A & B can complete a work in 24 days and 36 days respectively. A & B together start work and after 8 days C joined him. If remaining work complete by all three in `3(5)/(9)`days, then find in how many days B & C can complete the 75% of work together?

Options

A. A)9 days
B. B)8 days
C. C)`7(1)/(3)`days
D. D)`6(1)/(3)`days

Answer

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

### Step 1: Determine the work done by A and B individually.
- A can complete the work in 24 days, so A's work rate (efficiency) is:
\[ \text{Efficiency of A} = \frac{1 \text{ work}}{24 \text{ days}} = \frac{1}{24} \text{ work/day} \]
- B can complete the work in 36 days, so B's work rate (efficiency) is:
\[ \text{Efficiency of B} = \frac{1 \text{ work}}{36 \text{ days}} = \frac{1}{36} \text{ work/day} \]

### Step 2: Calculate the combined efficiency of A and B.
- The combined efficiency of A and B is:
\[ \text{Combined efficiency of A and B} = \frac{1}{24} + \frac{1}{36} \]
- To add these fractions, find a common denominator (LCM of 24 and 36 is 72):
\[ \frac{1}{24} = \frac{3}{72}, \quad \frac{1}{36} = \frac{2}{72} \]
- Thus,
\[ \text{Combined efficiency of A and B} = \frac{3}{72} + \frac{2}{72} = \frac{5}{72} \text{ work/day} \]

### Step 3: Calculate the work done by A and B in the first 8 days.
- In 8 days, A and B together will complete:
\[ \text{Work done by A and B in 8 days} = 8 \times \frac{5}{72} = \frac{40}{72} = \frac{5}{9} \text{ of the work} \]

### Step 4: Calculate the remaining work.
- The total work is considered as 1 (or 72 units if we scale it up). Therefore, the remaining work after 8 days is:
\[ \text{Remaining work} = 1 - \frac{5}{9} = \frac{4}{9} \text{ of the work} \]

### Step 5: Determine the time taken by A, B, and C to complete the remaining work.
- It is given that A, B, and C together complete the remaining work in \(3 \frac{5}{9}\) days, which can be converted to an improper fraction:
\[ 3 \frac{5}{9} = \frac{32}{9} \text{ days} \]

### Step 6: Set up the equation for the remaining work.
- Let C's efficiency be \(c\). The combined efficiency of A, B, and C is:
\[ \frac{5}{72} + c \]
- The work done by A, B, and C in \( \frac{32}{9} \) days is:
\[ \left(\frac{5}{72} + c\right) \times \frac{32}{9} = \frac{4}{9} \]

### Step 7: Solve for C's efficiency.
- Rearranging the equation gives:
\[ \frac{5 \times 32}{72 \times 9} + c \times \frac{32}{9} = \frac{4}{9} \]
- Simplifying:
\[ \frac{160}{648} + c \times \frac{32}{9} = \frac{4}{9} \]
- Convert \(\frac{4}{9}\) to have a common denominator:
\[ \frac{4}{9} = \frac{288}{648} \]
- Thus,
\[ c \times \frac{32}{9} = \frac{288}{648} - \frac{160}{648} = \frac{128}{648} \]
- Solving for \(c\):
\[ c = \frac{128}{648} \times \frac{9}{32} = \frac{128 \times 9}{648 \times 32} = \frac{1152}{20736} = \frac{1}{18} \text{ work/day} \]

### Step 8: Calculate the efficiency of B and C together.
- B's efficiency is \(\frac{1}{36}\) and C's efficiency is \(\frac{1}{18}\):
\[ \text{Efficiency of B and C} = \frac{1}{36} + \frac{1}{18} = \frac{1}{36} + \frac{2}{36} = \frac{3}{36} = \frac{1}{12} \text{ work/day} \]

### Step 9: Calculate 75% of the total work.
- 75% of the total work is:
\[ 0.75 \times 1 = \frac{3}{4} \text{ of the work} \]

### Step 10: Calculate the time taken by B and C to complete 75% of the work.
- Time taken by B and C to complete \(\frac{3}{4}\) of the work:
\[ \text{Time} = \frac{\frac{3}{4}}{\frac{1}{12}} = \frac{3}{4} \times 12 = 9 \text{ days} \]

Thus, the final answer is:
\[ \text{B and C can complete 75% of the work together in } 9 \text{ days.} \]