A and B can do a work in 12 days .B and C in 15 days .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 need to determine how long it will take for A, B, and C to complete the work together based on the information given about their combined work rates.

### Step 1: Determine the work rates of A+B, B+C, and C+A
- A and B can complete the work in 12 days, so their combined work rate is:
\[ A + B = \frac{1}{12} \text{ (work per day)} \]

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

- C and A can complete the work in 20 days, so 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 together:
\[ (A + B) + (B + C) + (C + A) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]
This simplifies to:
\[ 2A + 2B + 2C = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]

### Step 3: Find a common denominator
To add the fractions on the right side, we need a common denominator. The least common multiple (LCM) of 12, 15, and 20 is 60. We convert each fraction:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
Now, we can add these fractions:
\[ \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \]

### Step 4: Solve for A + B + C
From the equation \(2A + 2B + 2C = \frac{1}{5}\), we can divide both sides by 2:
\[ A + B + C = \frac{1}{10} \text{ (work per day)} \]

### Step 5: Calculate the time taken for A, B, and C to complete the work together
If A + B + C can complete \(\frac{1}{10}\) of the work in one day, then the total 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
Therefore, A, B, and C together can complete the work in **10 days**.

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