Solving a Catch-Up Uniform Motion Problem Where One Driver Leaves 15 Minutes Later
Apply the seven-step problem-solving strategy for systems of linear equations to a uniform motion catch-up scenario where a fractional time delay is involved.
Problem: Charlie left his mother’s house traveling at an average speed of miles per hour. His sister Sally left minutes ( hour) later traveling the same route at an average speed of miles per hour. How long before Sally catches up to Charlie?
- Read the problem and draw a diagram showing both drivers traveling the same route. Create a rate–time–distance table.
- Identify: The travel time for each driver.
- Name: Let = Charlie's driving time (in hours) and = Sally's driving time (in hours). Identify the rates: Charlie's rate is mph, Sally's rate is mph. Multiply rate by time to fill in the distance column:
| Rate (mph) | Time (hrs) | Distance (miles) | |
|---|---|---|---|
| Charlie | |||
| Sally |
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Translate into a system of equations. Sally catches Charlie when they have traveled the same distance: . Since Sally left hour later, her time is hour less than Charlie's: . The system is:
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Solve by substitution. Substitute into the second equation: . Distribute: (or ). Subtract from both sides: . Divide by : . Substitute into the first equation: .
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Check: Charlie: miles. Sally: miles. Both distances are equal.
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Answer: Sally will catch up to Charlie in 1.5 hours.
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