A and B can do a work in 12 days, B and C in 15 days and C and A in 20 days. If A, B and C work together, they will complete the work in :

Options

A. 5 days
B. `7(5)/(6)` days
C. 10 days
D. `15(2)/(3)` days

Answer

To solve the problem step by step, we will first determine the work rates of A, B, and C based on the information given.

### Step 1: Determine the work rates of A and B, B and C, and C and A.

- A and B can complete the work in 12 days. Therefore, their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1}{12} \text{ (work per day)} \]

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

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

### Step 2: Set up equations for the work rates.

Let the work rates of A, B, and C be represented as \( a \), \( b \), and \( c \) respectively. We can write the following equations based on the work rates:

1. \( a + b = \frac{1}{12} \) (Equation 1)
2. \( b + c = \frac{1}{15} \) (Equation 2)
3. \( c + a = \frac{1}{20} \) (Equation 3)

### Step 3: Add all three equations.

Adding all three equations together gives us:
\[ (a + b) + (b + c) + (c + a) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]
This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]

### Step 4: Calculate the right-hand side.

To add the fractions on the right-hand side, we need a common denominator. The least common multiple of 12, 15, and 20 is 60.

Converting each fraction:
- \( \frac{1}{12} = \frac{5}{60} \)
- \( \frac{1}{15} = \frac{4}{60} \)
- \( \frac{1}{20} = \frac{3}{60} \)

Now, adding these fractions:
\[ \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \]

### Step 5: Substitute back into the equation.

Now substituting back, we have:
\[ 2(a + b + c) = \frac{1}{5} \]
Dividing both sides by 2:
\[ a + b + c = \frac{1}{10} \]

### Step 6: Find the time taken by A, B, and C to complete the work together.

If A, B, and C together can complete \(\frac{1}{10}\) of the work in one day, then the time taken to complete the entire work is the reciprocal of their combined work rate:
\[ \text{Time} = \frac{1}{\frac{1}{10}} = 10 \text{ days} \]

### Final Answer:
If A, B, and C work together, they will complete the work in **10 days**.

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