A driver completed the first 20 miles of a 40-mile trip at an average speed of 50 miles per hour....

1. Translate the problem requirements: We need to find the speed for the second 20 miles such that the total 40-mile trip averages 60 mph. This means determining what speed is needed for the remaining portion to achieve the overall target average.
2. Calculate time constraints from the target average: Use the desired 60 mph average to find the total time allowed for the entire 40-mile trip.
3. Determine time already used: Calculate how much time was spent on the first 20 miles at 50 mph.
4. Find remaining time available: Subtract the time used from the total time allowed to find how much time is left for the final 20 miles.
5. Calculate required speed: Use the remaining distance and available time to determine the necessary speed for the second half.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're really being asked. We have a 40-mile trip that we want to complete at an average speed of 60 mph for the entire journey. The driver already completed the first 20 miles at 50 mph, and now we need to figure out how fast they must drive the remaining 20 miles to hit that 60 mph overall average.

Think of it like this: the driver has already "spent" some time on the first half of the trip. To achieve the desired overall average, they have a limited amount of time left for the second half. We need to find what speed will use up exactly that remaining time.

Process Skill: TRANSLATE - Converting the problem setup into a clear mathematical question about time and speed relationships

2. Calculate time constraints from the target average

If we want to average 60 mph for the entire 40-mile trip, let's figure out how much total time we have available.

Using everyday reasoning: if you travel 40 miles at 60 miles per hour, how long does that take? Well, 60 miles would take 1 hour, so 40 miles takes \(\frac{40}{60} = \frac{2}{3}\) of an hour.

Converting to minutes to make this concrete: \(\frac{2}{3}\) hour = \(\frac{2}{3} \times 60\) minutes = 40 minutes total.

So the entire 40-mile trip must be completed in exactly 40 minutes to achieve a 60 mph average.

3. Determine time already used

Now let's calculate how much time the driver already spent on the first 20 miles at 50 mph.

Thinking through this step by step: if the driver goes 50 miles in 1 hour, then 20 miles takes \(\frac{20}{50} = \frac{2}{5}\) of an hour.

Converting to minutes: \(\frac{2}{5}\) hour = \(\frac{2}{5} \times 60\) minutes = 24 minutes.

So the driver has already used 24 minutes of their total 40-minute budget.

4. Find remaining time available

This is straightforward subtraction. The driver started with 40 minutes total and has already used 24 minutes.

Remaining time = \(40 - 24 = 16\) minutes

So the driver has exactly 16 minutes to complete the final 20 miles.

5. Calculate required speed

Now we can find the required speed: 20 miles must be covered in 16 minutes.

Let's convert back to hours first: 16 minutes = \(\frac{16}{60}\) hours = \(\frac{4}{15}\) hours.

Using the relationship that speed = distance ÷ time:
Required speed = \(20 \div \frac{4}{15} = 20 \times \frac{15}{4} = \frac{300}{4} = 75\) mph

Technical summary: Speed = \(20 \div \frac{4}{15} = 20 \times \frac{15}{4} = 75\) mph

6. Final Answer

The driver must complete the remaining 20 miles at an average speed of 75 mph.

Looking at our answer choices, this matches choice (D) 75 mph.

We can verify this makes sense: 24 minutes + 16 minutes = 40 minutes total, and 40 miles in 40 minutes gives us exactly 60 mph average for the entire trip.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "average speed" means

Students often think they can simply average the two speeds (50 mph and the unknown speed) to get 60 mph. This leads them to set up: \(\frac{50 + x}{2} = 60\), solving for x = 70 mph. However, average speed is total distance divided by total time, not the arithmetic mean of individual speeds. This conceptual error stems from confusing average speed with average of speeds.

2. Incorrectly assuming equal time periods

Some students assume that since both segments are 20 miles each, the driver should spend equal time on each segment. They might think: "If I want 60 mph average, and I went 50 mph for the first half, I need to go 70 mph for the second half to balance it out." This ignores the fact that different speeds result in different time periods, even for equal distances.

3. Setting up the wrong constraint equation

Students might incorrectly set up the problem by focusing on individual segment averages rather than the overall trip constraint. They may try to work with ratios of speeds instead of understanding that the total time for 40 miles at 60 mph creates a fixed time budget that constrains the second segment.

Errors while executing the approach

1. Time unit conversion errors

When converting between hours and minutes (or staying in fractional hours), students frequently make arithmetic mistakes. For example, converting \(\frac{2}{5}\) hour to minutes might be calculated as \(\frac{2}{5} \times 100 = 40\) minutes instead of \(\frac{2}{5} \times 60 = 24\) minutes. These unit conversion errors propagate through the entire solution.

2. Fraction arithmetic mistakes

The problem involves several fraction calculations (\(\frac{2}{3}\) hour, \(\frac{2}{5}\) hour, \(\frac{4}{15}\) hour). Students often struggle with operations like \(20 \div \frac{4}{15}\), forgetting to flip the fraction when dividing, or making errors in multiplication like \(20 \times \frac{15}{4}\). A common error is calculating \(20 \times \frac{15}{4}\) as \(\frac{20 \times 15}{4} = \frac{300}{4}\) incorrectly.

3. Incorrect time budget calculation

Students may correctly calculate individual time segments but then make subtraction errors when finding remaining time. For instance, if they calculate total time as 40 minutes and first segment time as 24 minutes, they might compute the remaining time incorrectly due to rushed arithmetic.

Errors while selecting the answer

1. Choosing a "reasonable-sounding" wrong answer

After making calculation errors, students often end up with values like 70 mph (from the averaging misconception) and see it matches choice (C), giving them false confidence. They may not double-check their work because 70 mph "feels reasonable" as it's higher than 50 mph but not extremely high.

2. Units confusion in final answer

Some students might calculate correctly but then second-guess their units. If they calculated speed in miles per minute or some other unit during intermediate steps, they might accidentally convert incorrectly for the final answer, leading them to select a different multiple choice option.