If Naomi were to paint her living room alone, it would take 5 hours. Her sister Denise could do the job in 8 hours. How many hours would it take them working together? Express your answer as a fraction reduced to lowest terms

To solve this problem, we can use the concept of combined work rates. Let’s break it down step by step:

1. **Naomi's rate of work**:
   Since Naomi can complete the entire job (painting the living room) in 5 hours, her work rate is:
   \[
   \text{Naomi's rate} = \frac{1}{5} \text{ of the job per hour}.
   \]

2. **Denise's rate of work**:
   Denise can complete the same job in 8 hours, so her work rate is:
   \[
   \text{Denise's rate} = \frac{1}{8} \text{ of the job per hour}.
   \]

3. **Combined work rate**:
   When Naomi and Denise work together, their rates are additive. The combined rate is:
   \[
   \text{Combined rate} = \frac{1}{5} + \frac{1}{8}.
   \]
   To add these fractions, we need a common denominator. The least common denominator of 5 and 8 is 40, so we rewrite the fractions:
   \[
   \frac{1}{5} = \frac{8}{40}, \quad \frac{1}{8} = \frac{5}{40}.
   \]
   Now, add the fractions:
   \[
   \text{Combined rate} = \frac{8}{40} + \frac{5}{40} = \frac{13}{40}.
   \]

4. **Time to complete the job**:
   The combined rate of \(\frac{13}{40}\) means that together, Naomi and Denise can complete \(\frac{13}{40}\) of the job in 1 hour. The time it takes to complete the entire job is the reciprocal of their combined rate:
   \[
   \text{Time} = \frac{1}{\frac{13}{40}} = \frac{40}{13}.
   \]

Thus, it will take them \(\frac{40}{13}\) hours to complete the job working together.

Would you like a further breakdown or clarification? 

Here are 5 related questions:
1. How would the time change if Naomi worked twice as fast?
2. What if Denise could only work for half the time before taking a break?
3. How would their combined rate change if Denise’s speed decreased by 25%?
4. What is the percentage of the job completed after 1 hour of working together?
5. How long would it take if a third person joined, working at half the rate of Naomi?

**Tip**: When solving work rate problems, remember that the rates of individuals working together add up, not their times.

If Naomi were to paint her living room alone, it would take 5 hours. Her sister Denise could do the job in 8 hours. How many hours would it take them working together? Express your answer as a fraction reduced to lowest terms.

Discover how to calculate the time it takes for Naomi and Denise to paint a living room together, using work rate and fraction addition. Naomi takes 5 hours, while Denise takes 8 hours, and their combined effort is solved step by step. Ideal for students in Grades 6-8.

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