Problem 14 · 2016 AMC 8
Medium
Ratios, Rates & Proportions
distance-speed-timeunit-rate
Karl's car uses a gallon of gas every 35 miles, and his gas tank holds 14 gallons when it is full. One day, Karl started with a full tank of gas, drove 350 miles, bought 8 gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day?
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Answer: A — 525 miles.
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Hint 1 of 2
The miles after the gas stop aren't given directly — but the GAS is. Stop chasing miles; instead bookkeep the tank's gallons at every stage, and the unknown miles fall out of the gallons burned.
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Hint 2 of 2
March the gauge forward: full → (drive 350) → (buy 8) → half-full at the end. The only mystery leg is the last one; the gallons burned there = (gallons after buying) − (gallons left), and gallons × 35 gives those miles.
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Approach: bookkeep gallons stage by stage; convert gallons burned to miles
- Start full = 14 gal. Driving 350 mi burns 350 ÷ 35 = 10 gal, leaving 14 − 10 = 4 gal.
- Buy 8 gal → tank now holds 4 + 8 = 12 gal. He arrives half full = 7 gal, so the last leg burned 12 − 7 = 5 gal = 5 × 35 = 175 miles.
- Total distance = 350 + 175 = 525 miles.
- Why this transfers: in any "resource used up over a journey" problem (fuel, battery, money), track the level of the RESOURCE step by step — the hidden distance/time pops out of the amount consumed.
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