Question

A motorcyclist started riding at highway marker \(\mathrm{A},\) drove 120 miles to highway marker \(\mathrm{B}\), and then, without pausing, continued to highway marker \(C,\) where she stopped. The average speed of the motorcyclist, over the course of the entire trip, was 45 miles per hour. If the ride from marker \(A\) to marker \(B\) lasted 3 times as many hours as the rest of the ride, and the distance from marker \(\mathrm{B}\) to marker \(\mathrm{C}\) was half of the distance from marker \(\mathrm{A}\) to marker \(\mathrm{B},\) what was the average speed, in miles per hour, of the motorcyclist while driving from marker \(\mathrm{B}\) to marker \(\mathrm{C} ?\) A 40 B 45 C 50 D 55 E 60

Short answer

The average speed from B to C is 60 mph. Choice E.

Step-by-step solution

- Understand the distances

The motorcyclist drove 120 miles from marker A to marker B. The distance from marker B to marker C is half of the distance from A to B. Therefore, the distance from B to C is \( \frac{120}{2} = 60 \) miles.

- Calculate total distance

The total distance of the entire trip is the sum of the distances from A to B and B to C. Therefore, total distance = 120 miles + 60 miles = 180 miles.

- Write the total trip equation

The average speed over the entire trip is given as 45 miles per hour. Let t be the total time for the entire trip. \[ \frac{180}{t} = 45 \] Solving for t, \[ t = \frac{180}{45} = 4 \text{ hours} \]

- Calculate time from A to B

The ride from marker A to marker B lasted 3 times as many hours as the rest of the ride. Let x be the time for the ride from B to C, then the time from A to B is 3x. Since the total time t is 4 hours, we have: \[ 3x + x = 4 \] \[ 4x = 4 \] \[ x = 1 \text{ hour} \]

- Calculate time from B to C

Substituting, the time from B to C is x = 1 hour and the time from A to B is 3x = 3 hours.

- Determine average speed from B to C

The distance from B to C is 60 miles and it took 1 hour. Therefore, the average speed from B to C is \[ \text{Average speed} = \frac{\text{Distance}}{\text{Time}} = \frac{60 \text{ miles}}{1 \text{ hour}} = 60 \text{ mph} \]

Key concepts

Average Speed Calculation

In many GMAT problems, you might need to calculate the average speed over a journey. Average speed is defined as the total distance traveled divided by the total time taken. In mathematical terms, average speed (\text{AS}) can be expressed as: \[ \text{AS} = \frac{\text{Total Distance}}{\text{Total Time}} \] In the given problem, the motorcyclist's entire journey covered a total distance of 180 miles, and the total time taken was 4 hours. Therefore, we calculate the average speed as follows: \[ \text{AS} = \frac{180 \text{ miles}}{4 \text{ hours}} = 45 \text{ mph} \] To find the average speed for different portions of the journey, the same formula applies but only using the specific segment distance and time. For instance, from marker B to marker C, the distance was 60 miles and the time was 1 hour: \[ \text{Average speed from B to C} = \frac{60 \text{ miles}}{1 \text{ hour}} = 60 \text{ mph} \]

Distance and Time Relationship

Distance, speed, and time have a fundamental relationship that you need to understand, reflected in the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \] and its rearrangements like \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] and \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] In this problem, the total distance from marker A to marker C is 180 miles, calculated as: \[ \text{Total distance} = \text{Distance from A to B} + \text{Distance from B to C} = 120 \text{ miles} + 60 \text{ miles} = 180 \text{ miles} \] The next step is using the given average speed (45 mph) to find total travel time: \[ \text{Total Time} = \frac{180 \text{ miles}}{45 \text{ mph}} = 4 \text{ hours} \] Part of the problem involved understanding time segments. Let’s say the time to travel from B to C is x hours. Hence, the time from A to B is 3x hours, making the total travel time: \[ \text{Total time} = 3x + x = 4 \text{ hours} \] Solving for x gives: \[ 4x = 4 \] \[ x = 1 \text{ hour} \]

Ratio and Proportion in Time-Distance Problems

Often, GMAT problems involve ratios and proportions to understand relationships between portions of a journey. In this problem, the ride from marker A to marker B took three times as long as the ride from marker B to marker C. This ratio involves time segments and can be expressed as: \[ \text{Time from A to B} = 3 \times \text{Time from B to C} \] Letting time from B to C be denoted by x, therefore, time from A to B is 3x. The total time for the trip then becomes: \[ \text{Total Time} = 3x + x = 4 \text{ hours} \] Solving this equation results in: \[ 4x = 4 \] \[ x = 1 \text{ hour} \] Thus, the time from A to B is 3 hours, and from B to C is 1 hour. These ratios help in the next step of determining specific segment speeds: \[ \text{Average speed from B to C} = \frac{60 \text{ miles}}{1 \text{ hour}} = 60 \text{ mph} \]