Question

A passenger train leaves the station, traveling west at 120 miles per hour. A freight train leaves the same station 2 hours later traveling 80 miles per hour. If \(t\) represents the time in hours that the first train has traveled, which equation represents a situation where the two trains are 640 miles apart? A. \(120 t+80(t-2)=640\) B. \(120(t+2)+80 t=640\) C. \(120 t+80(t+2)=640\) D. \(120(t-2)+80 t=640\)

Short answer

A: \(120t + 80(t-2) = 640\)

Step-by-step solution

Define the variable

Let the variable t represent the time in hours that the passenger train has been traveling.

Distance traveled by the passenger train

The passenger train travels at 120 miles per hour. Hence, in t hours, the distance traveled by the passenger train can be represented by the equation: Distance_passenger = 120t miles.

Time and distance traveled by the freight train

The freight train starts 2 hours later, so it travels for (t-2) hours. The freight train travels at 80 miles per hour. Hence, the distance traveled by the freight train in (t-2) hours can be represented by the equation: Distance_freight = 80(t-2) miles.

Total distance apart

The problem states that the two trains are 640 miles apart. Therefore, the sum of the distances traveled by both trains should equal 640 miles. This gives us the equation: 120t + 80(t-2) = 640

Combine like terms to verify the equation

Simplify the equation by distributing and combining like terms: 120t + 80t - 160 = 640. Combine the terms: 200t - 160 = 640.

Solve the equation

Add 160 to both sides to solve for t: 200t = 800, t = 4 hours (not needed for the multiple-choice part).

Identify the correct option

Compare the derived equation with the options given. The correct option is: A. \(120t + 80(t-2) = 640\).

Key concepts

Distance-Time Relationship

Understanding the relationship between distance, time, and speed is crucial for solving such problems. The basic formula you need to remember is: Distance = Speed × Time. This tells us that the total distance covered by an object is the product of its speed and the time it has been traveling. In our problem, the passenger train travels at 120 miles per hour, so the distance it travels in \(t\) hours is given by \(120t\). Similarly, the freight train's distance, with a speed of 80 miles per hour and traveling for \((t-2)\) hours, is calculated as \(80(t-2)\). Understanding this relationship makes it easier to set up the equations correctly.

Solving Linear Equations

To find when the two trains are 640 miles apart, we need to solve a linear equation. A linear equation is an equation where the highest power of the variable is 1. In our case, the equation is \(120t + 80(t-2) = 640\). First, distribute 80 to both terms in the parenthesis to get \(120t + 80t - 160 = 640\). Then, combine the like terms \(120t\) and \(80t\) to get \(200t - 160\). Now you have \(200t - 160 = 640\). To isolate \(t\), add 160 to both sides: \(200t = 800\). Finally, divide by 200, giving \(t = 4\). This steps you through how to solve for \(t\), but remember: in real problems, make sure to carefully check your work at each step to prevent mistakes.

Train Travel Word Problems

Train travel word problems often involve a few essential elements: different speeds, different start times, and a relationship between the distances covered. Key steps to solve these problems include:

• Defining variables clearly.
• Writing equations based on the given speeds and times.
• Setting up an equation to find the point of interest (like when they meet or distance apart).

For our problem, it’s the time differential and varying speeds that create the equation \(120t + 80(t-2) = 640\). Always ensure you understand the setup before diving into calculations, and keep an eye on the units (hours, miles, etc.) throughout the problem.

GED Test Preparation

Preparing for the GED test requires understanding and practicing problems that are likely to appear on the exam. Focus areas for GED math often include:

• Basic algebra and linear equations.
• Word problems involving distance, speed, and time.
• Real-world applications of math concepts.

When practicing, consistently use the method of defining variables, setting up equations, and solving step-by-step as shown. Proper practice includes going over a variety of problems and familiarizing yourself with common traps and mistakes. Resources like past papers, online practice tests, and how-to problem guides can be very useful. Remember: understanding the methods rather than just memorizing solutions will help you succeed on the test day.