Problem 298 - Elementary Algebra

Question

In the following exercises, solve. Julian rides his bike uphill for 45 minutes, then turns around and rides back downhill. It takes him 15 minutes to get back to where he started. His uphill speed is 3.2 miles per hour slower than his downhill speed. Find Julian's uphill and downhill speed.

Step-by-Step Solution

Answer

Julian's downhill speed is 4.8 mph, and his uphill speed is 1.6 mph.

Step 1 - Define Variables

Let Julian's downhill speed be denoted as \( d \) miles per hour. Then, his uphill speed will be \( d - 3.2 \) miles per hour.

Step 2 - Convert Time to Hours

Julian rides uphill for 45 minutes and downhill for 15 minutes. Convert these times to hours: 45 minutes equals 0.75 hours, and 15 minutes equals 0.25 hours.

Step 3 - Set Up Distance Equations

The distance Julian travels uphill is equal to his uphill speed times the uphill travel time, and the distance he travels downhill is equal to his downhill speed times the downhill travel time. Set up the equations: \[ \text{Uphill Distance} = (d - 3.2) \times 0.75 \] \[ \text{Downhill Distance} = d \times 0.25 \] Since these distances are equal, we can set them equal to each other: \[ (d - 3.2) \times 0.75 = d \times 0.25 \]

Step 4 - Solve for Downhill Speed

Distribute and simplify the equation: \[ 0.75d - 2.4 = 0.25d \] Subtract 0.25d from both sides: \[ 0.75d - 0.25d - 2.4 = 0 \] Simplify further: \[ 0.5d - 2.4 = 0 \] Add 2.4 to both sides: \[ 0.5d = 2.4 \] Divide both sides by 0.5: \[ d = 4.8 \] Thus, Julian's downhill speed is 4.8 miles per hour.

Step 5 - Determine Uphill Speed

Since Julian's uphill speed is 3.2 miles per hour slower than his downhill speed, subtract 3.2 from 4.8: \[ \text{Uphill Speed} = 4.8 - 3.2 = 1.6 \]Thus, Julian's uphill speed is 1.6 miles per hour.

Key Concepts

speed and distance

Understanding the relationship between speed, distance, and time is fundamental in solving many real-world problems. In Julian's biking scenario, we have two key distances—uphill and downhill—which are the same. Speed is how fast an object moves, often measured in miles per hour (mph). Distance is the total length of the journey, measured in miles. Time is how long the journey takes, measured here in hours.
When calculating distance, we use the formula:

• Distance = Speed × Time

Applying this to Julian’s trip, we had to consider both his uphill and downhill travels. His uphill speed is slower due to the additional effort required for climbing—a common scenario in real-life biking.

solving equations

Solving equations is a step-by-step process that allows us to find an unknown variable by applying various algebraic methods. In the given problem, we needed to find Julian's speeds both uphill and downhill.
First, we defined the variables:

• Let Julian's downhill speed be denoted as d (in miles per hour).
• Julian's uphill speed is d - 3.2.

Next, we set up the equations based on the distance calculation. Because the distances uphill and downhill are equal, we used this equality to set up our main equation:

• (d - 3.2) × 0.75 = d × 0.25

We then distributed and simplified the terms to isolate the variable d and solved for Julian's downhill speed, which turned out to be 4.8 miles per hour. Finally, we calculated the uphill speed by subtracting 3.2 from the downhill speed.

time conversion

Time conversion is crucial here since Julian's travel times were initially given in minutes. To incorporate these into our equations properly, we must convert minutes to hours.
Here's how we did it:

• 45 minutes becomes 0.75 hours because \(45 ÷ 60 = 0.75\).
• 15 minutes becomes 0.25 hours because \(15 ÷ 60 = 0.25\).

This conversion allows us to use the same time unit (hours) consistently in our equations, aligning with the speeds given in miles per hour. Consistency in units ensures that our calculation remains accurate throughout the problem-solving process.
Understanding and performing time conversions accurately is essential not only in math problems but also in many practical situations.