the work be \(X\) days.

### Step 2: Express A's Time
Since A takes 60% more time than A and B together, the time taken by A alone to complete the work is:
\[ A's \, time = X + 0.6X = 1.6X \]

### Step 3: Determine B and C's Efficiency
We know that B and C together can complete the work in \(13 \frac{1}{3}\) days, which is equivalent to \(\frac{40}{3}\) days. Therefore, their combined work rate is:
\[ \text{Work rate of B and C} = \frac{1}{\frac{40}{3}} = \frac{3}{40} \text{ work per day} \]

### Step 4: Set Up B and C's Individual Rates
Let the work rate of B be \(b\) and that of C be \(c\). Since C is twice as efficient as B, we have:
\[ c = 2b \]
Thus, the combined work rate of B and C can be expressed as:
\[ b + c = b + 2b = 3b \]
Setting this equal to the combined work rate we found:
\[ 3b = \frac{3}{40} \]
From this, we can solve for \(b\):
\[ b = \frac{1}{40} \text{ work per day} \]

### Step 5: Calculate C's Rate
Using \(b\) to find \(c\):
\[ c = 2b = 2 \times \frac{1}{40} = \frac{1}{20} \text{ work per day} \]

### Step 6: Find A and B's Combined Rate
Now, we can find the combined work rate of A and B. Since A's work rate is unknown, we denote it as \(a\). The combined work rate of A and B is:
\[ a + b = a + \frac{1}{40} \]

### Step 7: Relate A's Rate to X
Since A and B together complete the work in \(X\) days, their combined work rate is:
\[ \frac{1}{X} = a + \frac{1}{40} \]
From this, we can express \(a\):
\[ a = \frac{1}{X} - \frac{1}{40} \]

### Step 8: Substitute A's Time
Now substitute \(a\) into the equation for A's time:
\[ 1.6X = \frac{1}{a} = \frac{1}{\left(\frac{1}{X} - \frac{1}{40}\right)} \]
Taking the reciprocal:
\[ 1.6X = \frac{X \cdot 40}{40 - X} \]

### Step 9: Solve for X
Cross-multiplying gives:
\[ 1.6X(40 - X) = 40X \]
Expanding:
\[ 64X - 1.6X^2 = 40X \]
Rearranging:
\[ 1.6X^2 - 24X = 0 \]
Factoring out \(X\):
\[ X(1.6X - 24) = 0 \]
Thus, \(X = 0\) or \(X = 15\).

### Step 10: Calculate A's Time
Now substituting \(X = 15\) back into the equation for A's time:
\[ A's \, time = 1.6X = 1.6 \times 15 = 24 \text{ days} \]

### Final Answer
A can finish the work alone in **24 days**.

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