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2016 AMC 8

Problem 16

Problem 16 · 2016 AMC 8 Medium
Ratios, Rates & Proportions ratiooff-by-one

Annie and Bonnie are running laps around a 400-meter oval track. They started together, but Annie has pulled ahead because she runs 25% faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?

Show answer
Answer: D — 5 laps.
Show hints
Hint 1 of 2
"First passes" on a loop track doesn't mean catching up — they started together, so Annie passes Bonnie when she's a WHOLE LAP ahead. The question is really about the GAP between them growing to 400 m, not their absolute positions.
Still stuck? Show hint 2 →
Hint 2 of 2
25% faster means speeds are in ratio 5 : 4, so Annie gains 1/4 of a lap on Bonnie for each lap Bonnie runs. Ask: how many of Bonnie's laps until that gain reaches one full lap?
Show solution
Approach: track the GAP: Annie gains a quarter-lap per Bonnie-lap until it's a full lap
  1. Annie : Bonnie speeds = 125 : 100 = 5 : 4, so while Bonnie runs 1 lap, Annie runs 5/4 laps — Annie's lead grows by 1/4 lap each Bonnie-lap.
  2. To pull a full lap ahead (the moment she "passes"), the lead must reach 1 lap: that takes 4 quarter-laps, i.e. Bonnie runs 4 laps.
  3. In that same stretch of time Annie runs 5/4 × 4 = 5 laps.
  4. Watch the trap: answer 4 is Bonnie's lap count, not Annie's — the question asks how far ANNIE has run. Track whose laps you're reporting.
  5. Why this transfers: for two bodies moving the same direction, work with the RELATIVE speed (the gap's growth rate); "lapping" happens when the gap equals one full lap.
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