Let's break down the problem step-by-step to find the total distance Ted travels.

1. Define Variables:

   • Let $d$ be the distance (in miles) of the uphill path.
   • Ted's climbing rate (uphill) is 3 miles per hour.
   • Ted's running rate (downhill) is 5 miles per hour.
   • It takes Ted 40 minutes longer to climb up than to run down.

2. Formulate Equations:

   • Time taken to climb up the hill: $\frac{d}{3}$ hours.
   • Time taken to run down the hill: $\frac{d}{5}$ hours.
   • The difference in time between climbing up and running down is 40 minutes, which is $\frac{40}{60} = \frac{2}{3}$ hours.

   So, we set up the equation: $\frac{d}{3} - \frac{d}{5} = \frac{2}{3}$

3. Solve for $d$:

   • Find a common denominator to combine the terms on the left side. $\frac{5d - 3d}{15} = \frac{2}{3}$ $\frac{2d}{15} = \frac{2}{3}$
   • Cross-multiply to solve for $d$: $2d = 10$ $d = 5$

4. Calculate Total Distance:

   • The distance up the hill is $d = 5$ miles.
   • The total distance Ted travels (up and down) is: $5 + 5 = 10 \text{ miles}$

Answer: Ted travels a total of 10 miles.

Would you like a further breakdown of any part of this solution?

Follow-Up Questions:

1. How would the total distance change if Ted's speed downhill increased?
2. What if the time difference was different, say 30 minutes, how would that affect the total distance?
3. Could this problem be solved using a different method, such as using proportions?
4. What is the effect of Ted’s uphill and downhill speeds on the time difference?
5. How would we set up this problem if Ted’s rate uphill and downhill were both variables instead of constants?

Tip:

For problems involving different speeds and time differences, always convert time differences to consistent units (like hours) and set up an equation based on distance = rate × time.