Problem 46 - Prealgebra and Introductory Algebra

Question

As part of a conditioning program, a jogger ran \(8 \mathrm{mi}\) in the same amount of time it took a cyclist to ride \(20 \mathrm{mi}\). The rate of the cyclist was \(12 \mathrm{mph}\) faster than the rate of the jogger. Find the rate of the jogger and the rate of the cyclist.

Step-by-Step Solution

Answer

The jogger's speed is 8 mph and the cyclist's speed is 20 mph.

Step 1: Understand the relationship

The relationship between speed, time and distance is given by, Speed = Distance/Time. From the problem, we understand that the time for both jogger and cyclist is the same. Let the speed of jogger be \(x\) mph. Since the cyclist's speed is 12 mph faster, it will be \(x+12\) mph.

Step 2: Set up the equations

We now write the equation based on the speed, distance and time relation for both jogger and cyclist. For jogger, we have \(Time = Distance / Speed = 8 / x\). Similarly, for cyclist, \(Time = Distance / Speed = 20 / (x+12)\). Since their times are equal, we can set these two equations equal to each other, \(8 / x = 20 / (x + 12)\)

Step 3: Simplify the equation

Cross multiply the fractions in the equation from step 2, we get, \(8(x+12) = 20x\). Distribute 8 on left, giving \(8x + 96 = 20x\). Rearrange terms to get \(12x = 96\)

Step 4: Solve for the unknown

Divide both sides of the equation by 12 to find the jogger's speed. \(x = 96/12 = 8\) mph. To find the cyclist's speed, add 12 to the jogger's speed. So, cyclist's speed = \(8+12 = 20\) mph

Step 5: Interpret the solution

The solution to the equations indicates that the jogger's speed is 8 mph and the cyclist's speed is 20 mph

Key Concepts

Distance-Speed-Time Relationship

Understanding the distance-speed-time relationship is crucial in solving problems like the one given in the exercise. The fundamental equation is:

• Speed = Distance / Time

This equation helps us determine how fast an object is moving or how far it goes in a certain amount of time. In the exercise, both the jogger and the cyclist cover different distances but in the same time period.
By rewriting the formula, we can express distance or time in terms of the other two variables. For example:

• Distance = Speed × Time
• Time = Distance / Speed

In our scenario, since the jogger runs 8 miles and the cyclist rides 20 miles but for the same time, these formulas allow us to compare their speeds. This is a practical use of the distance-speed-time relationship to find unknown quantities given certain conditions.

Equation Solving

Equation solving is about finding the value of an unknown variable by using mathematical techniques. The exercise requires setting up and solving equations to find the jogger's and the cyclist's speed. Firstly, we establish the speed relationship for both individuals using variables.
Let the jogger's speed be represented by \( x \) mph. Therefore, the cyclist's speed, which is 12 mph faster, is given by \( x + 12 \) mph.

• Jogger's time: \( \frac{8}{x} \)
• Cyclist's time: \( \frac{20}{x+12} \)

Since the time for both is the same, we set these equal: \[ \frac{8}{x} = \frac{20}{x + 12} \]Cross-multiplying helps us remove the fractions, leading to a simpler form. Solving this equation ultimately gives us the value for \( x \), the jogger's speed.
This step-by-step application involves basic algebra rules and is fundamental in finding solutions to problems involving multiple unknowns. It enhances our problem-solving skills by allowing multiple approaches for reaching the desired solution.

Linear Equations

Linear equations are equations of the first degree, meaning they involve no exponents higher than one. In our exercise, the relationship defined by the equation \( 8(x+12) = 20x \) is linear. Linear equations like this have a specific format: \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. Here’s how we handle the equation:

• Expand and distribute: \( 8x + 96 = 20x \)
• Rearrange by isolating terms: \( 20x - 8x = 96 \)
• Simplify: \( 12x = 96 \)

This rearrangement allows us to solve the equation for \( x \), representing the jogger's speed.
Once solved, the outcome \( x = 8 \) directly gives the jogger's speed in mph. Meanwhile, adding 12 to this value because of the problem's conditions provides us with the cyclist's speed.
Solving linear equations is a foundational skill in algebra, and understanding this process is key to tackling more complex mathematical challenges.