If a salesperson drove 220 miles nonstop in 4 hours and 15 minutes, which of the following is closest to...

1. Translate the problem requirements: We need to find average speed in miles per hour. \(\mathrm{Average\ speed} = \mathrm{total\ distance} ÷ \mathrm{total\ time}\). We have 220 miles driven in 4 hours and 15 minutes, and we need the closest answer from the given choices.
2. Convert time to consistent units: Transform 4 hours and 15 minutes into decimal hours to make the division straightforward.
3. Apply the speed relationship: Divide total distance by total time to get average speed in miles per hour.
4. Match with closest answer choice: Compare our calculated result with the given options to find the best approximation.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in everyday terms. When we talk about average speed, we're asking: "If the salesperson traveled at a constant speed for the entire trip, what would that speed be?"

Think of it like this: imagine you're telling a friend about your road trip. You say "I drove 220 miles and it took me 4 hours and 15 minutes." Your friend asks "How fast were you going?" To answer this, you need to figure out how many miles you covered per hour on average.

The relationship we need is simple: \(\mathrm{Average\ speed} = \mathrm{How\ far\ you\ went} ÷ \mathrm{How\ long\ it\ took}\)

In our case: \(\mathrm{Average\ speed} = 220 \mathrm{\ miles} ÷ (4 \mathrm{\ hours\ and\ } 15 \mathrm{\ minutes})\)

We need our final answer in miles per hour, and we need to pick the closest answer from the given choices.

Process Skill: TRANSLATE

2. Convert time to consistent units

Right now our time is in two different units: hours and minutes. To make our division work smoothly, let's convert everything to just hours.

We have: 4 hours and 15 minutes

The 4 hours part is already in the right unit. Now let's convert the 15 minutes to hours:

• There are 60 minutes in 1 hour
• So \(15 \mathrm{\ minutes} = \frac{15}{60} \mathrm{\ hours} = \frac{1}{4} \mathrm{\ hours} = 0.25 \mathrm{\ hours}\)

Therefore: \(4 \mathrm{\ hours\ and\ } 15 \mathrm{\ minutes} = 4 + 0.25 = 4.25 \mathrm{\ hours}\)

This makes our calculation much cleaner!

3. Apply the speed relationship

Now we can apply our simple relationship using consistent units:

\(\mathrm{Average\ speed} = \mathrm{Total\ distance} ÷ \mathrm{Total\ time}\)
\(\mathrm{Average\ speed} = 220 \mathrm{\ miles} ÷ 4.25 \mathrm{\ hours}\)

To calculate \(220 ÷ 4.25\), let's make this easier by converting 4.25 to a fraction:
\(4.25 = 4\frac{1}{4} = \frac{17}{4}\)

So: \(220 ÷ \frac{17}{4} = 220 × \frac{4}{17} = \frac{220 × 4}{17} = \frac{880}{17}\)

Now: \(\frac{880}{17} ≈ 51.76 \mathrm{\ miles\ per\ hour}\)

(To check: \(17 × 51 = 867\), and \(880 - 867 = 13\), so \(\frac{13}{17} ≈ 0.76\))

4. Match with closest answer choice

Our calculated speed is approximately 51.76 miles per hour.

Looking at our answer choices:

1. 47.2 - too low (difference of about 4.5)
2. 48.3 - too low (difference of about 3.4)
3. 50.6 - close but still low (difference of about 1.2)
4. 51.8 - very close! (difference of only about 0.04)
5. 55.5 - too high (difference of about 3.7)

Choice D (51.8) is clearly the closest to our calculated value of 51.76.

Final Answer

The salesperson's average speed is closest to 51.8 miles per hour.

Answer: D

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the speed formula

Students might confuse the relationship and think average speed = time ÷ distance instead of distance ÷ time. This fundamental misunderstanding would lead them to calculate \(4.25 ÷ 220\), giving an extremely small decimal result that doesn't match any answer choice.

2. Not recognizing the need for unit conversion

Some students might attempt to use the time as given ("4 hours and 15 minutes") without recognizing that they need to convert everything to a single unit. They might try to divide 220 by 4, then separately handle the 15 minutes, leading to confusion and incorrect calculations.

Errors while executing the approach

1. Incorrect time conversion

When converting 15 minutes to hours, students commonly make the error of thinking 15 minutes = 0.15 hours (confusing decimal representation with actual fraction). This gives 4.15 hours instead of 4.25 hours, leading to \(220 ÷ 4.15 ≈ 53.0\), which might make them choose answer choice E (55.5).

2. Arithmetic errors in division

The division \(\frac{880}{17}\) requires careful calculation. Students might make errors such as: incorrectly calculating \(17 × 50 = 850\) and getting \(\frac{880-850}{17} = \frac{30}{17} ≈ 1.8\), leading to \(50 + 1.8 = 51.8\) by coincidence, or making other computational mistakes that lead to wrong intermediate results.

3. Fraction conversion mistakes

When converting 4.25 to the fraction 17/4, students might incorrectly convert it as 16/4 or 18/4, or make errors when flipping the fraction for division, leading to significantly different final answers.

Errors while selecting the answer

1. Not selecting the closest answer

After calculating approximately 51.76, students might focus too much on getting an "exact" match and second-guess themselves. They might incorrectly choose C (50.6) thinking it's "close enough" or worry that their calculation must be wrong since 51.8 seems "too close" to their answer, not realizing that answer choices are designed to test approximation skills.