Problem 19 · 2026 AMC 8
Hard
Ratios, Rates & Proportions
relative-distance
Miguel and his dog Luna start together at a park entrance. Miguel throws a ball straight ahead to a tree and keeps walking at a steady pace. Luna sprints to the ball and immediately brings it back to Miguel. Luna runs 5 times as fast as Miguel walks. What fraction of the entrance-to-tree distance does Miguel cover by the time Luna brings him the ball?
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Answer: D — 1/3.
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Hint 1 of 2
They run for the same amount of time, so forget the clock entirely — in equal time, distance is just proportional to speed. Luna covers 5 times as far as Miguel, whatever that distance turns out to be.
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Hint 2 of 2
Picture Luna's actual route: entrance → tree → back to wherever Miguel has walked to. Set that whole length equal to 5 times Miguel's short walk and solve.
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Approach: same time ⇒ Luna's path length = 5 × Miguel's
- Because they move for the same time, the time cancels: Luna's total distance is simply 5 times Miguel's. So set their path lengths in a 5-to-1 ratio — no need for actual speeds or seconds.
- Let the entrance-to-tree distance be 1 and Miguel's walk be d. Luna runs to the tree (1) and back to Miguel, who has moved up to d, so the return leg is 1 − d. Luna's path = 1 + (1 − d) = 2 − d.
- Equal time means Luna's path = 5d: 5d = 2 − d ⇒ 6d = 2 ⇒ d = 1/3.
- Why this transfers: when two movers share the same time window, drop the clock — their distances are in the ratio of their speeds, turning a rate problem into a one-line distance equation.
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