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.