If A can do a work in 8 days and B can do it in 12 days. Thay work together for 3

days and then B left the work. In how many days can A finish the remaining work?

To solve this problem, we first need to determine how much work A and B can accomplish together in 3 days, and then figure out how much of the work remains for A to finish alone.

### Step-by-Step Solution:

1. **Calculate the work rate of A and B:**
   - A can complete the work in 8 days, so A's work rate is \(\frac{1}{8}\) of the work per day.
   - B can complete the work in 12 days, so B's work rate is \(\frac{1}{12}\) of the work per day.

2. **Calculate the combined work rate of A and B:**
   \[
   \text{Combined work rate} = \frac{1}{8} + \frac{1}{12}
   \]

   To add these fractions, find a common denominator (which is 24):
   \[
   \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24}
   \]
   \[
   \text{Combined work rate} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24} \text{ of the work per day.}
   \]

3. **Calculate the amount of work done in 3 days by A and B together:**
   \[
   \text{Work done in 3 days} = 3 \times \frac{5}{24} = \frac{15}{24} = \frac{5}{8} \text{ of the work.}
   \]

4. **Determine the remaining work:**
   Since \(\frac{5}{8}\) of the work is completed, the remaining work is:
   \[
   1 - \frac{5}{8} = \frac{3}{8} \text{ of the work.}
   \]

5. **Calculate the time for A to finish the remaining work:**
   A's work rate is \(\frac{1}{8}\) of the work per day. To find out how many days A will take to finish the remaining \(\frac{3}{8}\) of the work:
   \[
   \text{Time for A} = \frac{\text{Remaining work}}{\text{A's work rate}} = \frac{\frac{3}{8}}{\frac{1}{8}} = 3 \text{ days.}
   \]

### Conclusion:

A will take **3 days** to finish the remaining work.

---

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions you might find interesting:
1. What if B had worked for 5 days instead of 3 before leaving?
2. How long would it take for A and B to complete the work together?
3. If C could complete the work in 6 days, how long would it take for A, B, and C to complete the work together?
4. What is the combined work rate if three people with different work rates are involved?
5. How does changing one person's work rate affect the total time to complete the task?

**Tip:** When working with rates, it helps to think of them as fractions of the work done per unit of time, which can make it easier to combine and compare them!

If A can do a work in 8 days and B can do it in 12 days. They work together for 3 days and then B left the work. In how many days can A finish the remaining work?

This problem demonstrates how to calculate the time required for A to finish the remaining work after A and B worked together for 3 days. Learn how to use work rates and fractions to solve this real-life problem. Suitable for students in Grades 6-8.

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