Problem 6 · 2018 AMC 8
Medium
Ratios, Rates & Proportions
distance-speed-time
On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?
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Answer: C — 80 minutes.
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Hint 1 of 2
The one fact you're handed (10 miles in 30 minutes) is secretly a speed. Pin that down and the highway speed comes free — it's just three times bigger.
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Hint 2 of 2
The technique: when given distance and time for one stretch, get a speed, then use that speed (or a scaled version) to find the missing time on the other stretch. Time = distance ÷ speed.
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Approach: find each leg's time separately
- Coastal leg: 10 miles in 30 min, so his coastal speed is 1/3 mile per minute. Highway is 3× faster: 1 mile per minute.
- Highway time: 50 miles ÷ 1 mile/min = 50 min.
- Total trip: 30 + 50 = 80 minutes.
Another way — compare the legs directly (no speeds needed):
- Going 3× faster on the highway means each highway mile takes 1/3 the time of a coastal mile. The highway is 5× longer (50 vs 10 miles) but 3× quicker per mile, so the highway takes 5/3 of the coastal time.
- Coastal took 30 min, so highway takes (5/3)(30) = 50 min, and the total is 30 + 50 = 80 minutes — reached entirely by scaling the one known time.
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