Cindy and Jenna working together and clean the house in six hours working alone. Jenna takes twice as long as Cindy how long does it take Cindy to clean the house alone

Let's solve the problem step by step.

### Step 1: Define Variables
Let:
- \( t_C \) be the time it takes Cindy to clean the house alone (in hours).
- \( t_J \) be the time it takes Jenna to clean the house alone (in hours).

### Step 2: Relationship Between Times
We are told that Jenna takes twice as long as Cindy:
\[
t_J = 2t_C
\]

### Step 3: Work Rates
The work rate is the fraction of the house cleaned per hour. Therefore:
- Cindy's work rate: \( \frac{1}{t_C} \) of the house per hour.
- Jenna's work rate: \( \frac{1}{t_J} = \frac{1}{2t_C} \) of the house per hour.

### Step 4: Combined Work Rate
When they work together, their combined work rate is:
\[
\frac{1}{t_C} + \frac{1}{2t_C} = \frac{1 + \frac{1}{2}}{t_C} = \frac{\frac{3}{2}}{t_C} = \frac{3}{2t_C}
\]

### Step 5: Use the Given Information
We are told that together they can clean the house in 6 hours. This means:
\[
\frac{3}{2t_C} = \frac{1}{6}
\]

### Step 6: Solve for \( t_C \)
To find \( t_C \), solve the equation:
\[
\frac{3}{2t_C} = \frac{1}{6}
\]

Multiply both sides by \( 2t_C \):
\[
3 = \frac{2t_C}{6}
\]

Simplify the right side:
\[
3 = \frac{t_C}{3}
\]

Now, multiply both sides by 3:
\[
t_C = 9 \text{ hours}
\]

### Conclusion
Cindy takes **9 hours** to clean the house alone.

Would you like further details or have any questions?

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Here are 5 related questions you might find interesting:
1. If Cindy and Jenna worked together with a third person who took 4 hours to clean the house alone, how long would it take all three to clean the house together?
2. What would be the time if Jenna took three times as long as Cindy to clean the house?
3. If Cindy improved her cleaning speed by 25%, how long would it take them to clean the house together?
4. How would the solution change if they could only work for 4 hours together?
5. How long would it take if Cindy and Jenna each worked for half of the time they would take alone?

**Tip:** When dealing with combined work rate problems, always add their individual rates to find the combined rate.

Cindy and Jenna working together and clean the house in six hours working alone. Jenna takes twice as long as Cindy how long does it take Cindy to clean the house alone.

Solve the work rate problem where Cindy and Jenna clean a house together in 6 hours, with Jenna taking twice as long as Cindy. Learn how to calculate the time it takes Cindy to clean the house alone with step-by-step guidance. Suitable for students in Grades 7-9.

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