Carlos can travel 40 mi on his motorbike in the same time it...

Carlos can travel 40 $\mathrm{mi}$ on his motorbike in the same time it takes Paul to travel 15 $\mathrm{mi}$ on his bicycle. If Paul rides his bike 20 $\mathrm{mi} / \mathrm{h}$ slower than Carlos rides his motorbike, find the speed for each bike.

Key Concepts

System of Equations

A system of equations is used when multiple relationships between unknown quantities need to be resolved together. In motion problems like these, one equation may represent the ratio of distances (or speeds) while another represents the constant difference between the speeds. Solving the system allows for finding the values of the unknowns in a coherent manner.

Relative Speed Differences

Relative speed differences involve comparing the speeds of two moving objects by expressing one speed as a certain amount faster or slower than the other. This concept introduces an algebraic relationship between the speeds, which is used to establish a connection between the variables and solve for their values.

Proportional Relationships in Motion

When two objects travel for the same duration, the ratio of their distances is directly proportional to the ratio of their speeds. This proportionality allows for the creation of equations that connect the movements of different objects, making it possible to compare and relate their speeds based on the distances they cover in equal time.

Rate-Time-Distance Relationship

This concept refers to the fundamental relationship in motion problems where distance is defined as the product of speed and time. When the time is known or fixed, the speed can be determined by dividing the distance traveled by the time taken, and vice versa. This relationship is essential for setting up equations in problems involving movement over a given period.

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