A car traveled 462 miles per tankful of gasoline on the highway and 336 miles per tankful of gasoline in...

1. Translate the problem requirements: We need to find city miles per gallon. We know the car travels 462 miles per tankful on highway, 336 miles per tankful in city, and gets 6 fewer miles per gallon in the city than on the highway. Since the tank size doesn't change, we can relate miles per gallon to miles per tankful.
2. Set up the fundamental relationship: Establish that miles per gallon equals miles per tankful divided by tank capacity (in gallons), and since tank capacity stays the same, we can work with this relationship.
3. Express the 6 mpg difference using our known values: Write an equation showing that highway mpg minus city mpg equals 6, substituting our miles per tankful values divided by the unknown tank capacity.
4. Solve for tank capacity and then city mpg: Use the equation to find the tank capacity, then calculate city miles per gallon directly.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in simple terms:

• The car can go 462 miles on one full tank when driving on the highway
• The same car can go 336 miles on one full tank when driving in the city
• The car gets 6 fewer miles per gallon in the city compared to the highway
• We need to find how many miles per gallon the car gets in the city

The key insight here is that the tank size stays the same - it's the same car! The only thing that changes is how efficiently the car uses the gas in different driving conditions.

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

2. Set up the fundamental relationship

Let's think about this logically. If you have a tank that holds a certain number of gallons, and you can drive a certain number of miles on that tank, then:

Miles per gallon = Total miles driven ÷ Number of gallons in tank

Since we're using the same tank for both highway and city driving:

• Highway miles per gallon = \(\frac{462}{\text{tank capacity}}\)
• City miles per gallon = \(\frac{336}{\text{tank capacity}}\)

Let's call the tank capacity 'T' gallons. So:

• Highway mpg = \(\frac{462}{\mathrm{T}}\)
• City mpg = \(\frac{336}{\mathrm{T}}\)

3. Express the 6 mpg difference using our known values

We know that the highway mpg is 6 more than the city mpg. In mathematical terms:

Highway mpg - City mpg = 6

Substituting our expressions:

\(\frac{462}{\mathrm{T}} - \frac{336}{\mathrm{T}} = 6\)

We can combine the fractions since they have the same denominator:

\(\frac{462 - 336}{\mathrm{T}} = 6\)

\(\frac{126}{\mathrm{T}} = 6\)

4. Solve for tank capacity and then city mpg

Now we can solve for T (the tank capacity):

\(\frac{126}{\mathrm{T}} = 6\)

\(126 = 6\mathrm{T}\)

\(\mathrm{T} = 126 ÷ 6 = 21\)

So the tank holds 21 gallons.

Now we can find the city miles per gallon:

City mpg = \(336 ÷ 21 = 16\)

Let's verify: Highway mpg = \(462 ÷ 21 = 22\)

Difference = \(22 - 16 = 6\) ✓

4. Final Answer

The car travels 16 miles per gallon in the city.

Looking at our answer choices, this matches option (B) 16.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what stays constant: Students often fail to recognize that the tank capacity remains the same for both highway and city driving. They might try to set up separate variables for highway tank size and city tank size, leading to an unsolvable system of equations.

2. Confusing the relationship direction: Students may incorrectly interpret "6 fewer miles per gallon in the city" as city mpg being 6 more than highway mpg, setting up the equation as City mpg - Highway mpg = 6 instead of Highway mpg - City mpg = 6.

3. Setting up the wrong fundamental equation: Some students might try to work directly with gallons consumed rather than miles per gallon, or confuse the relationship between total miles, tank capacity, and efficiency, leading to incorrect initial equations.

Errors while executing the approach

1. Fraction arithmetic errors: When combining fractions \(\frac{462}{\mathrm{T}} - \frac{336}{\mathrm{T}}\), students might incorrectly subtract denominators as well as numerators, getting \(\frac{462-336}{\mathrm{T}-\mathrm{T}} = \frac{126}{0}\), or make other fraction manipulation errors.

2. Division mistakes: Students often make arithmetic errors when calculating \(126 ÷ 6 = 21\) for the tank capacity, or when calculating \(336 ÷ 21 = 16\) for the final city mpg, potentially getting values like 20 or 15.

3. Solving for the wrong variable: After finding T = 21, students might mistakenly think this is the final answer (city mpg) rather than the tank capacity, forgetting to complete the calculation of \(336 ÷ 21\).

Errors while selecting the answer

1. Selecting highway mpg instead of city mpg: After correctly calculating both city mpg = 16 and highway mpg = 22, students might accidentally select answer choice (D) 22, which represents highway efficiency rather than the requested city efficiency.

2. Selecting tank capacity as the answer: Students who calculated the tank capacity correctly as 21 gallons might mistakenly think this is the final answer and select choice (C) 21, not realizing they need to continue the calculation to find city mpg.