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Finding Speeds for Consecutive Trip Segments Using a Known Total Distance

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Example: Finding Running and Biking Speeds Using a Known Total Distance

Apply the distance, rate, and time problem-solving strategy to find two unknown speeds for consecutive trip segments when their distances add up to a known total.

Problem: Cruz is training to compete in a triathlon. He left his house at 6:00 and ran until 7:30. Then he rode his bike until 9:45. He covered a total distance of $51$ miles. His speed when biking was $1.6$ times his speed when running. Find Cruz's biking and running speeds.

Read and draw: Sketch the situation with one path labeled "run" (from 6:00 to 7:30) and another labeled "bike" (from 7:30 to 9:45, $1.6$ times faster). The total distance is $51$ miles.
Identify: The running speed and the biking speed.
Name: Let $r$ $r$ = the running speed. The biking speed is $1.6r$ $1.6 r$ . Calculate elapsed times: running from 6:00 to 7:30 is $1.5$ $1.5$ hours; biking from 7:30 to 9:45 is $2.25$ $2.25$ hours. Multiply rate by time to find distance expressions:
- Run: Rate = $r$ , Time = $1.5$ , Distance = $1.5r$
- Bike: Rate = $1.6r$ , Time = $2.25$ , Distance = $2.25(1.6r)$
Translate: The sum of both distances equals the total distance: $1.5r + 2.25(1.6r) = 51$
Solve: Simplify the terms: $1.5r + 3.6r = 51$ $5.1r = 51$ $r = 10$ The running speed is $10$ miles per hour. The biking speed is $1.6 \cdot 10 = 16$ miles per hour.
Check: The running distance is $10 \cdot 1.5 = 15$ miles. The biking distance is $16 \cdot 2.25 = 36$ miles. The total distance is $15 + 36 = 51$ miles. $\checkmark$
Answer: Cruz's running speed is $10$ mph and his biking speed is $16$ mph.

Updated 2026-04-22

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OpenStax Intermediate Algebra

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