A alone can complete a task in 10 days A,B worked together for 3 days after which C replaced B. A was `50%` more efficient than C, After A and C had worked for three days, `13%` task was incomplete. In approximately how many days B (working alone) can complete the entire work?

Options

A. 24
B. 64
C. 12
D. 43

Answer

To solve the problem step by step, we will break down the information given and calculate the required values.

### Step 1: Determine A's efficiency
A can complete the task in 10 days. Therefore, A's work done in one day is:
\[ \text{Work done by A in 1 day} = \frac{1}{10} \]

### Step 2: Determine C's efficiency
It is given that A is 50% more efficient than C. If we denote C's efficiency as \( x \), then:
\[ A = 1.5 \times C \implies \frac{1}{10} = 1.5x \implies x = \frac{1}{15} \]
Thus, C can complete the task in 15 days.

### Step 3: Determine B's efficiency
Let B's efficiency be \( y \). Since we do not know B's efficiency yet, we will keep it as \( \frac{1}{y} \).

### Step 4: Calculate the work done by A and B together in 3 days
In 3 days, the work done by A and B together is:
\[ \text{Work done by A in 3 days} = 3 \times \frac{1}{10} = \frac{3}{10} \]
The work done by B in 3 days is:
\[ \text{Work done by B in 3 days} = 3 \times \frac{1}{y} = \frac{3}{y} \]
Thus, the total work done by A and B together in 3 days is:
\[ \text{Total work in 3 days} = \frac{3}{10} + \frac{3}{y} \]

### Step 5: Calculate the work done by A and C together in the next 3 days
After 3 days, C replaces B, and A and C work together for another 3 days. The work done by A and C in 3 days is:
\[ \text{Work done by A in 3 days} = 3 \times \frac{1}{10} = \frac{3}{10} \]
\[ \text{Work done by C in 3 days} = 3 \times \frac{1}{15} = \frac{1}{5} \]
Thus, the total work done by A and C together in 3 days is:
\[ \text{Total work in 3 days} = \frac{3}{10} + \frac{1}{5} = \frac{3}{10} + \frac{2}{10} = \frac{5}{10} = \frac{1}{2} \]

### Step 6: Calculate total work done and remaining work
The total work done by A and B in the first 3 days and A and C in the next 3 days is:
\[ \text{Total work done} = \left(\frac{3}{10} + \frac{3}{y}\right) + \frac{1}{2} \]
We know that 13% of the task is incomplete, meaning 87% of the task is complete. Therefore:
\[ \left(\frac{3}{10} + \frac{3}{y}\right) + \frac{1}{2} = \frac{87}{100} \]

### Step 7: Solve for B's efficiency
Convert all fractions to a common denominator:
\[ \frac{3}{10} = \frac{30}{100}, \quad \frac{1}{2} = \frac{50}{100} \]
Thus, we have:
\[ \frac{30}{100} + \frac{3}{y} + \frac{50}{100} = \frac{87}{100} \]
This simplifies to:
\[ \frac{80}{100} + \frac{3}{y} = \frac{87}{100} \]
Subtracting \( \frac{80}{100} \) from both sides gives:
\[ \frac{3}{y} = \frac{7}{100} \]
Cross-multiplying gives:
\[ 3 \times 100 = 7y \implies 300 = 7y \implies y = \frac{300}{7} \approx 42.86 \]

### Step 8: Conclusion
Thus, B can complete the entire work alone in approximately 43 days.