1Cademy - Solving a Uniform Motion Application: Driving Rates on Different Roads

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Problem-Solving Strategy for Distance, Rate, and Time Applications

Example

Solving a Uniform Motion Application: Driving Rates on Different Roads

Apply the distance, rate, and time problem-solving strategy to find unknown speeds when a trip is divided into segments with a known total time.

Problem: Joon drove $4$ hours to his home, driving $208$ miles on the interstate and $40$ miles on country roads. If he drove $15$ mph faster on the interstate than on the country roads, what was his rate on the country roads?

Read and draw: Sketch the two segments of the trip: $208$ miles on the interstate and $40$ miles on country roads. Create a rate-time-distance table.
Identify: Joon's driving rate on the country roads.
Name: Let $r$ = Joon's rate on country roads. His interstate rate is $r + 15$ . The time spent on the interstate is $\frac{208}{r + 15}$ and the time spent on country roads is $\frac{40}{r}$ .
Translate: The total driving time is $4$ hours, meaning the sum of the times is $4$ : $\frac{208}{r + 15} + \frac{40}{r} = 4$
Solve: Multiply both sides by the least common denominator, $r(r + 15)$ : $208r + 40(r + 15) = 4r(r + 15)$ $208r + 40r + 600 = 4r^2 + 60r$ $248r + 600 = 4r^2 + 60r$ $0 = 4r^2 - 188r - 600$ Divide by $4$ : $0 = r^2 - 47r - 150$ $0 = (r - 50)(r + 3)$ The solutions are $r = 50$ and $r = -3$ . Speed cannot be negative, so $r = 50$ mph.
Check: Interstate time: $\frac{208}{50 + 15} = \frac{208}{65} = 3.2$ hours. Country road time: $\frac{40}{50} = 0.8$ hours. Total time: $3.2 + 0.8 = 4$ hours. $\checkmark$
Answer: Joon's rate on the country roads was $50$ mph.

Updated 2026-05-01

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OpenStax Intermediate Algebra

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