Problem 2 · AMC 8 Stretch
Core
Arithmetic & Operations
Logic & Word ProblemsRatios, Rates & Proportions
identifying-relevant-datavisual-representationlogical-reasoning
A worker bikes to work. The trip is 3 km and he usually rides at 15 km/h. One day, after going 1 km, he gets a flat tire, so he pushes his bike the rest of the way, arriving 20 minutes late. At work he fixed the tire and rode all the way home as usual. Over the whole round trip (there and back), how many more kilometers did he ride than he walked?
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Answer: 2 km more by bike
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Hint 1 of 4
Draw the trip as a straight line: home to work, then work back home. Mark the flat (1 km from home). Shade the parts he rode and the parts he walked.
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Hint 2 of 4
Going to work: he rode the first 1 km, then walked the last 2 km. Coming home: he rode all 3 km. Now compare total riding to total walking.
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Hint 3 of 4
The 2 km from the flat to work was walked once (going) and ridden once (coming home), so those cancel. What part of the route did he ride but never walk?
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Approach: Picture the round trip and cancel the matching segments
- Picture the round trip: home —(ride 1 km)— flat —(walk 2 km)— work, then work —(ride 3 km)— home.
- He rode 1 km going + 3 km coming = 4 km. He walked 2 km. So he rode \(4 - 2 = 2\) km more than he walked.
- A neat way to see it: the 2 km from the flat to work was walked once and ridden once, so it cancels. Only the first 1 km was ridden both directions and never walked, counting twice (\(2 \times 1 = 2\) km).
- So he rode 2 km more than he walked. (Notice everything except that 1 km — even the speed and lateness — is unnecessary for this question.)
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