A can finished one-third of a work in 5 days, B can finish `underset( 5) overset( 2) `th of the same workin 10days and C can finish 75% of the same work in 15 days. They work together for 6 days. The remaining workk will be finished by B along in `:`

Options

A. 3 days
B. 2 days
C. 5 days
D. `1(1)/(2)` days

Answer

To solve the problem, let's break it down step by step.

### Step 1: Determine the total work done by A, B, and C.

1. **A's Work**: A can finish \( \frac{1}{3} \) of the work in 5 days.
- Therefore, in 15 days (which is 3 times 5 days), A can finish the entire work.
- A's efficiency = \( \frac{1}{15} \) of the work per day.

2. **B's Work**: B can finish \( \frac{2}{5} \) of the work in 10 days.
- Therefore, in 25 days (which is 2.5 times 10 days), B can finish the entire work.
- B's efficiency = \( \frac{1}{25} \) of the work per day.

3. **C's Work**: C can finish 75% of the work in 15 days.
- 75% is equivalent to \( \frac{3}{4} \), so in 20 days (which is \( \frac{15}{0.75} \)), C can finish the entire work.
- C's efficiency = \( \frac{1}{20} \) of the work per day.

### Step 2: Calculate the combined efficiency of A, B, and C.

- A's efficiency = \( \frac{1}{15} \)
- B's efficiency = \( \frac{1}{25} \)
- C's efficiency = \( \frac{1}{20} \)

To find the combined efficiency, we need to find a common denominator:
- The LCM of 15, 25, and 20 is 300.

Now, convert each efficiency to have a denominator of 300:
- A's efficiency = \( \frac{20}{300} \)
- B's efficiency = \( \frac{12}{300} \)
- C's efficiency = \( \frac{15}{300} \)

Combined efficiency = \( \frac{20 + 12 + 15}{300} = \frac{47}{300} \) of the work per day.

### Step 3: Calculate the work done in 6 days.

- Work done in 6 days = \( 6 \times \frac{47}{300} = \frac{282}{300} \) of the work.

### Step 4: Calculate the remaining work.

- Total work = 1 (whole work)
- Work done in 6 days = \( \frac{282}{300} \)
- Remaining work = \( 1 - \frac{282}{300} = \frac{300 - 282}{300} = \frac{18}{300} = \frac{3}{50} \) of the work.

### Step 5: Calculate how long B will take to finish the remaining work.

- B's efficiency = \( \frac{1}{25} \) of the work per day.
- Time taken by B to finish the remaining work = Remaining work / B's efficiency
- Time = \( \frac{\frac{3}{50}}{\frac{1}{25}} = \frac{3}{50} \times 25 = \frac{3 \times 25}{50} = \frac{75}{50} = \frac{3}{2} \) days or \( 1 \frac{1}{2} \) days.

### Final Answer:
The remaining work will be finished by B alone in \( 1 \frac{1}{2} \) days.

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