A and B together complete a work in `20` days, B and C together complete same work in `30` days, C and A together complete the same work `24` days, In how many days A, B and C together can complete the same work?

Options

A. `8`
B. `16`
C. `12`
D. `15`

Answer

To solve the problem, we need to find out how many days A, B, and C together can complete the work based on the information given about their combined work rates.

### Step-by-Step Solution:

1. **Identify the work rates of each pair:**
- A and B together complete the work in 20 days.
- B and C together complete the work in 30 days.
- C and A together complete the work in 24 days.

2. **Calculate the total work:**
- The total work can be represented as the least common multiple (LCM) of the days taken by each pair.
- Calculate LCM of 20, 30, and 24.
- LCM(20, 30, 24) = 120 units.

3. **Determine the work done by each pair in one day:**
- A and B together: \( \frac{120 \text{ units}}{20 \text{ days}} = 6 \text{ units/day} \)
- B and C together: \( \frac{120 \text{ units}}{30 \text{ days}} = 4 \text{ units/day} \)
- C and A together: \( \frac{120 \text{ units}}{24 \text{ days}} = 5 \text{ units/day} \)

4. **Set up equations for work rates:**
- Let the work done by A in one day be \( a \), by B be \( b \), and by C be \( c \).
- From the pairs, we can write:
- \( a + b = 6 \) (1)
- \( b + c = 4 \) (2)
- \( c + a = 5 \) (3)

5. **Solve the equations:**
- From equation (1): \( b = 6 - a \)
- Substitute \( b \) in equation (2):
\[ (6 - a) + c = 4 \implies c = 4 - 6 + a \implies c = a - 2 \quad (4) \]
- Substitute \( c \) in equation (3):
\[ (a - 2) + a = 5 \implies 2a - 2 = 5 \implies 2a = 7 \implies a = 3.5 \]
- Now substitute \( a \) back to find \( b \) and \( c \):
- From (1): \( b = 6 - 3.5 = 2.5 \)
- From (4): \( c = 3.5 - 2 = 1.5 \)

6. **Calculate the total work rate of A, B, and C together:**
- Total work rate \( a + b + c = 3.5 + 2.5 + 1.5 = 7.5 \text{ units/day} \)

7. **Determine the total time taken by A, B, and C together to complete the work:**
- Time taken = \( \frac{\text{Total Work}}{\text{Total Work Rate}} = \frac{120 \text{ units}}{7.5 \text{ units/day}} = 16 \text{ days} \)

### Final Answer:
A, B, and C together can complete the work in **16 days**.