A can complete a work in 15 days working 8 hours a day B can complete the same work in 10 days working 9 hours a day. If both A and B work together, working 8 hours a day, in how many days can they complete the work?

To solve this problem, let's start by calculating the work done by A and B individually in one hour.

### Step 1: Calculate A's work rate
- A can complete the work in 15 days, working 8 hours each day.
- Total work hours for A = \( 15 \times 8 = 120 \) hours.
- Work rate of A = \( \frac{1}{120} \) of the work per hour.

### Step 2: Calculate B's work rate
- B can complete the work in 10 days, working 9 hours each day.
- Total work hours for B = \( 10 \times 9 = 90 \) hours.
- Work rate of B = \( \frac{1}{90} \) of the work per hour.

### Step 3: Calculate the combined work rate
- When A and B work together for 8 hours a day, their combined work rate per hour is the sum of their individual work rates:
  \[
  \text{Combined work rate} = \frac{1}{120} + \frac{1}{90}
  \]

To find the combined rate, we calculate:
\[
\frac{1}{120} = \frac{1}{120} \quad \text{and} \quad \frac{1}{90} = \frac{1}{90}
\]

To add these fractions, we need a common denominator:
\[
\frac{1}{120} + \frac{1}{90} = \frac{3}{360} + \frac{4}{360} = \frac{7}{360}
\]

So, the combined work rate is \( \frac{7}{360} \) of the work per hour.

### Step 4: Calculate the total time to complete the work
- Since they work 8 hours a day together, the amount of work they complete in one day is:
  \[
  \text{Work per day} = 8 \times \frac{7}{360} = \frac{56}{360} = \frac{7}{45}
  \]

- The total time required to complete the work is:
  \[
  \text{Total days} = \frac{1}{\frac{7}{45}} = \frac{45}{7} \approx 6.43 \text{ days}
  \]

Thus, A and B working together for 8 hours a day can complete the work in approximately **6.43 days**.

Would you like more details, or do you have any other questions?

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Here are 5 related questions:
1. How would the completion time change if they worked 10 hours a day?
2. If B worked alone for the first 2 days, how long would it take them together to finish the remaining work?
3. What if A works for 6 hours and B for 8 hours per day, how many days would they need?
4. How would the calculation change if A took 18 days to complete the work instead of 15?
5. What would be the work completion time if A and B could work different numbers of hours each day?

**Tip:** When solving work problems, breaking down the tasks into rates (work per unit time) helps simplify the calculations, especially when combining efforts.

Learn how to calculate the time required for A and B to complete a work together based on their individual work rates and combined effort. This problem involves understanding work and time calculations with specific rates and hours worked per day. Suitable for students in Grades 9-12.

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