A and B can do a piece of work in 10 days, B and C in 15 days and C and A in 20 days. C alone can do the work in :

Options

A. 60 days
B. 120 days
C. 80 days
D. 30 days

Answer

To solve the problem step by step, we will first define the work done by each pair of workers and then find out how long C alone takes to complete the work.

### Step-by-Step Solution:

1. **Define the Work Rates**:
- A and B can complete the work in 10 days. Therefore, their combined work rate is:
\[ \text{Work rate of A + B} = \frac{1}{10} \text{ (work per day)} \]
- B and C can complete the work in 15 days. Therefore, their combined work rate is:
\[ \text{Work rate of B + C} = \frac{1}{15} \text{ (work per day)} \]
- C and A can complete the work in 20 days. Therefore, their combined work rate is:
\[ \text{Work rate of C + A} = \frac{1}{20} \text{ (work per day)} \]

2. **Add the Work Rates**:
- Now, we will add the work rates of all three pairs:
\[ \text{(A + B)} + \text{(B + C)} + \text{(C + A)} = \frac{1}{10} + \frac{1}{15} + \frac{1}{20} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 10, 15, and 20 is 60.
- Converting each fraction:
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
- Adding these gives:
\[ \frac{6}{60} + \frac{4}{60} + \frac{3}{60} = \frac{13}{60} \]

3. **Combine the Work Rates**:
- The total work rate of A, B, and C combined is:
\[ 2(A + B + C) = \frac{13}{60} \]
- Therefore, the work rate of A + B + C is:
\[ A + B + C = \frac{13}{120} \text{ (work per day)} \]

4. **Find C's Work Rate**:
- To find C's work rate, we can use the equation:
\[ C = (A + B + C) - (A + B) \]
- We already know \(A + B = \frac{1}{10} = \frac{6}{60} = \frac{12}{120}\).
- Now substituting:
\[ C = \frac{13}{120} - \frac{12}{120} = \frac{1}{120} \]

5. **Calculate the Time for C Alone**:
- Since C's work rate is \(\frac{1}{120}\), this means C can complete the work in:
\[ \text{Time taken by C} = \frac{1}{\frac{1}{120}} = 120 \text{ days} \]

### Final Answer:
C alone can complete the work in **120 days**.

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