Problem 3 - Introductory Algebra for College Students

Question

Each exercise is a problem involving motion. A jogger runs 4 miles per hour faster downhill than uphill. If the jogger can run 5 miles downhill in the same time that it takes to run 3 miles uphill, find the jogging rate in each direction.

Step-by-Step Solution

Answer

The jogger runs at a speed of 6 mph uphill and 10 mph downhill.

Step 1: Define Variables

Let's denote the speed uphill as \(x\) miles per hour. Given that the speed downhill is 4 miles per hour faster, this can be defined as \(x + 4\) miles per hour.

Step 2: Set Up the Equations

We know that the time it takes to jog in each direction is the same. We also know that time equals distance divided by speed. Therefore, we can write the equation using this information: \(\frac{{3 miles}}{{x (mph)}} = \frac{{5 miles}}{{(x+4) mph}}\).

Step 3: Solve the Equation

Cross multiply: \(5x = 3(x + 4) \Longleftrightarrow 5x = 3x + 12 \Longleftrightarrow 2x = 12\). Solve for \(x\), we get: \(x = 6 mph\).

Step 4: Find the Speed in Each Direction

Therefore, the jogging rate uphill is \(x = 6 mph\). The jogging rate downhill is \(x + 4 = 6 + 4 = 10 mph\).

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In motion problems, algebra helps in establishing relationships between different quantities, such as speed, distance, and time. This is especially useful when dealing with data that is not directly observable, allowing for unknown values to be determined.

In our problem, we started by defining a variable. We let \(x\) represent the jogger's speed uphill in miles per hour. This is an example of using an algebraic variable to represent a quantity we want to find. By doing this, we can set up equations to solve for \(x\).

We also used some basic algebraic operations such as:

• Cross-multiplying to eliminate fractions
• Combining like terms
• Solving linear equations

These steps are crucial in simplifying and eventually solving the equation to find the jogger's speed uphill.

Speed and Distance

Understanding the relationship between speed, distance, and time is key to solving motion problems. The formula \(\text{speed} = \frac{\text{distance}}{\text{time}}\) is fundamental. When you know any two of the quantities, you can calculate the third.

In this exercise, we were given the distances both uphill and downhill, and that these distances are covered in the same time. By applying our speed-distance-time formula, we derived two expressions for time

• Uphill time: \(\frac{3 \, \text{miles}}{x \, \text{mph}}\)
• Downhill time: \(\frac{5 \, \text{miles}}{(x+4) \, \text{mph}}\)

By setting these expressions equal to each other, we formed an equation that allowed us to solve for the speed uphill, using the original information given.

Word Problems

Word problems require translating a text-based scenario into mathematical terms. This process involves identifying pertinent information, defining variables, and setting up an equation representative of the situation.

Key steps in solving word problems include:

• Reading the problem carefully to determine what information is given and what is being asked
• Defining variables to represent unknown quantities
• Constructing equations based on the relationships described in the problem
• Solving these equations using a variety of methods, such as algebraic manipulation

In this jogger scenario, we understood from the problem that the time taken to travel different distances was equal, even though the speeds differed. This crucial insight guided us in setting up the correct equation to solve for the unknowns. Word problems like this can initially seem complex, but with practice, breaking them down into manageable steps becomes more intuitive.