- Translate the problem requirements: We need to find the overall fuel efficiency (miles per gallon) for a combined trip. The truck gets 10 mpg in city driving and 25 mpg on highway. The trip consists of 20 city miles and 30 highway miles.
- Calculate fuel consumption for each segment: Determine how many gallons are used for the city portion and highway portion separately using the given efficiency rates.
- Find total trip metrics: Add up the total miles driven and total gallons consumed across both segments.
- Apply overall efficiency formula: Divide total miles by total gallons to get the combined fuel efficiency for the entire trip.
Execution of Strategic Approach
1. Translate the problem requirements
Let's understand what we're being asked to find. We have a truck that gets different fuel efficiency depending on where it's driven:
- In the city: 10 miles per gallon
- On the highway: 25 miles per gallon
The truck makes a specific trip:
- 20 miles in the city
- 30 miles on the highway
We need to find the overall fuel efficiency for this entire trip. Think of it this way: if someone asked "How many miles per gallon did your truck get on that trip?", what would be the answer?
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding
2. Calculate fuel consumption for each segment
Now let's figure out how much gas the truck used for each part of the trip. We'll use the basic relationship that miles per gallon tells us how far we can go on one gallon.
City driving:
If the truck gets 10 miles per gallon in the city, and it drove 20 miles, then:
Gallons used in city = \(20 \text{ miles} \div 10 \text{ miles per gallon} = 2 \text{ gallons}\)
Highway driving:
If the truck gets 25 miles per gallon on the highway, and it drove 30 miles, then:
Gallons used on highway = \(30 \text{ miles} \div 25 \text{ miles per gallon} = 1.2 \text{ gallons}\)
3. Find total trip metrics
Now we'll add up everything for the complete trip:
Total miles driven:
City miles + Highway miles = \(20 + 30 = 50 \text{ miles}\)
Total gallons consumed:
City gallons + Highway gallons = \(2 + 1.2 = 3.2 \text{ gallons}\)
4. Apply overall efficiency formula
To find the overall fuel efficiency, we use the basic definition of miles per gallon: it's simply the total miles divided by the total gallons used.
Overall efficiency = Total miles ÷ Total gallons
Overall efficiency = \(50 \text{ miles} \div 3.2 \text{ gallons} = 15.625 \text{ miles per gallon}\)
Since we need the closest answer from the choices, 15.625 rounds to approximately 16 miles per gallon.
Final Answer
The truck averages approximately 16 miles per gallon for this combined trip.
Looking at our answer choices:
- 16 ✓ (This matches our calculation)
- 18
- 20
- 23
- 25
The answer is A. 16
Common Faltering Points
Errors while devising the approach
- Misunderstanding what "average" means: Students often think they should take the simple average of the two efficiency rates \((10 + 25)/2 = 17.5 \text{ mpg}\), rather than understanding that we need to find total miles divided by total gallons for the actual trip. This conceptual error leads them to ignore the specific distances driven.
- Incorrectly setting up weighted averages: Some students attempt to use weighted averages but set up the weights incorrectly. They might weight by miles (20:30) when they should be considering the fuel consumption impact, or they might try to weight the efficiency rates directly without understanding the underlying fuel consumption relationship.
Errors while executing the approach
- Calculation errors with division: Students frequently make arithmetic mistakes when calculating \(30 \div 25 = 1.2 \text{ gallons}\) for highway consumption, often getting 1.5 or other incorrect values. This error propagates through the entire solution.
- Incorrect total fuel calculation: After calculating individual fuel consumption (2 gallons city + 1.2 gallons highway), students sometimes add incorrectly, getting 3.0 instead of 3.2 total gallons, which significantly affects the final answer.
- Final division errors: When computing \(50 \div 3.2\), students may struggle with the decimal division, getting values like 14.6 or 18.2 instead of the correct 15.625, leading to selection of wrong answer choices.
Errors while selecting the answer
- Poor rounding judgment: Students who correctly calculate 15.625 mpg might round to 15 or 16 but then second-guess themselves and select 18 (choice B) thinking it's "closer to the middle" of the given efficiency rates, rather than trusting their calculation that 15.625 is closest to 16.
