AMC 10 · 2011 · #15
Grade 7 rate-ratioRoy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?
Pick an answer.
AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A hybrid car runs on its battery for the first $40$ miles, then switches to gasoline for the rest of the trip. On gasoline it burns $0.02$ gallons for every mile. Across the whole trip the car averaged $55$ miles per gallon. Find the total length of the trip in miles.
Givens: The first $40$ miles use the battery and no gasoline; After that, the car uses gasoline at $0.02$ gallons per mile; The average for the entire trip is $55$ miles per gallon; Answer choices: (A) $140$, (B) $240$, (C) $440$, (D) $640$, (E) $840$
Unknowns: The total length of the trip in miles
Understand
Restated: A hybrid car runs on its battery for the first $40$ miles, then switches to gasoline for the rest of the trip. On gasoline it burns $0.02$ gallons for every mile. Across the whole trip the car averaged $55$ miles per gallon. Find the total length of the trip in miles.
Givens: The first $40$ miles use the battery and no gasoline; After that, the car uses gasoline at $0.02$ gallons per mile; The average for the entire trip is $55$ miles per gallon; Answer choices: (A) $140$, (B) $240$, (C) $440$, (D) $640$, (E) $840$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #8 Analyze the Units, #13 Convert to Algebra, #3 Eliminate Possibilities
The trip splits into a battery part and a gasoline part, and only the gasoline part uses fuel, so the single thing worth naming is the gasoline distance. Tool #4 (Introduce a Variable) names that unknown, $x$, and lets every other quantity be written in terms of it. Tool #8 (Analyze the Units) is what makes the whole problem work: '$55$ miles per gallon' is a rate, total miles divided by total gallons, so tracking miles and gallons separately is the key move. Tool #13 (Convert to Algebra) turns the sentence 'the average was $55$ mpg' into one equation. Tool #3 (Eliminate Possibilities) is the backup: since the choices are given, each can simply be tested against the $55$ mpg rule.
Execute — Answer: C
6.RP.A.3 Step 1 Read the units of the average
- 'Miles per gallon' means miles divided by gallons.
- So the trip average of $55$ mpg is the whole trip's distance divided by all the gasoline burned.
- Only the gasoline part of the trip burns fuel, so those are the two totals to track.
💡 The word 'per' is a division sign, so a rate always tells you which two totals to put over each other.
6.EE.B.6 Step 2 Name the gasoline distance
- Let $x$ be the number of miles the car ran on gasoline.
- The first $40$ miles were on the battery, so the total trip is $40 + x$ miles.
- Gasoline is burned at $0.02$ gallons per mile over those $x$ miles, so the gasoline used is $0.02x$ gallons.
💡 Naming just the gasoline miles lets both totals be written from that one letter.
7.RP.A.3 Step 3 Write the average as an equation
- Put the two totals into the rate from Step 1.
- The whole-trip average of $55$ mpg equals the total miles $40 + x$ divided by the total gallons $0.02x$.
💡 Once the two totals are in terms of $x$, the rate sentence becomes one true equation.
7.EE.B.4 Step 4 Solve for the gasoline miles
- Multiply both sides by $0.02x$ to clear the fraction: $40 + x = 55 \times 0.02x$.
- Since $55 \times 0.02 = 1.1$, this is $40 + x = 1.1x$.
- Subtract $x$ from both sides to get $40 = 0.1x$, then divide by $0.1$ to find $x = 400$.
💡 Clearing the fraction turns the rate equation into a one-step decimal equation.
7.NS.A.3 Step 5 Add the two parts of the trip
- The gasoline part is $x = 400$ miles and the battery part is $40$ miles, so the total trip is $40 + 400 = 440$ miles.
- This matches choice (C).
💡 The answer is just the battery miles plus the gasoline miles added back together.
6.RP.A.3 'Miles per gallon' means miles divided by gallons. So the trip average of $55$ m 6.EE.B.6 Let $x$ be the number of miles the car ran on gasoline. The first $40$ miles wer 7.RP.A.3 Put the two totals into the rate from Step 1. The whole-trip average of $55$ mpg 7.EE.B.4 Multiply both sides by $0.02x$ to clear the fraction: $40 + x = 55 \times 0.02x$ 7.NS.A.3 The gasoline part is $x = 400$ miles and the battery part is $40$ miles, so the Review
Reasonableness: Check the story with a $440$-mile trip: $400$ of those miles use gasoline at $0.02$ gal/mi, so the car burns $400 \times 0.02 = 8$ gallons. The average is $440 \div 8 = 55$ mpg, exactly the given value, so $\textbf{(C)}$ is correct. The answer should also beat $55$ mpg overall, since $40$ free battery miles pull the average up above the gasoline-only rate of $1 \div 0.02 = 50$ mpg — and $55 > 50$, which is consistent.
Alternative: Test the choices directly (Tool #3, Eliminate Possibilities). For each choice, subtract the $40$ battery miles to get the gasoline miles, multiply by $0.02$ for gallons, then divide the total distance by that. (A) $140$: $100$ gas miles, $2$ gal, $140 \div 2 = 70$ mpg — too high. (B) $240$: $200$ gas miles, $4$ gal, $240 \div 4 = 60$ mpg — still too high. (C) $440$: $400$ gas miles, $8$ gal, $440 \div 8 = 55$ mpg — matches, so $\textbf{(C)}$.
CCSS standards used (min grade 7)
6.RP.A.3Use ratio and rate reasoning to solve real-world problems (Reading '$55$ miles per gallon' as total miles divided by total gallons of gasoline.)6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Naming the gasoline distance $x$ and writing total miles $40 + x$ and total gallons $0.02x$.)7.RP.A.3Use proportional relationships to solve multistep ratio and percent problems (Turning the whole-trip rate of $55$ mpg into the equation $(40 + x) / (0.02x) = 55$.)7.EE.B.4Solve word problems leading to equations of the form px + q = r (Solving $40 + x = 1.1x$ to get $x = 400$ gasoline miles.)7.NS.A.3Solve real-world problems involving the four operations with rational numbers (Working with the decimals $0.02$ and $1.1$ and adding $40 + 400 = 440$.)
⭐ Miles per gallon is total miles over total gallons, so name the fuel-burning miles and set that fraction equal to the average.
⭐ Miles per gallon is total miles over total gallons, so name the fuel-burning miles and set that fraction equal to the average.
More like this
Same archetype — closest grade level first.