A and B together can do a piece of work in 10 days, B and C together can do it in 15 days while C and A together can do it in 20 days. Together they will complete the work in:

Options

A. 12 days
B. `3 1/5` days
C. 16 days
D. `9 3/13` days

Answer

To solve the problem step by step, we will first determine the work done by each pair of workers (A and B, B and C, C and A) and then find the total work done by A, B, and C together.

### Step-by-Step Solution:

1. **Identify the work done by each pair:**
- A and B together can complete the work in 10 days.
- B and C together can complete the work in 15 days.
- C and A together can complete the work in 20 days.

2. **Calculate the work done by each pair in one day:**
- Work done by A and B in one day = \( \frac{1}{10} \)
- Work done by B and C in one day = \( \frac{1}{15} \)
- Work done by C and A in one day = \( \frac{1}{20} \)

3. **Set up equations for the work done:**
- Let the work done by A in one day be \( a \).
- Let the work done by B in one day be \( b \).
- Let the work done by C in one day be \( c \).

From the information given:
- \( a + b = \frac{1}{10} \) (Equation 1)
- \( b + c = \frac{1}{15} \) (Equation 2)
- \( c + a = \frac{1}{20} \) (Equation 3)

4. **Add all three equations:**
\[ (a + b) + (b + c) + (c + a) = \frac{1}{10} + \frac{1}{15} + \frac{1}{20} \]
This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{10} + \frac{1}{15} + \frac{1}{20} \]

5. **Calculate the right-hand side (RHS):**
- Find the LCM of 10, 15, and 20, which is 60.
- Convert each fraction:
- \( \frac{1}{10} = \frac{6}{60} \)
- \( \frac{1}{15} = \frac{4}{60} \)
- \( \frac{1}{20} = \frac{3}{60} \)

Thus,
\[ \frac{1}{10} + \frac{1}{15} + \frac{1}{20} = \frac{6 + 4 + 3}{60} = \frac{13}{60} \]

6. **Substituting back:**
\[ 2(a + b + c) = \frac{13}{60} \]
Therefore,
\[ a + b + c = \frac{13}{120} \]

7. **Calculate the total work done by A, B, and C together in one day:**
- The total work done by A, B, and C together in one day is \( \frac{13}{120} \).

8. **Calculate the number of days they will take to complete the work together:**
- The total time taken to complete the work is the reciprocal of the work done in one day:
\[ \text{Time} = \frac{1}{\frac{13}{120}} = \frac{120}{13} \]

9. **Convert to mixed number:**
- \( \frac{120}{13} = 9 \frac{3}{13} \) days.

### Final Answer:
Together, A, B, and C will complete the work in \( 9 \frac{3}{13} \) days.