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, we need to find out how long C alone can do the work given the combined work rates of A, B, and C. Let's break it down step by step:

### Step 1: Determine the work rates of A + B, B + C, and C + A
- A and B can complete the work in 10 days. Therefore, their combined work rate is:
\[ A + B = \frac{1}{10} \text{ (work per day)} \]

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

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

### Step 2: Add the three equations
Now we can add the three equations:
\[ (A + B) + (B + C) + (C + A) = \frac{1}{10} + \frac{1}{15} + \frac{1}{20} \]

This simplifies to:
\[ 2A + 2B + 2C = \frac{1}{10} + \frac{1}{15} + \frac{1}{20} \]

### Step 3: Find the LCM and simplify the right side
To add the fractions on the right side, we need to find the LCM of 10, 15, and 20. The LCM is 60.

Now we convert 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:
\[ \frac{6}{60} + \frac{4}{60} + \frac{3}{60} = \frac{13}{60} \]

### Step 4: Solve for A + B + C
Now we have:
\[ 2A + 2B + 2C = \frac{13}{60} \]
Dividing both sides by 2:
\[ A + B + C = \frac{13}{120} \text{ (work per day)} \]

### Step 5: Find C's work rate
We know:
\[ A + B = \frac{1}{10} = \frac{12}{120} \]
Now we can find C's work rate:
\[ C = (A + B + C) - (A + B) = \frac{13}{120} - \frac{12}{120} = \frac{1}{120} \]

### Step 6: Calculate the time taken by C alone to complete the work
If C can do \(\frac{1}{120}\) of the work in one day, then the time taken by C to complete the entire work is:
\[ \text{Time} = \frac{1}{\text{C's work rate}} = \frac{1}{\frac{1}{120}} = 120 \text{ days} \]

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

---