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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, 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: \[ 4x = 4 \] \[ x = 1 \text{ hour} \]

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: \[ \text{Average speed from B to C} = \frac{60 \text{ miles}}{1 \text{ hour}} = 60 \text{ mph} \]