Question

Did Jon complete a journey of 40 kilometers in less time than it took Ann to complete the same journey? (A) Jon traveled at an average speed of 30 kilometers per hour for the first 10 kilometers and then at an average speed of 15 kilometers per hour for the rest of the journey. (B) Ann traveled at an average speed of 20 kilometers per hour for the entire journey.

Short answer

No, Jon took more time than Ann.

Step-by-step solution

- Calculate Jon's Time for First Part

Jon traveled the first 10 kilometers at an average speed of 30 kilometers per hour. Time is calculated by dividing the distance by the speed. Therefore, Jon's time for the first 10 kilometers is: \[ \text{Time}_{1} = \frac{10 \text{ km}}{30 \text{ km/h}} = \frac{1}{3} \text{ hours} \text{ or 20 minutes} \tag{1} \]

- Calculate Jon's Time for the Remaining Part

Jon traveled the remaining 30 kilometers at an average speed of 15 kilometers per hour. Therefore, Jon's time for the remaining 30 kilometers is: \[ \text{Time}_{2} = \frac{30 \text{ km}}{15 \text{ km/h}} = 2 \text{ hours} \tag{2} \]

- Calculate Total Time for Jon

Add the time for both parts of Jon's journey to obtain the total time: \[ \text{Total Time}_{\text{Jon}} = \text{Time}_{1} + \text{Time}_{2} = \frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3} \text{ hours} = 2 \text{ hours and } 20 \text{ minutes} \tag{3} \]

- Calculate Ann's Time for the Entire Journey

Ann traveled the entire 40 kilometers at an average speed of 20 kilometers per hour. Therefore, Ann's time for the 40 kilometers is: \[ \text{Total Time}_{\text{Ann}} = \frac{40 \text{ km}}{20 \text{ km/h}} = 2 \text{ hours} \tag{4} \]

- Compare Jon's and Ann's Times

Jon took 2 hours and 20 minutes while Ann took 2 hours. Therefore, Jon took more time than Ann to complete the journey.

Key concepts

average speed calculation

Understanding how to calculate average speed is important for solving many time-distance problems. Average speed is calculated by dividing the total distance traveled by the total time taken. For example, if you travel 40 kilometers in 2 hours, your average speed is \(\frac{40 \text{ km}}{2 \text{ hours}} = 20 \text{ km/h}\). In more complex journeys where speeds vary, you can divide the journey into parts and calculate the time for each part separately, then add up those times to find the total travel time.
Let’s say Jon traveled the first 10 kilometers at 30 km/h and the remaining 30 kilometers at 15 km/h. You can use the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\) to calculate the time for each segment of the journey and then add them together. This approach ensures you accurately account for the variations in speed over different parts of the journey.

distance-time relationship

The relationship between distance and time is crucial for understanding how long a trip will take. This relationship is usually defined by the formula \(\text{Distance} = \text{Speed} \times \text{Time}\) or rearranged as \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\). For example, if a car travels at a constant speed of 20 km/h for 40 kilometers, the travel time is \(\frac{40 \text{ km}}{20 \text{ km/h}} = 2 \text{ hours}\).
When dealing with variable speeds, as with Jon's journey, you break the trip into segments where each segment has a constant speed. Calculate the time for each segment: \(\text{First Segment} = \frac{10 \text{ km}}{30 \text{ km/h}} = \frac{1}{3} \text{ hours (or 20 minutes)}\) and \(\text{Second Segment} = \frac{30 \text{ km}}{15 \text{ km/h}} = 2 \text{ hours}\). Add these times together to get the total time: \(\frac{1}{3} \text{ hours} + 2 \text{ hours} = 2 \text{ hours and } 20 \text{ minutes}\).

comparison of travel times

Comparing travel times allows you to see who traveled more efficiently. In the given problem, Jon's travel time is broken into two parts due to his changing speeds, resulting in a total of 2 hours and 20 minutes. Ann’s travel time is simpler to calculate since she traveled at a constant speed.
Let's compare Jon and Ann's travel times: Ann's total time is \(\frac{40 \text{ km}}{20 \text{ km/h}} = 2 \text{ hours}\), whereas Jon's total time from the steps calculated is 2 hours and 20 minutes. Hence, Ann was quicker. This comparison highlights the effect of consistent speed over varying speeds on total travel time. By breaking the journey into segments based on speed and calculating each segment's time, you can accurately compare different travel scenarios.