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

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Work Rates
A and B together can complete the work in 12 days. This means that their combined work rate is:
\[ \text{Work rate of A + B} = \frac{1 \text{ work}}{12 \text{ days}} = \frac{1}{12} \]

B alone can complete the work in 30 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1 \text{ work}}{30 \text{ days}} = \frac{1}{30} \]

### Step 2: Calculate A's Work Rate
To find A's work rate, we can use the relationship:
\[ \text{Work rate of A} = \text{Work rate of A + B} - \text{Work rate of B} \]
Substituting the known values:
\[ \text{Work rate of A} = \frac{1}{12} - \frac{1}{30} \]

### Step 3: Find a Common Denominator
To perform the subtraction, we need a common denominator. The least common multiple (LCM) of 12 and 30 is 60. Now we can rewrite the fractions:
\[ \frac{1}{12} = \frac{5}{60} \quad \text{and} \quad \frac{1}{30} = \frac{2}{60} \]
Now substitute these back into the equation:
\[ \text{Work rate of A} = \frac{5}{60} - \frac{2}{60} = \frac{3}{60} \]

### Step 4: Simplify A's Work Rate
Now simplify \(\frac{3}{60}\):
\[ \text{Work rate of A} = \frac{1}{20} \]

### Step 5: Calculate Time Taken by A Alone
If A can do \(\frac{1}{20}\) of the work in one day, then the time taken by A to finish the entire work is the reciprocal of the work rate:
\[ \text{Time taken by A} = \frac{1}{\frac{1}{20}} = 20 \text{ days} \]

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

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