Problem 14 - Algebra 2 and Trigonometry

Question

Aaron can ride his bicycle to school at an average rate that is three times that of the rate at which he can walk to school. How does the time that it takes Aaron to ride to school compare with the time that it takes him to walk to school?

Step-by-Step Solution

Answer

Aaron takes one-third the time to ride a bicycle compared to walking.

Step 1: Define the Variables

Let the rate at which Aaron walks be \( x \) (miles per hour). Then, the rate at which he rides his bicycle is \( 3x \) (miles per hour).

Step 2: Understand the Relationship Between Rate, Distance, and Time

Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Rate}} \), we can express the time it takes Aaron to walk and ride the bicycle in terms of distance \( d \).

Step 3: Calculate Time to Walk

The time taken for Aaron to walk to school is \( \frac{d}{x} \).

Step 4: Calculate Time to Ride

The time taken for Aaron to ride his bicycle to school is \( \frac{d}{3x} \).

Step 5: Compare the Times

To compare, we look at \( \frac{d}{x} \) versus \( \frac{d}{3x} \). \( \frac{d}{3x} \) is \( \frac{1}{3} \) of \( \frac{d}{x} \), meaning riding is faster. Thus, the time to ride is \( \frac{1}{3} \) of the time to walk.

Key Concepts

Rates of Speed

In understanding how fast Aaron travels to school, we need to talk about rates of speed. Rate of speed refers to the distance traveled over a certain period. It's usually expressed in terms of miles per hour (mph) or kilometers per hour (kph). In Aaron's case, his walking speed is defined as a basic rate of speed, let's call it \( x \) mph. When Aaron rides his bicycle, he moves at three times this rate. So, the rate at which he cycles becomes \( 3x \) mph.
Imagine measuring how fast you are going by counting how many units you cover per time unit, like miles in one hour. Knowing Aaron's rates helps in understanding the time it might take him to travel a given distance.

• Aaron's walking rate: \( x \) mph
• Aaron's biking rate: \( 3x \) mph

Time Calculation

Calculating how long it takes to travel any distance requires understanding the relationship between time, distance, and speed. The formula to use here is \( \text{Time} = \frac{\text{Distance}}{\text{Rate}} \). This equation shows that time taken is inversely proportional to the rate when the distance is kept constant. In simple words, the faster you travel, the less time it takes to cover the same distance.
For Aaron, the time to walk to school is \( \frac{d}{x} \). This is derived by dividing the distance \( d \) by his walking rate \( x \). Meanwhile, when he rides his bicycle, the time becomes \( \frac{d}{3x} \), as the same distance is now covered at a speed three times faster.

• Walking Time: \( \frac{d}{x} \)
• Riding Time: \( \frac{d}{3x} \)

This calculation nicely sets the stage to understand how changing the rate of speed affects travel time.

Comparison of Travel Times

Comparing travel times gives insight into how different rates of speed can influence the overall time taken to travel the same route. Aaron's situation provides a simple yet effective picture of how time reduces as speed increases.
We already established that Aaron's bike riding speed is faster than his walking speed. So, let's compare his travel times: - The time taken to walk to school is \( \frac{d}{x} \).- The time taken to bike is \( \frac{d}{3x} \).
Here, \( \frac{d}{3x} \) is exactly one third of \( \frac{d}{x} \). In easier terms, Aaron takes three times longer to walk the same distance than he does biking it. Hence, riding is much quicker than walking when covering the same distance.
By reducing the travel time by a factor of three, Aaron has efficiently utilized his increased speed to shorten his commute.