A can do `4/5` of a work in 20 days and B can do `3/4` of the same work in 15 days. They work together for 10 days. C alone completes the remaining work in 1 day. B and C together can complete `3/4` of the same work in:

Options

A. 8 days
B. 5 days
C. 4 days
D. 6 days

Answer

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

### Step 1: Calculate the total work done by A and B
- A can do \( \frac{4}{5} \) of the work in 20 days.
- Therefore, A can complete the entire work in \( \frac{20 \text{ days}}{\frac{4}{5}} = 25 \text{ days} \).
- This means A's work rate is \( \frac{1}{25} \) of the work per day.

- B can do \( \frac{3}{4} \) of the work in 15 days.
- Therefore, B can complete the entire work in \( \frac{15 \text{ days}}{\frac{3}{4}} = 20 \text{ days} \).
- This means B's work rate is \( \frac{1}{20} \) of the work per day.

### Step 2: Calculate the combined work rate of A and B
- The combined work rate of A and B is:
\[ \text{Work rate of A} + \text{Work rate of B} = \frac{1}{25} + \frac{1}{20} \]
- To add these fractions, we find a common denominator, which is 100:
\[ \frac{1}{25} = \frac{4}{100}, \quad \frac{1}{20} = \frac{5}{100} \]
- Therefore,
\[ \text{Combined work rate} = \frac{4}{100} + \frac{5}{100} = \frac{9}{100} \]

### Step 3: Calculate the work done by A and B in 10 days
- In 10 days, A and B together can complete:
\[ 10 \times \frac{9}{100} = \frac{90}{100} = 0.9 \text{ (or 90% of the work)} \]

### Step 4: Calculate the remaining work
- Since A and B completed 90% of the work, the remaining work is:
\[ 1 - 0.9 = 0.1 \text{ (or 10% of the work)} \]

### Step 5: Determine the work done by C
- C completes the remaining 10% of the work in 1 day. Therefore, C's work rate is:
\[ \text{C's work rate} = 0.1 \text{ work per day} \]
- This means C can complete the entire work in \( \frac{1}{0.1} = 10 \text{ days} \).

### Step 6: Calculate the combined work rate of B and C
- B's work rate is \( \frac{1}{20} \) and C's work rate is \( 0.1 \).
- To find the combined work rate of B and C:
\[ \text{Work rate of B} + \text{Work rate of C} = \frac{1}{20} + 0.1 \]
- Converting \( 0.1 \) to a fraction gives us \( \frac{1}{10} \):
\[ \frac{1}{20} + \frac{1}{10} = \frac{1}{20} + \frac{2}{20} = \frac{3}{20} \]

### Step 7: Calculate the time taken by B and C to complete \( \frac{3}{4} \) of the work
- To find out how long it takes for B and C to complete \( \frac{3}{4} \) of the work:
\[ \text{Time} = \frac{\text{Work}}{\text{Combined work rate}} = \frac{\frac{3}{4}}{\frac{3}{20}} = \frac{3}{4} \times \frac{20}{3} = 5 \text{ days} \]

### Final Answer
B and C together can complete \( \frac{3}{4} \) of the work in **5 days**.