Example: Finding Upstream and Downstream Boat Speeds
Apply the distance, rate, and time problem-solving strategy to find two unknown uniform motion speeds when both the upstream and downstream trips cover the same distance, replacing simple rates with relative expressions and converting units.
Problem: Llewyn takes minutes to drive his boat upstream from the dock to his favorite fishing spot. It takes him minutes to drive the boat back downstream to the dock. The boat's speed going downstream is four miles per hour faster than its speed going upstream. Find the boat's upstream and downstream speeds.
- Read and draw: Sketch the route. The upstream trip ( minutes) and the downstream back trip ( minutes, mph faster) cover an identical distance. Create a rate–time–distance table.
- Identify: The upstream speed and the downstream speed.
- Name: Let = upstream speed in mph. Then the downstream speed is . Convert the times to hours: minutes hour, and minutes hour. Multiply rate by time to find the distance expressions:
| Rate (mph) | Time (hrs) | Distance (miles) | |
|---|---|---|---|
| Upstream | |||
| Downstream |
- Translate: Since the boat travels to the fishing spot and returns to the dock, the distances are identical:
- Solve: Clear the fractions by multiplying both sides by the LCD, which is :
Subtract from both sides: . The upstream speed is mph. The downstream speed is mph.
-
Check: Upstream: miles. Downstream: miles. The distance traveled each way is miles.
-
Answer: The boat's upstream speed is mph and its downstream speed is mph.
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Intermediate Algebra @ OpenStax
Ch.2 Solving Linear Equations - Intermediate Algebra @ OpenStax
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Example: Finding Uphill and Downhill Hiking Speeds
Example: Finding Upstream and Downstream Boat Speeds