AMC 10 · 2011 · #15

Grade 7 rate-ratio
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
rateratio-proportionlinear-equations-one-var convert-to-algebradimensional-analysis ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights
Problem
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.

Pick an answer.

(A)
$\ 140$
(B)
$\ 240$
(C)
$\ 440$
(D)
$\ 640$
(E)
$\ 840$

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

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.

1STEP 1

Read the units of the average

'Miles per gallon' means miles divided by gallons, so 55 is the whole trip's distance over all the gasoline burned.

55 = (total miles)/(total gallons of gasoline)
2STEP 2

Name the gasoline distance

Let x be the miles run on gasoline. The trip is then 40 + x miles long, and the gasoline used is 0.02x gallons.

total miles = 40 + x, total gallons = 0.02x
3STEP 3

Write the average as an equation

Put both totals into the rate from Step 1: the average 55 equals the miles 40 + x divided by the gallons 0.02x.

(40 + x)/0.02x = 55
4STEP 4

Solve for the gasoline miles

Clear the fraction: 40 + x = 1.1x. Subtracting x gives 40 = 0.1x, so x = 400 gasoline miles.

40 + x = 1.1x → 40 = 0.1x → x = 400
5STEP 5

Add the two parts of the trip

Add the battery 40 miles to the gasoline 400 miles: the whole trip is 440 miles, choice (C).

40 + 400 = 440 → (C) 440
Answer
440
Check the story with a 440-mile trip: 400 of those miles use gasoline at 0.02 gal/mi, so the car burns 400 × 0.02 = 8 gallons. The average is 440 ÷ 8 = 55 mpg, exactly the given value, so (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 ÷ 0.02 = 50 mpg — and 55 ＞ 50, which is consistent.
💡Key takeaway

Miles per gallon is total miles over total gallons, so name the fuel-burning miles and set that fraction equal to the average.

  • Read the units of the average
  • Name the gasoline distance
  • Write the average as an equation
  • Solve for the gasoline miles
  • Add the two parts of the trip

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