Problem 55 - Algebra for College Students

Question

Solve each of Problems \(47-62\) by setting up. Juan starts walking at 4 miles per hour. An hour and a half later, Cathy starts jogging along the same route at 6 miles per hour. How long will it take Cathy to catch up with Juan?

Step-by-Step Solution

Answer

3 hours

Step 1: Define the variables

Let \( t \) be the time in hours that Cathy is jogging to catch up with Juan. We need to find \( t \).

Step 2: Write the equation for Juan's distance

Juan started 1.5 hours earlier. In that time, he has walked \( 4 \times 1.5 = 6 \) miles. Therefore, if he walks for \( t + 1.5 \) hours, the total distance he covers is \( 4(t + 1.5) \) miles.

Step 3: Write the equation for Cathy's distance

Cathy is jogging at 6 miles per hour for \( t \) hours, so the distance she covers is \( 6t \) miles.

Step 4: Set the distances equal

To find when Cathy catches up with Juan, set Cathy's distance equal to Juan's distance: \( 6t = 4(t + 1.5) \).

Step 5: Solve for \( t \)

Distribute the 4 in the equation: \[ 6t = 4t + 6 \]. Subtract \( 4t \) from both sides to get: \[ 2t = 6 \]. Divide both sides by 2: \[ t = 3 \].

Step 6: Conclude the solution

It will take Cathy 3 hours to catch up with Juan.

Key Concepts

Algebra Word Problems

Algebra word problems can seem daunting at first, but they are just real-world scenarios that need mathematical solutions. Problems often involve finding unknown values based on given information, similar to puzzles. In this exercise, Juan and Cathy's speeds and starting times provide the information needed to find when Cathy catches up to Juan.

The key is to translate words into mathematical expressions. This involves identifying what you need to find—like Cathy's catch-up time—and expressing it with variables. It's important to pick variables that clearly represent the problem, like using \(t\) for Cathy's time jogging. By systematically going through each part of the problem, you can create equations that relate all the pieces of information, making it easier to solve.

Solving Equations

Solving equations is a central skill in algebra, crucial for solving problems like the one involving Juan and Cathy. Once equations are established, the next step is solving them by isolating the variable of interest.

For this problem, set distances equal because Cathy catches up when they have traveled the same distance. For Cathy, the equation is \(6t\) miles, and for Juan: \(4(t + 1.5)\) miles. Equating them gives:

• \(6t = 4(t + 1.5)\)
• Simplify to: \(6t = 4t + 6\)
• Subtract \(4t\) from both sides resulting in: \(2t = 6\)
• Finally, divide by 2 to find \(t = 3\)

This result tells us Cathy needs 3 hours to meet Juan. Solving these equations involves careful algebraic manipulation and checking your work to avoid mistakes.

Mathematical Modeling

Mathematical modeling involves using mathematical equations to represent real-world situations. This process makes complex problems more manageable and allows for predictions and insights.

In this scenario, mathematical modeling helps us understand the distance-rate-time relationship between Juan and Cathy. By representing each person's journey with simple equations, we can accurately predict when Cathy will catch up. It's about taking real-world variables —such as speed, time, and distance— and incorporating them into equations. With practice, anyone can be adept at setting up these models to solve various real-life problems.

Mathematical modeling is not just about numbers but understanding and representing the relationship between different elements of a situation to find solutions, making decision-making and problem-solving more efficient.