- Translate the problem requirements: Marcus drives 2 miles total - first mile in 1 minute, second mile in less time. His average speed for the entire trip is 90 mph. We need to find his speed during the second mile only.
- Calculate total time for the entire trip: Use the overall average speed of 90 mph for 2 miles to determine how long the entire test took.
- Determine time spent on the second mile: Subtract the known first mile time (1 minute) from the total trip time to find how long the second mile took.
- Calculate speed for the second mile: Apply speed = distance ÷ time using the second mile distance and its time to get the final answer.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what Marcus did during his test drive:
- He drove a total of 2 miles
- First mile: took exactly 1 minute
- Second mile: took less time than the first mile (so less than 1 minute)
- Overall performance: averaged 90 mph for the entire 2-mile trip
We need to find Marcus's speed during just the second mile.
Process Skill: TRANSLATE - Converting the story into mathematical relationships we can work with
2. Calculate total time for the entire trip
Think about it this way: if Marcus averaged 90 mph over 2 miles, how long did the entire trip take?
When you drive at an average speed, you can find the total time by asking: "How long does it take to cover this distance at this speed?"
For 2 miles at 90 mph:
- Time = Distance ÷ Speed
- Time = \(2 \text{ miles} ÷ 90 \text{ mph}\)
- Time = \(\frac{2}{90} \text{ hours}\)
- Time = \(\frac{1}{45} \text{ hours}\)
To convert this to minutes (since the first mile time was given in minutes):
- \(\frac{1}{45} \text{ hours} × 60 \frac{\text{minutes}}{\text{hour}} = \frac{60}{45} \text{ minutes} = \frac{4}{3} \text{ minutes}\)
So the entire 2-mile trip took \(\frac{4}{3}\) minutes.
3. Determine time spent on the second mile
Now we can find how long the second mile took:
- Total time for both miles = \(\frac{4}{3}\) minutes
- Time for first mile = 1 minute
- Time for second mile = \(\frac{4}{3} - 1 = \frac{4}{3} - \frac{3}{3} = \frac{1}{3} \text{ minutes}\)
This makes sense! The second mile took \(\frac{1}{3}\) minute, which is indeed less than the 1 minute for the first mile.
4. Calculate speed for the second mile
Now we can find Marcus's speed during the second mile:
We know:
- Distance of second mile = 1 mile
- Time for second mile = \(\frac{1}{3}\) minute
Speed = Distance ÷ Time
- Speed = \(1 \text{ mile} ÷ (\frac{1}{3} \text{ minute})\)
- Speed = \(1 × 3 = 3 \text{ miles per minute}\)
Converting to miles per hour:
- \(3 \frac{\text{miles}}{\text{minute}} × 60 \frac{\text{minutes}}{\text{hour}} = 180 \text{ miles per hour}\)
Final Answer
Marcus's average speed for the second mile was 180 mph.
This matches answer choice C. 180.
Let's verify: If Marcus drove 1 mile in 1 minute (60 mph) and 1 mile in \(\frac{1}{3}\) minute (180 mph), his total time is \(\frac{4}{3}\) minutes for 2 miles, giving an average speed of \(2 ÷ (\frac{4}{3} ÷ 60) = 90 \text{ mph}\). ✓
Common Faltering Points
Errors while devising the approach
- Misinterpreting what "averaged 60 mph" means for the first mile: Students might think this gives them the time directly, forgetting that if Marcus drove 1 mile in 1 minute, his speed was actually 60 mph (since 1 mile per minute = 60 miles per hour). The key insight is recognizing that "1 mile in 1 minute" and "averaged 60 mph" are giving the same information in different units.
- Confusion about what speed to calculate: Students might try to find the average speed for both miles combined (which is already given as 90 mph) instead of focusing on finding the speed for just the second mile. The question specifically asks for "Marcus's average speed for the second mile," not the overall average.
- Setting up incorrect relationships: Students might try to use the constraint "second mile took less time" as a mathematical equation rather than understanding it's just a logical check. This constraint helps verify the answer makes sense but isn't needed in the calculation setup.
Errors while executing the approach
- Unit conversion errors: Students often struggle converting between hours and minutes consistently throughout the problem. For example, calculating total time as \(\frac{2}{90}\) hours but forgetting to convert to minutes, or mixing units when calculating the final speed (getting 3 miles per minute but forgetting to convert to miles per hour).
- Fraction arithmetic mistakes: When calculating the time for the second mile (\(\frac{4}{3} - 1 = \frac{1}{3}\) minutes), students frequently make errors with fraction subtraction, potentially getting \(\frac{3}{4}\) or other incorrect values, which would lead to wrong final speeds.
- Division by fractions errors: When calculating speed = distance ÷ time = \(1 ÷ (\frac{1}{3})\), students often forget that dividing by a fraction means multiplying by its reciprocal, instead calculating \(\frac{1}{3}\) directly and getting 0.33 instead of 3.
Errors while selecting the answer
- Forgetting final unit conversion: Students might correctly calculate 3 miles per minute for the second mile but forget to convert this to miles per hour, selecting a non-existent answer choice or getting confused about units in the final step.
- Selecting intermediate calculations: Students might choose 60 (the speed of the first mile) or 90 (the overall average speed) thinking these represent the answer to what was asked, without completing the full calculation for the second mile's speed.
