A certain car averages 25 miles per gallon of gasoline when driven in the city and 40 miles per gallon...

1. Translate the problem requirements: We need to find the overall average miles per gallon for a combined trip of 10 city miles (at 25 mpg) plus 50 highway miles (at 40 mpg). Average mpg means total miles driven divided by total gallons used.
2. Calculate fuel consumption for each segment: Determine how many gallons were used in the city portion and highway portion separately using the given efficiency rates.
3. Find total distance and total fuel used: Add up the distances and gallons to get overall totals for the entire trip.
4. Calculate overall average efficiency: Divide total miles by total gallons to find the combined miles per gallon, then match to closest answer choice.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking us to find. We have a car that gets different gas mileage depending on where it's driven:

• City driving: 25 miles per gallon
• Highway driving: 40 miles per gallon

The car makes a specific trip: 10 miles in the city, then 50 miles on the highway. We need to find the overall average miles per gallon for this entire 60-mile trip.

Think of it this way: if you wanted to know your average speed for a trip, you'd take the total distance and divide by the total time. Similarly, average fuel efficiency means we take the total distance traveled and divide by the total amount of gas used.

Process Skill: TRANSLATE - Converting the problem language into a clear mathematical understanding

2. Calculate fuel consumption for each segment

Now let's figure out how much gas the car used for each part of the trip.

For the city portion:

• Distance: 10 miles
• Efficiency: 25 miles per gallon
• Gas used = Distance ÷ Efficiency = \(10 \div 25 = 0.4\) gallons

For the highway portion:

• Distance: 50 miles
• Efficiency: 40 miles per gallon
• Gas used = Distance ÷ Efficiency = \(50 \div 40 = 1.25\) gallons

Notice how we're using the basic relationship: if you know miles per gallon, then gallons used equals miles driven divided by miles per gallon.

3. Find total distance and total fuel used

Let's add up our totals for the entire trip:

Total distance traveled:

• City miles + Highway miles = \(10 + 50 = 60\) miles

Total fuel consumed:

• City gas + Highway gas = \(0.4 + 1.25 = 1.65\) gallons

These are the two key numbers we need for our final calculation.

4. Calculate overall average efficiency

Now we can find the overall average miles per gallon using our fundamental principle:

Average mpg = Total miles ÷ Total gallons
Average mpg = \(60 \div 1.65\)

Let's calculate this division:
\(60 \div 1.65 = 60 \div \frac{165}{100} = 60 \times \frac{100}{165} = \frac{6000}{165}\)

To simplify: \(6000 \div 165 \approx 36.36\)

Looking at our answer choices: (A) 28, (B) 30, (C) 33, (D) 36, (E) 38

Our calculated value of 36.36 is closest to 36.

Final Answer

The answer is (D) 36. The car averages approximately 36 miles per gallon for the combined trip of 10 city miles and 50 highway miles.

Common Faltering Points

Errors while devising the approach

• Misunderstanding what "average" means in this context: Students might think they can simply average the two fuel efficiency rates \((25 + 40) \div 2 = 32.5\) mpg, without considering that different distances are traveled at each rate. This weighted average approach ignores the fact that more miles are driven on the highway.
• Confusing the direction of the fuel efficiency calculation: Some students might try to calculate gallons per mile instead of miles per gallon, leading them to add \(\frac{1}{25} + \frac{1}{40}\) and then try to convert back, which creates unnecessary complexity and potential for errors.
• Not recognizing this as a weighted average problem: Students may not realize that since different distances are covered at different efficiencies, they need to find total distance divided by total fuel consumed, rather than trying to apply simple averaging formulas.

Errors while executing the approach

• Arithmetic errors in division: When calculating gas used for each segment, students might make errors like \(10 \div 25 = 0.25\) instead of 0.4, or \(50 \div 40 = 1.5\) instead of 1.25. These small errors cascade into an incorrect final answer.
• Decimal calculation mistakes: The final division \(60 \div 1.65\) requires careful arithmetic. Students might incorrectly calculate this as 30 or 40, especially if they make rounding errors in intermediate steps or struggle with dividing by a decimal.
• Adding fractions incorrectly: If students choose to work with fractions (\(\frac{10}{25} + \frac{50}{40}\)), they might struggle with finding common denominators or make errors when adding \(\frac{2}{5} + \frac{5}{4}\), leading to incorrect total fuel consumption.

Errors while selecting the answer

• Premature rounding: Students might round their calculated answer of 36.36 to 36.4 or 36 and then second-guess themselves, thinking that 38 might be closer, especially if they made small arithmetic errors that shifted their result slightly.
• Selecting a 'reasonable-sounding' middle value: Knowing the answer should be between 25 and 40, students might gravitate toward middle options like 30 or 33 without carefully checking which is actually closest to their calculated result.