Question

A truck driver drove for 2 days. On the second day, he drove 3 hours longer and at an average speed of 15 miles per hour faster than he drove on the first day. If he drove a total of 1,020 miles and spent 21 hours driving during the 2 days, what was his average speed on the first day, in miles per hour? A 25 B 30 C 35 D 40 E 45

Short answer

The average speed on the first day was 40 miles per hour.

Step-by-step solution

- Define Variables

Let the average speed on the first day be denoted as \( s \) miles per hour. Let the time spent driving on the first day be denoted as \( t \) hours.

- Expressions for Second Day

On the second day, the driver drove 3 hours longer than the first day and at an average speed of 15 miles per hour faster. Therefore, the time spent driving on the second day is \( t + 3 \) hours, and the average speed is \( s + 15 \) miles per hour.

- Total Driving Time

Combine the times for both days: \[ t + (t + 3) = 21 \, \text{hours} \] Simplifying, the equation becomes: \[ 2t + 3 = 21 \] Solve for \( t \): \[ 2t = 18 \] \[ t = 9 \, \text{hours} \]

- Total Distance Driven

Define the total distance equations for both days and sum them up: \[ s \cdot t + (s + 15) \cdot (t + 3) = 1020 \, \text{miles} \] Substitute \( t = 9 \): \[ s \cdot 9 + (s + 15) \cdot 12 = 1020 \] Simplify and solve for \( s \): \[ 9s + 12s + 180 = 1020 \] \[ 21s + 180 = 1020 \] \[ 21s = 840 \] \[ s = 40 \, \text{miles per hour} \]

Key concepts

algebraic equations

In this exercise, we will use algebraic equations to solve a real-world problem. The first step is to set up variables to represent unknown quantities. Let’s denote the average speed on the first day as \(s\) miles per hour and the time spent driving on the first day as \(t\) hours. Setting up these variables helps us to create algebraic expressions for both the time and distance.

The equations formed include:

• The total driving time equation: \(t + (t + 3) = 21\) hours
• The total distance equation: \(s \cdot t + (s + 15) \cdot (t + 3) = 1020\) miles

Solving these equations step-by-step allows us to find the values of \(t\) and \(s\), which are crucial to determining the solution. This practice demonstrates the power of algebra in breaking down complex word problems into manageable parts.

average speed calculations

Calculating average speed is an essential skill in many GMAT math problems. Average speed is computed using the formula: \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \].

In our problem, we need to calculate the driver’s average speed for each day. On the first day, the average speed is \(s\) miles per hour. On the second day, the average speed is \(s + 15\) miles per hour as given by the problem statement.
By knowing the total distance (1,020 miles) and total time (21 hours), we were able to create equations that allow us to calculate the average speed for each day once we determined the travel times. This helps ensure the driver’s speeds and travel times make logical sense according to the provided information.

distance-time relationships

Understanding distance-time relationships is crucial for solving related math problems. The relationship can be summed up with the equation: \[ \text{Distance} = \text{Speed} \cdot \text{Time} \].

For the truck driver:

• The driving distance on the first day: \(s \cdot t \)
• The driving distance on the second day: \((s + 15) \cdot (t + 3)\)
• Total distance: \(s \cdot t + (s + 15) \cdot (t + 3) = 1020\) miles

By correctly applying these relationships to the variables and given total values, we can solve for unknowns like the driver’s average speed. This method helps to visualize how changes in speed or travel time affect the overall distance traveled.

test preparation

For test preparation, focus on practicing different types of algebra problems, particularly those involving distance, speed, and time. Here are some tips:

• Create a study schedule: Allocate regular time blocks specifically for math practice.
• Understand key formulas: Memorize crucial formulas like \[ \text{Distance} = \text{Speed} \cdot \text{Time} \] and practice using them in various scenarios.
• Solve practice problems: Work on a variety of problems to develop problem-solving strategies.
• Review mistakes: Carefully review incorrect answers to understand where you went wrong and how to correct it.
• Simulate test conditions: Practice under timed conditions to get comfortable with the pace of the test.

By following these strategies, you can become more confident in solving complex math problems effectively. Regular practice can hone your abilities and improve your overall test-taking skills.