Problem 10 · 2005 AMC 8
Easy
Ratios, Rates & Proportions
speed-time-ratio
Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school?
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Answer: D — 8 minutes.
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Hint 1 of 2
Both halves cover the same distance, so you can compare them directly — no need to know the actual distance or speed.
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Hint 2 of 2
For a fixed distance, speed and time are flip sides: triple the speed means a third of the time.
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Approach: same distance ⇒ time scales by 1/speed
- The two halves are equal distances, so walking and running can be compared head-to-head. Walking the first half took 6 min.
- Running is 3× as fast over the same distance, so it takes ⅓ the time: 6 ÷ 3 = 2 min.
- Total: 6 + 2 = 8 min.
- Why this transfers: over a fixed distance, time is inversely proportional to speed — double speed → half time, triple speed → third the time. No distance value needed. (The decimal traps 7.3, 7.7, 8.3 punish anyone who fumbles the 'a third' step.)
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