Aerobic Exercise A woman keeps fit by bicycling and running...

Aerobic Exercise A woman keeps fit by bicycling and running every day. On Monday she spends $\frac{1}{2} h$ at each activity, covering a total of 12$\frac{1}{2}$ mi. On Tuesday she runs for 12 min and cycles for 45 min, covering a total of 16 $\mathrm{mi}$ . Assuming that her running and cycling speeds don't change from day to day, find these speeds.

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Key Concepts

Distance-Rate-Time Relationship

This key concept is based on the formula distance = speed × time. It allows one to express the distance covered in terms of the speed and the duration of an activity. This relationship is crucial in problems where varying times and speeds lead to different distances and helps in setting up the equations necessary to solve for unknown speeds.

System of Linear Equations

When a problem provides multiple scenarios leading to different outcomes (in this case, different combinations of running and cycling durations and distances), each scenario can be represented as a linear equation. The simultaneous equations form a system that can be solved using methods such as substitution, elimination, or matrix operations to determine the unknown variables, in this case, the speeds.

Unit Conversion

Unit conversion involves ensuring that all quantities in an equation are expressed in compatible units. In problems like this where time is given both in hours and minutes, converting units (e.g., minutes to hours) is essential to maintain consistency across the equations and to accurately solve for the rate or speed.

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