A and B can do a piece of work in 12 days, B and C in 15 days, C and A in 20 days. In what time can A do it separately?

Options

A. 45 days
B. 20 days
C. 60 days
D. 30 days

Answer

To solve the problem, we need to find out how much work A can do alone in a certain number of days. We'll start by determining the work rates of A, B, and C based on the information given.

### Step 1: Determine Work Rates
Let the total work be represented as 1 unit of work.

- A and B can complete the work in 12 days, so their combined work rate is:
\[ \text{Work rate of A + B} = \frac{1 \text{ unit}}{12 \text{ days}} = \frac{1}{12} \]

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

- C and A can complete the work in 20 days, so their combined work rate is:
\[ \text{Work rate of C + A} = \frac{1 \text{ unit}}{20 \text{ days}} = \frac{1}{20} \]

### Step 2: Set Up Equations
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: Solve the Equations
We can add all three equations together:
\[ (a + b) + (b + c) + (c + a) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]
This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]
Now, we need to find a common denominator for the right side. The least common multiple of 12, 15, and 20 is 60.

Calculating each fraction:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
Adding these gives:
\[ \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \]
Thus, we have:
\[ 2a + 2b + 2c = \frac{1}{5} \]
Dividing everything by 2:
\[ a + b + c = \frac{1}{10} \quad \text{(Equation 4)} \]

### Step 4: Find Individual Work Rates
Now, we can express \( c \) in terms of \( a \) and \( b \) using Equation 1:
From Equation 1:
\[ b = \frac{1}{12} - a \]
Substituting \( b \) into Equation 2:
\[ \left(\frac{1}{12} - a\right) + c = \frac{1}{15} \]
This simplifies to:
\[ c = \frac{1}{15} - \left(\frac{1}{12} - a\right) = a + \frac{1}{15} - \frac{1}{12} \]
Finding a common denominator for \(\frac{1}{15}\) and \(\frac{1}{12}\):
\[ \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
Thus:
\[ c = a + \left(\frac{4}{60} - \frac{5}{60}\right) = a - \frac{1}{60} \]

Now substituting \( c \) back into Equation 3:
\[ \left(a - \frac{1}{60}\right) + a = \frac{1}{20} \]
This simplifies to:
\[ 2a - \frac{1}{60} = \frac{3}{60} \]
So:
\[ 2a = \frac{3}{60} + \frac{1}{60} = \frac{4}{60} = \frac{1}{15} \]
Thus:
\[ a = \frac{1}{30} \]

### Step 5: Calculate Time for A to Complete the Work Alone
If A's work rate is \( \frac{1}{30} \), then the time taken by A to complete the work alone is:
\[ \text{Time} = \frac{1 \text{ unit}}{a} = \frac{1}{\frac{1}{30}} = 30 \text{ days} \]

### Final Answer
A can do the work alone in **30 days**.

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