Question:
A jet goes from City 1 to City 2 at an average speed of 600 miles per hour, and returns along the same path at an average speed of 300 miles per hour. What is the average speed, in miles per hour, for the trip?
A jet goes from City 1 to City 2 at an average speed of 600 miles per hour, and returns along the same path at an average speed of 300 miles per hour. What is the average speed, in miles per hour, for the trip?
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When an object travels the same distance at two different speeds, \(v_1\) and \(v_2\), the average speed is the harmonic mean of the two speeds, not the arithmetic mean. The formula is: \(\text{Average Speed} = \frac{2v_1v_2}{v_1+v_2}\). In this case: \(\frac{2(600)(300)}{600+300} = \frac{360000}{900} = 400\) mph.
Updated On: Oct 3, 2025
- 300 miles/hour
- 400 miles/hour
- 350 miles/hour
- 450 miles/hour
- 500 miles/hour
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
Average speed is calculated as Total Distance divided by Total Time. It is a common mistake to simply take the arithmetic average of the two speeds, especially when the time taken for each part of the journey is different.
Step 2: Key Formula or Approach:
Let \(d\) be the distance between City 1 and City 2.
The formula for time is \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).
The formula for average speed is \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \).
Step 3: Detailed Explanation:
Let's calculate the time for each leg of the journey.
Time to go from City 1 to City 2 (\(t_1\)):
\[ t_1 = \frac{d}{600} \text{ hours} \] Time to return from City 2 to City 1 (\(t_2\)):
\[ t_2 = \frac{d}{300} \text{ hours} \] Now, let's find the total distance and total time for the entire trip.
Total Distance = Distance to City 2 + Distance back = \(d + d = 2d\).
Total Time = \(t_1 + t_2 = \frac{d}{600} + \frac{d}{300}\).
To add the fractions, we find a common denominator, which is 600.
\[ \text{Total Time} = \frac{d}{600} + \frac{2d}{600} = \frac{3d}{600} = \frac{d}{200} \text{ hours} \] Now we can calculate the average speed:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{200}} \] \[ \text{Average Speed} = 2d \times \frac{200}{d} \] The \(d\) terms cancel out.
\[ \text{Average Speed} = 2 \times 200 = 400 \text{ miles/hour} \] Step 4: Final Answer:
The average speed for the entire trip is 400 miles/hour.
Average speed is calculated as Total Distance divided by Total Time. It is a common mistake to simply take the arithmetic average of the two speeds, especially when the time taken for each part of the journey is different.
Step 2: Key Formula or Approach:
Let \(d\) be the distance between City 1 and City 2.
The formula for time is \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).
The formula for average speed is \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \).
Step 3: Detailed Explanation:
Let's calculate the time for each leg of the journey.
Time to go from City 1 to City 2 (\(t_1\)):
\[ t_1 = \frac{d}{600} \text{ hours} \] Time to return from City 2 to City 1 (\(t_2\)):
\[ t_2 = \frac{d}{300} \text{ hours} \] Now, let's find the total distance and total time for the entire trip.
Total Distance = Distance to City 2 + Distance back = \(d + d = 2d\).
Total Time = \(t_1 + t_2 = \frac{d}{600} + \frac{d}{300}\).
To add the fractions, we find a common denominator, which is 600.
\[ \text{Total Time} = \frac{d}{600} + \frac{2d}{600} = \frac{3d}{600} = \frac{d}{200} \text{ hours} \] Now we can calculate the average speed:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{200}} \] \[ \text{Average Speed} = 2d \times \frac{200}{d} \] The \(d\) terms cancel out.
\[ \text{Average Speed} = 2 \times 200 = 400 \text{ miles/hour} \] Step 4: Final Answer:
The average speed for the entire trip is 400 miles/hour.
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