Both A and B together can complete a work in 20 days and B alone can complete the same work in 36 days. In how many days will A alone complete the work?

Options

A. 30
B. 50
C. 45
D. 42

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Understand the work done by A and B together
We know that A and B together can complete the work in 20 days. This means that their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1}{20} \text{ work per day} \]

### Step 2: Determine the work done by B alone
B can complete the work alone in 36 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{36} \text{ work per day} \]

### Step 3: Calculate the work rate of A
To find A's work rate, we can subtract B's work rate from the combined work rate of A and B:
\[ \text{Work rate of A} = \text{Work rate of A and B} - \text{Work rate of B} \]
\[ \text{Work rate of A} = \frac{1}{20} - \frac{1}{36} \]

### Step 4: Find a common denominator
The least common multiple of 20 and 36 is 180. We can convert both fractions:
\[ \frac{1}{20} = \frac{9}{180} \quad \text{and} \quad \frac{1}{36} = \frac{5}{180} \]

### Step 5: Subtract the fractions
Now we can subtract the two fractions:
\[ \text{Work rate of A} = \frac{9}{180} - \frac{5}{180} = \frac{4}{180} \]
\[ \text{Work rate of A} = \frac{1}{45} \text{ work per day} \]

### Step 6: Calculate the time taken by A to complete the work alone
If A's work rate is \(\frac{1}{45}\) work per day, then the time taken by A to complete the work alone is the reciprocal of the work rate:
\[ \text{Time taken by A} = \frac{1}{\frac{1}{45}} = 45 \text{ days} \]

### Final Answer
A alone will complete the work in **45 days**.

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