🇺🇸 AMC 8 ⇄ switch contest
1996 AJHSME

Problem 7

Problem 7 · 1996 AJHSME Medium
Number Theory powersexponential

Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, in how many months from that time will they have the same number of goldfish?

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Answer: B — 5 months.
Show hints
Hint 1 of 2
Brent starts at 4, way behind Gretel's 128 — but each month Brent multiplies by 4 while Gretel only doubles. So every month Brent's count gains an extra ×2 on Gretel's. Track how fast that closes the gap.
Still stuck? Show hint 2 →
Hint 2 of 2
Brent starts at 4 = Gretel ÷ 32, and each month Brent's ratio to Gretel doubles (×4 vs ×2). Step both forward — or count how many doublings turn 1/32 into 1.
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Approach: step both counts forward until they meet
  1. Each month Brent's ×4 outruns Gretel's ×2 by a factor of 2, so the ratio Brent:Gretel doubles every month. He starts at 4 vs 128 (a ratio of 1/32), and 32 is 2⁵ — so it takes 5 doublings to catch up.
  2. Run the counts to confirm. Brent quadruples: 4, 16, 64, 256, 1024, 4096. Gretel doubles: 128, 256, 512, 1024, 2048, 4096. They first match at 4096, after 5 months.
Another way — match powers of 2:
  1. As powers of 2: Brent has 4·4m = 22m+2, Gretel has 128·2m = 2m+7.
  2. Setting the exponents equal, 2m + 2 = m + 7, gives m = 5.
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