A and B together can do a piece of work in 12 days, while B alone can finish it in 30 days. A alone can finish the work in

Options

A. 15 days
B. 18 days
C. 20 days
D. 25 days

Answer

To solve the problem, we need to find out how many days A alone would take to finish the work. Let’s break it down step by step.

### Step 1: Determine the work rates of A and B
1. **Work done by A and B together in one day**: Since A and B together can complete the work in 12 days, their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1}{12} \text{ (work/day)} \]

2. **Work done by B alone in one day**: B can finish the work in 30 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{30} \text{ (work/day)} \]

### Step 2: Calculate the work rate of A
3. **Work rate of A**: We know the combined work rate of A and B, and we can find A's work rate by subtracting B's work rate from the combined work rate:
\[ \text{Work rate of A} = \text{Work rate of A and B} - \text{Work rate of B} \]
\[ \text{Work rate of A} = \frac{1}{12} - \frac{1}{30} \]

### Step 3: Find a common denominator and perform the subtraction
4. **Finding a common denominator**: The least common multiple of 12 and 30 is 60. We convert the fractions:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
Now we can subtract:
\[ \text{Work rate of A} = \frac{5}{60} - \frac{2}{60} = \frac{3}{60} = \frac{1}{20} \text{ (work/day)} \]

### Step 4: Calculate the time taken by A to finish the work alone
5. **Time taken by A alone**: Since A's work rate is \(\frac{1}{20}\) work/day, the time taken by A to finish the work alone is the reciprocal of the work rate:
\[ \text{Time taken by A} = \frac{1}{\text{Work rate of A}} = \frac{1}{\frac{1}{20}} = 20 \text{ days} \]

### Final Answer
A alone can finish the work in **20 days**.

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