A, B and C can complete a work in 2h. If A does the job alone in 6h and B in 5h, how long will it take for C to finish the job alone ?

Options

A. `6(1)/(2) h`
B. `7 (1)/(2)` h`
C. `9h`
D. `4 (1)/(2) h`

Answer

To solve the problem step by step, we will determine how long it will take for C to finish the job alone, given the efficiencies of A, B, and C.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- A, B, and C together can complete the work in 2 hours.
- A can complete the work alone in 6 hours.
- B can complete the work alone in 5 hours.

2. **Calculating the Efficiencies:**
- The efficiency of A (E_A) is calculated as:
\[ E_A = \frac{1 \text{ unit of work}}{6 \text{ hours}} = \frac{1}{6} \]
- The efficiency of B (E_B) is calculated as:
\[ E_B = \frac{1 \text{ unit of work}}{5 \text{ hours}} = \frac{1}{5} \]
- The efficiency of A, B, and C together (E_A + E_B + E_C) is:
\[ E_{A,B,C} = \frac{1 \text{ unit of work}}{2 \text{ hours}} = \frac{1}{2} \]

3. **Setting Up the Equation:**
- We know that:
\[ E_A + E_B + E_C = E_{A,B,C} \]
- Substituting the known values:
\[ \frac{1}{6} + \frac{1}{5} + E_C = \frac{1}{2} \]

4. **Finding a Common Denominator:**
- The least common multiple (LCM) of 6, 5, and 2 is 30. We will convert each fraction:
\[ \frac{1}{6} = \frac{5}{30}, \quad \frac{1}{5} = \frac{6}{30}, \quad \frac{1}{2} = \frac{15}{30} \]
- Now substituting these values into the equation:
\[ \frac{5}{30} + \frac{6}{30} + E_C = \frac{15}{30} \]

5. **Solving for E_C:**
- Combine the fractions:
\[ \frac{11}{30} + E_C = \frac{15}{30} \]
- Subtract \(\frac{11}{30}\) from both sides:
\[ E_C = \frac{15}{30} - \frac{11}{30} = \frac{4}{30} = \frac{2}{15} \]

6. **Calculating the Time Taken by C:**
- Since efficiency is inversely proportional to time, we can find the time taken by C (T_C):
\[ T_C = \frac{1 \text{ unit of work}}{E_C} = \frac{1}{\frac{2}{15}} = \frac{15}{2} = 7.5 \text{ hours} \]

### Final Answer:
C will take **7.5 hours** to finish the job alone.

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