A and B can complete a work in 20 days and A alone can finish that work in 30 days. In how many days B can complete the work?

Answer

To solve the problem step by step, we will use the information given about the work done by A and B.

### Step 1: Determine the total work
We know that A and B together can complete the work in 20 days. To find the total work, we can use the formula:
\[ \text{Total Work} = \text{Work Rate} \times \text{Time} \]

Since A and B together can complete the work in 20 days, we can calculate the total work as follows:
\[ \text{Total Work} = 1 \text{ (work)} \times 20 \text{ (days)} = 20 \text{ work units} \]

### Step 2: Determine A's work rate
A alone can finish the work in 30 days. Therefore, A's work rate (efficiency) can be calculated as:
\[ \text{A's Work Rate} = \frac{1 \text{ (work)}}{30 \text{ (days)}} = \frac{1}{30} \text{ work units per day} \]

### Step 3: Determine the combined work rate of A and B
Since A and B together can complete the work in 20 days, their combined work rate is:
\[ \text{Combined Work Rate (A + B)} = \frac{1 \text{ (work)}}{20 \text{ (days)}} = \frac{1}{20} \text{ work units per day} \]

### Step 4: Calculate B's work rate
Now, we can find B's work rate by subtracting A's work rate from the combined work rate:
\[ \text{B's Work Rate} = \text{Combined Work Rate} - \text{A's Work Rate} \]
\[ \text{B's Work Rate} = \frac{1}{20} - \frac{1}{30} \]

To perform this subtraction, we need a common denominator. The least common multiple of 20 and 30 is 60. Therefore, we convert the fractions:
\[ \frac{1}{20} = \frac{3}{60} \]
\[ \frac{1}{30} = \frac{2}{60} \]

Now, we can subtract:
\[ \text{B's Work Rate} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \text{ work units per day} \]

### Step 5: Calculate the time taken by B to complete the work alone
To find out how many days B can complete the work alone, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{B's Work Rate}} \]
\[ \text{Time} = \frac{1 \text{ (work)}}{\frac{1}{60} \text{ (work units per day)}} = 60 \text{ days} \]

### Final Answer
B can complete the work alone in **60 days**.

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