Problem 22 · 2022 AMC 8
Hard
Ratios, Rates & Proportions
distance-speed-timecasework
A bus takes 2 minutes to drive from one stop to the next, and waits 1 minute at each stop to let passengers board. Zia takes 5 minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus 3 stops behind. After how many minutes will Zia board the bus?
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Answer: A — 17 minutes.
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Hint 1 of 2
Zia only makes a decision at the instants she reaches a stop — every 5 minutes. So you don't need a continuous chase; just check the situation at t = 5, 10, 15, … and see where the bus is each time.
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Hint 2 of 2
The bus's rhythm is 2 min driving + 1 min waiting = 3 min per stop. Track which stop each of them is at on Zia's 5-minute marks.
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Approach: only check the few moments Zia can act — her 5-minute arrivals
- Insight: a step-by-step chase looks messy, but Zia only chooses to wait-or-walk when she arrives at a stop — at t = 5, 10, 15. Sampling just those moments (the bus runs on a tidy 3-min-per-stop cycle: 2 driving + 1 waiting) makes it a 3-line simulation. Number the stops from the bus's start (stop 0); at t = 0, Zia is at stop 3, bus at stop 0.
- t = 5: Zia at stop 4. Bus took 5 min → finished stop 1 (arrived at 2 min, left at 3 min, arrived at stop 2 at 5 min). Bus is at stop 2 — not yet at the previous stop (3), so Zia walks on.
- t = 10: Zia at stop 5. Bus: from t = 5 (at stop 2) waits 1 min (leaves at 6), drives 2 min to stop 3 (arrives at 8), waits till 9, drives to stop 4 (arrives at 11). So at t = 10, bus is mid-drive between stops 3 and 4 — not at the previous stop (4), so Zia walks on.
- t = 15: Zia at stop 6. Bus: arrives at stop 4 at 11, waits till 12, drives to stop 5 (arrives 14, waits till 15). At t = 15, bus is at stop 5 — the previous stop. Zia waits.
- Bus leaves stop 5 at t = 15 and drives 2 min to stop 6: arrives at t = 17.
- You'll see this again: when one mover only acts at fixed intervals, you don't have to track time continuously — jump straight to those decision instants and read off the other mover's state. Discretizing turns a chase into a short table.
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