A and B together can do a piece of work in 40 days. A, after working for 24 days, B completes the remaining work alone in 54 days. In how many days B alone can complete the whole work?

Let's solve this problem step by step.

1. Let the total work be denoted as \( W \).
2. The rate of work done by A and B together is \( \frac{1}{40} \) of the total work per day. This is because they can complete the whole work in 40 days.
   
   Therefore, A and B together complete \( \frac{W}{40} \) of the work per day.

3. Let the rate of work done by A alone be \( \frac{1}{A} \) of the work per day, and the rate of work done by B alone be \( \frac{1}{B} \) of the work per day.

From the first condition:
- A works for 24 days. During this time, A and B together would have completed \( 24 \times \frac{W}{40} = \frac{24W}{40} = \frac{3W}{5} \) of the total work.

So, after 24 days, \( 1 - \frac{3W}{5} = \frac{2W}{5} \) of the work remains.

4. According to the second condition, B alone completes the remaining \( \frac{2W}{5} \) of the work in 54 days.

Thus, the rate at which B works is \( \frac{\frac{2W}{5}}{54} = \frac{2W}{5 \times 54} = \frac{W}{135} \) of the work per day.

5. Now, we know that B completes \( \frac{W}{135} \) of the work in one day. Therefore, B alone can complete the entire work in 135 days.

Thus, B alone can complete the whole work in **135 days**.

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### Related Questions:
1. If A worked alone, how many days would it take for A to complete the work?
2. If A and B work together for 20 days and then B works alone, how many more days would B need to finish the work?
3. If A worked for 30 days and B worked for 40 days, what fraction of the work would be completed?
4. If A can finish the work in 60 days, how many days will it take for A and B together to finish the work?
5. How much more efficient is B compared to A based on their individual working days?

### Tip:
In such work-rate problems, converting everything into rates of work per day and then using fractions is the most straightforward approach!

Learn how to solve a work and time problem where A and B together finish a task in 40 days, but after 24 days, B alone completes the remaining work in 54 days. Find out how long it takes for B to complete the entire work alone using work rate formulas and fractions. Suitable for Grades 8-10 students.

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