Finding Flat and Uphill Biking Speeds Using a Known Total Distance
Apply the distance, rate, and time problem-solving strategy to find two unknown uniform speeds for consecutive trip segments when the segment distances add up to a known total.
Problem: Phuong left home on his bicycle at 10:00. He rode on the flat street until 11:15, then rode uphill until 11:45. He rode a total of miles. His speed riding uphill was times his speed on the flat street. Find his speed biking uphill and on the flat street.
- Read and draw: Sketch the route from home with two segments: flat street (10:00 to 11:15) and uphill (11:15 to 11:45). The total distance is miles. Create a rate–time–distance table.
- Identify: Phuong's biking speed on the flat street and his biking speed uphill.
- Name: Let = the speed on the flat street in mph. Because his uphill speed was times as fast, it equals . Convert the clock times to elapsed times: riding on the flat street lasts from 10:00 to 11:15, which is hours; riding uphill lasts from 11:15 to 11:45, which is hours. Multiply rate by time to fill in the distance column:
| Rate (mph) | Time (hrs) | Distance (miles) | |
|---|---|---|---|
| Flat | |||
| Uphill | |||
| Total |
- Translate: The flat distance plus the uphill distance equals the total of miles:
- Solve: Multiply : Combine like terms: . Divide both sides by : The speed on the flat street is mph. The uphill speed is mph.
- Check: Flat street: miles. Uphill: miles. Total distance: miles.
- Answer: Phuong's speed biking on the flat street was mph and his speed biking uphill was mph.
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