Problem 33 - Elementary Algebra

Question

The distance between Jacksonville and Miami is 325 miles. A freight train leaves Jacksonville and travels toward Miami at 40 miles per hour. At the same time, a passenger train leaves Miami and travels toward Jacksonville at 90 miles per hour. How long will it take the two trains to meet?

Step-by-Step Solution

Answer

The trains will meet in 2.5 hours.

Step 1: Understand the Problem

We are given the distance between Jacksonville and Miami, which is 325 miles. There are two trains moving towards each other, a freight train traveling at 40 mph and a passenger train traveling at 90 mph. We need to find the time it takes for them to meet.

Step 2: Set Up the Equation

Since both trains are moving towards each other, their speeds add up. If we denote the time it takes for them to meet as \( t \) hours, then the equation for the total distance is: \( 40t + 90t = 325 \).

Step 3: Combine the Speeds

Add the speeds of the two trains: \( 40 + 90 = 130 \). Thus, the equation becomes \( 130t = 325 \).

Step 4: Solve for Time

Divide both sides of the equation \( 130t = 325 \) by 130 to solve for \( t \): \( t = \frac{325}{130} \).

Step 5: Simplify the Result

Simplify the fraction: \( \frac{325}{130} = \frac{65}{26} = 2.5 \). Thus, \( t = 2.5 \).

Key Concepts

Distance-Rate-Time Problems

Distance-rate-time problems involve calculating the relationship between distance, speed, and time. These problems often appear in real-world scenarios, such as traveling.

To tackle these problems, use the formula:

• Distance = Rate × Time

where:

• Distance is how far an object travels.
• Rate (or speed) is how fast the object is traveling.
• Time is how long the object has been traveling.

In this exercise, the two trains move towards each other, so their combined speeds effectively reduce the time it takes to meet. It's essential to add the rates because both are closing the gap simultaneously.

Linear Equations

Linear equations form the foundation of solving distance-rate-time problems. They represent relationships where the graph is a straight line. In these exercises, they often involve variables that stand for unknown quantities, like time or speed.

For instance, in our problem, we represent the time it takes for the two trains to meet by the variable \( t \). By setting up the equation \( 40t + 90t = 325 \), we describe the relationship between the rates and the total distance.

• Coefficients such as 40 and 90 indicate the rates of each train.
• The sum of the products of these rates and time should equal the total distance, 325 miles.

Solving these equations involves combining like terms and isolating the variable to find its value.

Solving Equations

To solve equations, you perform operations to isolate the variable of interest. This involves several steps that simplify the expression.

Let's break down the steps to solve the linear equation from the exercise:

• Add the rates: First, add the speeds of both trains: \( 40 + 90 = 130 \). This represents the combined speed.
• Set up the equation: The equation becomes \( 130t = 325 \), illustrating that the combined speeds multiplied by the time they travel equals the total distance.
• Isolate the variable: To find \( t \), divide both sides of the equation by 130: \( t = \frac{325}{130} \).
• Simplify the expression: Simplify \( \frac{325}{130} \) which reduces to \( \frac{65}{26} = 2.5 \).

Thus, the time \( t \) it takes for the two trains to meet is 2.5 hours.