To solve this problem, we can use the concept of work rates.

Your rate of painting is $\frac{1}{10}$ of the garage per hour, and your brother's rate is $\frac{1}{15}$ of the garage per hour. When working together, the rates add up.

1. Your rate: $\frac{1}{10}$ (fraction of the garage per hour)
2. Your brother's rate: $\frac{1}{15}$ (fraction of the garage per hour)

Together, your combined rate is:

$\frac{1}{10} + \frac{1}{15}$

To add these fractions, first find a common denominator:

$\frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}$

So, together, you both can paint $\frac{1}{6}$ of the garage per hour.

The time it will take to paint the entire garage together is the reciprocal of your combined rate:

$\text{Time} = \frac{1}{\frac{1}{6}} = 6 \text{ hours}$

Answer: It will take 6 hours for both of you to paint the garage together.

Do you want more details or have any questions?

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Here are 5 related questions to consider:

1. What if you work faster, completing the garage in 8 hours instead of 10?
2. How long would it take if your brother had some help and finished in 12 hours instead of 15?
3. What if you work for 4 hours together, then finish the rest alone—how long would it take?
4. How much of the garage would be painted after 3 hours of working together?
5. What would happen if your brother took breaks and only worked half the time?

Tip: When combining work rates, always add the rates first, and then find the reciprocal to get the total time required for the job.