14 hours. If his friend Huck​ helps, they can paint the fence in 1 hour. How fast could Huck paint the fence by​ himself?

Let's solve this problem step by step.

### Step 1: Define the Variables
- Let \( R \) be the rate at which Huck paints the fence alone (in terms of fence per hour).
- Tom's rate is \( \frac{1}{14} \) of the fence per hour since he can paint the entire fence in 14 hours.
- Together, Tom and Huck can paint the fence in 1 hour, so their combined rate is 1 fence per hour.

### Step 2: Write the Equation
Since Tom and Huck are working together, their rates add up:
\[
\text{Tom's rate} + \text{Huck's rate} = \text{Combined rate}
\]
\[
\frac{1}{14} + R = 1
\]

### Step 3: Solve for Huck's Rate \( R \)
Subtract Tom's rate from both sides:
\[
R = 1 - \frac{1}{14}
\]
To simplify, convert 1 into a fraction with a denominator of 14:
\[
R = \frac{14}{14} - \frac{1}{14} = \frac{13}{14}
\]
So, Huck's rate is \( \frac{13}{14} \) of the fence per hour.

### Step 4: Find the Time Huck Takes to Paint the Fence
The time Huck would take to paint the fence by himself is the reciprocal of his rate:
\[
\text{Time} = \frac{1}{R} = \frac{1}{\frac{13}{14}} = \frac{14}{13} \text{ hours}
\]

### Final Answer:
Huck could paint the fence by himself in **\(\frac{14}{13}\) hours**, which is approximately **1.08 hours**.

Would you like any details or have any questions? 

Here are 5 related questions:
1. What if Tom could paint the fence in 10 hours? How long would it take Huck alone then?
2. How would the result change if both together could paint the fence in 2 hours?
3. How long would it take both of them together to paint the fence if Huck worked alone at 2 hours per fence?
4. If Huck slowed down and painted the fence in 20 hours, how would that affect the joint time?
5. How could we model the problem if a third friend joined in to help?

**Tip:** Always double-check your rate and time conversions to avoid small mistakes in problems like these!

14 hours. If his friend Huck helps, they can paint the fence in 1 hour. How fast could Huck paint the fence by himself?

Learn how to determine Huck's painting speed when he and his friend can paint a fence together in 1 hour. This algebra problem involves understanding rates, work problems, and how to calculate individual rates based on a combined effort. Suitable for students in Grades 7-9.

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