Problem 37 - Elementary Algebra

Question

Two trains leave at the same time, one traveling east and the other traveling west. At the end of \(9 \frac{1}{2}\) hours they are 1292 miles apart. If the rate of the train traveling east is 8 miles per hour faster than the rate of the other train, find their rates.

Step-by-Step Solution

Answer

The westbound train travels at 64 mph, and the eastbound train travels at 72 mph.

Step 1: Define Variables

Let's define the rate of the train traveling west as \( r \) miles per hour. Then, the rate of the train traveling east is \( r + 8 \) miles per hour because it's 8 miles per hour faster than the other train.

Step 2: Use the Distance Formula

The formula for distance is \( \text{Distance} = \text{Rate} \times \text{Time} \). Since both trains travel for \( 9.5 \) hours, the distance covered by the westbound train is \( r \times 9.5 \) and the distance covered by the eastbound train is \((r + 8) \times 9.5 \).

Step 3: Set Up the Equation

The sum of the distances covered by both trains is the total distance between them, which is 1292 miles. Thus, we have the equation: \( 9.5r + 9.5(r + 8) = 1292 \).

Step 4: Simplify the Equation

Expand and simplify the equation: \( 9.5r + 9.5r + 76 = 1292 \). Simplifying further: \( 19r + 76 = 1292 \).

Step 5: Solve for r

Subtract 76 from both sides: \( 19r = 1216 \). Then, divide both sides by 19 to find \( r \): \( r = \frac{1216}{19} = 64 \).

Step 6: Calculate the Rate of the Eastbound Train

Since the train traveling east is 8 miles per hour faster, its rate is \( r + 8 = 64 + 8 = 72 \).

Key Concepts

Understanding Rate Calculations

Rate calculations involve determining the speed at which an object is moving, which is typically expressed in units like miles per hour or kilometers per hour. When solving word problems involving rates, it is crucial to:

• Identify what each rate represents—such as the speed of an object heading in a particular direction.
• Use variables to express unknown rates.

In the context of the train problem, we define the rate of the train going west as \( r \) miles per hour. Since the train heading east goes faster, we express this rate as \( r + 8 \) miles per hour. By setting these variables, it becomes easier to insert them into equations to find the solution efficiently.
Rate calculations often provide a foundation that allows us to organize and solve more complex problems, enabling us to visualize how different speeds relate to each other.

Exploring the Distance Formula

The distance formula is an essential tool in these types of algebra problems. It allows us to relate distance, rate, and time through the equation:

• \( \text{Distance} = \text{Rate} \times \text{Time} \)

This formula helps you to calculate how far an object has traveled over a time period when you know its speed.
In our train example, both trains travel for \( 9.5 \) hours, so the westbound train covers \( r \times 9.5 \) miles, while the eastbound train covers \((r + 8) \times 9.5 \) miles.
With this understanding, we can develop equations that help explain the relationship between their rates and the total distance they are apart. Remember, the formula is highly versatile and can be rearranged if you need to solve for distance, rate, or time individually.

Using Simultaneous Equations

Simultaneous equations involve finding the values of unknown variables that satisfy multiple equations at the same time. In many word problems, you have multiple relationships that must be solved simultaneously to find the correct answer.
In our train problem, we combine the distances traveled by each train. We form a single equation: \( 9.5r + 9.5(r + 8) = 1292 \). This equation expresses that the sum of the two distances equals the total distance apart.
Solving such equations typically involves:

• First, simplifying the equation by distributing and combining like terms.
• Next, isolating the variable by performing algebraic operations like addition or division.

For instance, after simplifying, you are left with \( 19r + 76 = 1292 \). Continuing to solve this gives \( r = 64 \), which is the rate of the westbound train. You can also find the eastbound train's rate using this result.
Understanding how to set up and solve simultaneous equations is crucial for tackling complex word problems efficiently.