Question

A car traveled from Town \(A\) to Town \(B\). The car traveled the first \(\frac{3}{8}\) of the distance from Town \(A\) to Town \(B\) at an average speed of \(x\) miles per hour, where \(x>0 .\) The car traveled the remaining distance at an average speed of \(y\) miles per hour, where \(y>0 .\) The car traveled the entire distance from Town \(A\) to Town \(B\) at an average speed of \(z\) miles per hour. Which of the following equations gives \(y\) in terms of \(x\) and \(z ?\) a. \(y=\frac{3 x+5 z}{8}\) b. \(y=\frac{5 x z}{8 x+3 z}\) c. \(y=\frac{8 x-3 z}{5 x z}\) d. \(y=\frac{3 x z}{8 x-5 z}\) e. \(y=\frac{5 x z}{8 x-3 z}\)

Short answer

The equation for y in terms of x and z is \ \( y = \frac{5xz}{8x - 3z} \ \). The correct answer is e.

Step-by-step solution

Define total distance

Let the total distance from Town A to Town B be denoted as D.

Define first part of the distance

The first part of the distance is \ \( \frac{3}{8}D \ \) at a speed of \ \( x \ \) miles per hour.

Define remaining part of the distance

The remaining part of the distance is \ \( \frac{5}{8}D \ \) at a speed of \ \( y \ \) miles per hour.

Calculate time for the first part of the trip

Time spent to cover the first part is \ \( \frac{\frac{3}{8}D}{x} \ \). Simplified, it is \ \( \frac{3D}{8x} \ \).

Calculate time for the remaining part of the trip

Time spent to cover the remaining part is \ \( \frac{\frac{5}{8}D}{y} \ \). Simplified, it is \ \( \frac{5D}{8y} \ \).

Calculate total travel time

Total time for the entire trip is \ \( \frac{3D}{8x} + \frac{5D}{8y} \ \).

Express average speed for the entire trip

The average speed \ \( z \ \) for the entire trip is defined as \ \( z = \frac{D}{\text{total time}} \ \). Thus, total time is \ \( \frac{D}{z} \ \).

Set equation for total time

Set the total time equations equal to each other: \ \( \frac{D}{z} = \frac{3D}{8x} + \frac{5D}{8y} \ \).

Simplify and solve for y

Cancel \ \( D \ \) from both sides: \ \( \frac{1}{z} = \frac{3}{8x} + \frac{5}{8y} \ \). Rearrange and solve for \ \( y \ \): \ \( \frac{5}{8y} = \frac{1}{z} - \frac{3}{8x} \ \) Simplify further: \ \( \frac{5}{8y} = \frac{8x - 3z}{8xz} \ \) \ \( \Rightarrow y = \frac{5xz}{8x - 3z} \ \).

Key concepts

Distance and Speed Calculations

Understanding how to calculate distance and speed is crucial for solving average speed problems. Let's break this down step-by-step to fully grasp the concept. When dealing with distance, speed, and time, remember these key formulas:

• Distance = Speed × Time
• Speed = Distance ÷ Time
• Time = Distance ÷ Speed

In the given problem, we have a car traveling from Town A to Town B. First, we split the journey into two parts: the first \(\frac{3}{8} \) of the distance at speed \( x \) and the remaining \(\frac{5}{8} \) at speed \( y \). To find the time taken for each part of the trip, use the time formula: Time = Distance ÷ Speed. For the first part, the time taken is \( \frac{3D}{8x} \). For the second part, it's \( \frac{5D}{8y} \).
The total time for the journey is the sum of these two times. The average speed for the entire trip is defined by the formula Average Speed = Total Distance ÷ Total Time.

GMAT Math Practice

When preparing for the GMAT, it's essential to practice problems that involve algebraic equations and average speed calculations. This particular problem is a typical GMAT-style question that tests your ability to apply basic math concepts.
GMAT questions often require you to:

• Break down the problem into manageable parts.
• Use algebra to express relationships.
• Simplify complicated expressions step-by-step.

In this example, you first define the total distance \(\text{D} \), then calculate the time for each part of the trip. By setting up equations that relate time, distance, and speed, you find the average speed over the entire journey. With that information, you can solve for the unknown variable. Applying these strategies consistently will help you improve your problem-solving skills for the GMAT.

Algebraic Equations

Algebraic equations are powerful tools to solve problems like the one we've discussed. They help in expressing complex relationships in a simplified manner. In this problem, we use algebra to set up the relationship between the speeds and distances.
First, recall the relationship we derived:
\( \frac{D}{z} = \frac{3D}{8x} + \frac{5D}{8y} \).
By solving this equation, we isolate \( y \) to find its expression in terms of x and z. Here’s how we do it step-by-step:

1. Simplify the equation by cancelling D:
\( \frac{1}{z} = \frac{3}{8x} + \frac{5}{8y} \)
2. Isolate \( y \):
Move terms involving y to one side
\( \frac{5}{8y} = \frac{1}{z} - \frac{3}{8x} \)
3. Further simplify:
\( \frac{5}{8y} = \frac{8x - 3z}{8xz} \)
4. Solve for y:
\( y = \frac{5xz}{8x - 3z} \)
Understanding how to manipulate algebraic equations and solve for unknowns is crucial. This skill extends beyond just speed and distance problems. It is foundational for diverse topics in algebra and math at large.