For a 30-mile trip, the fuel consumption of a certain car was 15 miles per gallon for the first 10...

1. Translate the problem requirements: We need to find the overall fuel consumption (miles per gallon) for a 30-mile trip that had different fuel efficiencies for different segments - 15 mpg for first 10 miles and 45 mpg for remaining 20 miles.
2. Calculate fuel consumed in each segment: Find how many gallons were used in the first 10 miles and how many gallons were used in the remaining 20 miles using the given fuel efficiencies.
3. Find total fuel consumption: Add up the gallons used in both segments to get total gallons for the entire trip.
4. Calculate overall fuel efficiency: Use the relationship that overall mpg equals total miles divided by total gallons used for the entire 30-mile trip.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in plain English:

- We have a car that traveled 30 miles total
- For the first 10 miles, the car got 15 miles per gallon
- For the remaining miles (\(30 - 10 = 20\) miles), the car got 45 miles per gallon
- We need to find the overall fuel consumption for the entire trip

Think of it like this: if you're calculating your average speed for a trip, you can't just average the two speeds - you need to consider how much time was spent at each speed. Similarly, for fuel consumption, we can't just average 15 mpg and 45 mpg. We need to find out how much fuel was actually used for the entire trip.

Process Skill: TRANSLATE - Converting the problem setup into clear mathematical relationships

2. Calculate fuel consumed in each segment

Now let's figure out how many gallons of fuel were used in each part of the trip.

For the first segment (first 10 miles at 15 mpg):
If the car travels 15 miles on 1 gallon, then for 10 miles it uses:
\(10 \text{ miles} ÷ 15 \text{ miles per gallon} = \frac{10}{15} = \frac{2}{3} \text{ gallon}\)

For the second segment (remaining 20 miles at 45 mpg):
If the car travels 45 miles on 1 gallon, then for 20 miles it uses:
\(20 \text{ miles} ÷ 45 \text{ miles per gallon} = \frac{20}{45} = \frac{4}{9} \text{ gallon}\)

Notice how we're using the relationship: Gallons used = Miles traveled ÷ Miles per gallon

3. Find total fuel consumption

Now we add up all the fuel used during the entire trip:
Total gallons used = \(\frac{2}{3} + \frac{4}{9}\)

To add these fractions, we need a common denominator. The least common multiple of 3 and 9 is 9:
\(\frac{2}{3} = \frac{6}{9}\)

So: Total gallons = \(\frac{6}{9} + \frac{4}{9} = \frac{10}{9}\) gallons

4. Calculate overall fuel efficiency

Now we can find the overall fuel efficiency using the basic relationship:
Overall mpg = Total miles ÷ Total gallons

Overall mpg = \(30 \text{ miles} ÷ \frac{10}{9} \text{ gallons}\)

When dividing by a fraction, we multiply by its reciprocal:
Overall mpg = \(30 × \frac{9}{10} = \frac{270}{10} = 27\) miles per gallon

Let's verify this makes sense: 27 mpg is between our two segment efficiencies of 15 mpg and 45 mpg, which is exactly what we'd expect for an average.

Final Answer

The car's fuel consumption for the trip was 27 miles per gallon.

Answer: B. 27

Common Faltering Points

Errors while devising the approach

1. Incorrectly averaging the two fuel efficiency rates

Many students mistakenly think they can simply average 15 mpg and 45 mpg to get \(\frac{15 + 45}{2} = 30\) mpg. This is incorrect because it doesn't account for the fact that different amounts of fuel were consumed during each segment of the trip. Just like averaging speeds over different time periods, you cannot directly average fuel efficiency rates without considering the actual fuel consumed.

2. Misunderstanding what "fuel consumption" means in the context

Some students may confuse "fuel consumption" with "fuel used" and try to calculate total gallons consumed rather than the overall miles per gallon for the entire trip. The question asks for fuel consumption "in miles per gallon," which means we need the overall efficiency rate, not the total amount of fuel used.

Errors while executing the approach

1. Arithmetic errors when working with fractions

Students often make mistakes when adding fractions \(\frac{2}{3} + \frac{4}{9}\). Common errors include: not finding the correct common denominator, incorrectly converting \(\frac{2}{3}\) to \(\frac{6}{9}\), or making calculation mistakes when adding \(\frac{6}{9} + \frac{4}{9} = \frac{10}{9}\).

2. Division error when calculating final miles per gallon

When dividing 30 by \(\frac{10}{9}\), students may forget to multiply by the reciprocal and instead try to divide 30 by 10 and then by 9 separately, leading to incorrect calculations. The correct approach is \(30 × \frac{9}{10} = 27\).

3. Incorrectly calculating fuel used for each segment

Students may confuse the relationship between miles, gallons, and mpg. For example, they might multiply instead of divide: incorrectly calculating 10 × 15 gallons for the first segment instead of 10 ÷ 15 gallons.

Errors while selecting the answer

No likely faltering points - the calculation leads directly to 27, which matches answer choice B exactly.