If A and B together can finish a piece of work in 20 days, B and C in 10 days and C and A in 12 days, then A, B, C jointly can finish the same work in

Options

A. `4(2)/(7)` days
B. 30 days
C. `8(4)/(7)` days
D. `(7)/(60)` days

Answer

To solve the problem, we need to find out how long A, B, and C together can finish the work based on the information given about their individual and combined efficiencies.

### Step-by-Step Solution:

1. **Understanding the Work Rates**:
- Let the total work be represented in terms of units. We can assume the total work is 60 units (this is a common multiple of the days given).
- A and B together can finish the work in 20 days, so their combined work rate is:
\[ \text{Work rate of A and B} = \frac{60 \text{ units}}{20 \text{ days}} = 3 \text{ units/day} \]
- B and C together can finish the work in 10 days, so their combined work rate is:
\[ \text{Work rate of B and C} = \frac{60 \text{ units}}{10 \text{ days}} = 6 \text{ units/day} \]
- C and A together can finish the work in 12 days, so their combined work rate is:
\[ \text{Work rate of C and A} = \frac{60 \text{ units}}{12 \text{ days}} = 5 \text{ units/day} \]

2. **Setting Up Equations**:
- Let the work rates of A, B, and C be \(a\), \(b\), and \(c\) respectively.
- From the above work rates, we can set up the following equations:
\[ a + b = 3 \quad \text{(1)} \]
\[ b + c = 6 \quad \text{(2)} \]
\[ c + a = 5 \quad \text{(3)} \]

3. **Solving the Equations**:
- We can add all three equations:
\[ (a + b) + (b + c) + (c + a) = 3 + 6 + 5 \]
This simplifies to:
\[ 2a + 2b + 2c = 14 \]
Dividing by 2 gives:
\[ a + b + c = 7 \quad \text{(4)} \]

4. **Finding Individual Work Rates**:
- Now we can use equation (4) to find the individual work rates:
- From equation (1):
\[ c = 7 - (a + b) = 7 - 3 = 4 \]
- From equation (2):
\[ a = 7 - (b + c) = 7 - 6 = 1 \]
- From equation (3):
\[ b = 7 - (c + a) = 7 - 5 = 2 \]
- Thus, we have:
\[ a = 1, \quad b = 2, \quad c = 4 \]

5. **Calculating the Combined Work Rate**:
- Now, we can find the combined work rate of A, B, and C:
\[ a + b + c = 1 + 2 + 4 = 7 \text{ units/day} \]

6. **Calculating the Time Taken by A, B, and C Together**:
- To find out how many days A, B, and C together can finish the work:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Work Rate}} = \frac{60 \text{ units}}{7 \text{ units/day}} = \frac{60}{7} \text{ days} \]
- This can be expressed as:
\[ 8 \frac{4}{7} \text{ days} \]

### Final Answer:
A, B, and C together can finish the work in \(8 \frac{4}{7}\) days.