Problem 12 · 2024 AMC 8
Medium
Algebra & Patterns
substitutionsum-constraint
Rohan keeps a total of 90 guppies in 4 fish tanks.
- There is 1 more guppy in the 2nd tank than in the 1st tank.
- There are 2 more guppies in the 3rd tank than in the 2nd tank.
- There are 3 more guppies in the 4th tank than in the 3rd tank.
How many guppies are in the 4th tank?
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Answer: E — 26 guppies.
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Hint 1 of 2
The clues chain off each other, so anchor everything to ONE tank. From tank 1, each later tank is tank 1 plus a running total: +1, then +1+2 = +3, then +1+2+3 = +6.
Still stuck? Show hint 2 →
Hint 2 of 2
Technique: write all four as tank 1 + (0, 1, 3, 6). Their sum is 4·(tank 1) + 10 = 90 — one equation for one unknown.
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Approach: express every tank in terms of tank 1
- The differences chain, so pin everything to tank 1 = x. Then tank 2 = x + 1, tank 3 = (x+1) + 2 = x + 3, tank 4 = (x+3) + 3 = x + 6.
- Adding: 4x + (1 + 3 + 6) = 4x + 10 = 90, so x = 20.
- The question wants tank 4, not tank 1 — so finish the job: tank 4 = 20 + 6 = 26. Watch out: solving for x = 20 and stopping is the classic trap; always re-read what's being asked.
Another way — level the tanks against the total:
- If all four tanks matched tank 1, the total would be 4·(tank 1). The real total is 10 more (the built-in extras 0+1+3+6), so 4·(tank 1) = 90 − 10 = 80 → tank 1 = 20.
- Tank 4 carries the full extra +6: 20 + 6 = 26.
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