1. sam can paint a house in 5 hours. Gary can do it in 4 hours. How long will it take the two working together?

To find out how long it will take Sam and Gary to paint the house together, we can use the formula for combined work rates. 

### Formula:
\[
\frac{1}{T} = \frac{1}{t_1} + \frac{1}{t_2}
\]
where:
- \( T \) is the time it takes for both Sam and Gary to finish the job together,
- \( t_1 \) is Sam's time to finish the job alone (5 hours),
- \( t_2 \) is Gary's time to finish the job alone (4 hours).

### Step-by-step solution:

1. Sam's rate of work is \(\frac{1}{5}\) of the house per hour.
2. Gary's rate of work is \(\frac{1}{4}\) of the house per hour.
3. Adding their work rates together:
   \[
   \frac{1}{T} = \frac{1}{5} + \frac{1}{4}
   \]

4. To add the fractions, find a common denominator:
   \[
   \frac{1}{T} = \frac{4}{20} + \frac{5}{20} = \frac{9}{20}
   \]

5. Invert to solve for \( T \):
   \[
   T = \frac{20}{9} \text{ hours}
   \]

### Final Answer:
It will take Sam and Gary \(\frac{20}{9}\) hours, or approximately **2.22 hours**, to paint the house together.

Would you like more details on any part of this? 

Here are 5 related questions for practice:
1. If Sam works for 2 hours alone and Gary finishes the job, how long will Gary take?
2. If both work together for 1 hour and then Gary leaves, how much longer will Sam need to finish the job?
3. What if Sam can paint the house in 7 hours and Gary in 4? How long together now?
4. If Gary works 25% slower than normal, how long will it take them both?
5. How long will it take if Sam works twice as fast, but Gary stays the same?

**Tip**: Always check if the units of the work rate are consistent when combining tasks like this!

Sam can paint a house in 5 hours. Gary can do it in 4 hours. How long will it take the two working together?

Discover how long it will take Sam and Gary to paint a house together using work rate formulas. This step-by-step solution covers key concepts like fraction addition and reciprocal rates, ideal for students in Grades 6-8.

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