Efficiency of A is `(3)/(4)` of B's efficiency and B's efficiency is 80% of C's efficiency. If A takes 120 more days than that of B and C working together to finish a certain piece of work. Then in how many days working together they will finish the work?

Answer

To solve the problem step by step, let's denote the efficiencies of A, B, and C as follows:

1. **Define the efficiencies**:
- Let B's efficiency be \( b \).
- Then A's efficiency is \( \frac{3}{4}b \).
- B's efficiency is 80% of C's efficiency, so \( b = 0.8c \) or \( c = \frac{b}{0.8} = \frac{5}{4}b \).

2. **Calculate combined efficiency of B and C**:
- The efficiency of B is \( b \).
- The efficiency of C is \( \frac{5}{4}b \).
- Therefore, the combined efficiency of B and C is:
\[ b + \frac{5}{4}b = \frac{4}{4}b + \frac{5}{4}b = \frac{9}{4}b \]

3. **Determine the time taken by A and B+C**:
- Let \( T \) be the time taken by B and C to complete the work together.
- The work done by B and C in \( T \) days is:
\[ \text{Work} = \text{Efficiency} \times \text{Time} = \frac{9}{4}b \times T \]
- A takes 120 days more than B and C, so the time taken by A is \( T + 120 \).
- The work done by A in \( T + 120 \) days is:
\[ \text{Work} = \text{Efficiency} \times \text{Time} = \frac{3}{4}b \times (T + 120) \]

4. **Set the work equations equal**:
- Since both expressions represent the same total work, we can set them equal:
\[ \frac{9}{4}b \times T = \frac{3}{4}b \times (T + 120) \]

5. **Cancel \( b \) and simplify**:
- Dividing both sides by \( \frac{3}{4}b \) (assuming \( b \neq 0 \)):
\[ \frac{9}{4}T = T + 120 \]
- Multiply through by 4 to eliminate the fraction:
\[ 9T = 4T + 480 \]
- Rearranging gives:
\[ 9T - 4T = 480 \implies 5T = 480 \implies T = \frac{480}{5} = 96 \]

6. **Calculate total time working together**:
- The total time taken by B and C working together is \( T = 96 \) days.
- Therefore, the total time taken by A, B, and C working together is:
\[ T + 120 = 96 + 120 = 216 \text{ days} \]

7. **Final answer**:
- The total time taken by A, B, and C working together to finish the work is **216 days**.