AMC 8 · 2024 · #12

Grade 2 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

sequences-arithmeticlinear-equations-one-var systematic-enumeration ↑ Prerequisites: multi-digit-arithmeticmental-arithmetic

📏 Medium solution 💡 3 insights

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Problem

Rohan has 4 fish tanks holding 90 guppies in total. Each tank has a few more guppies than the one before: tank 2 has 1 more than tank 1, tank 3 has 2 more than tank 2, and tank 4 has 3 more than tank 3. We need the number of guppies in tank 4.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

This is a multiple-choice problem and the unknown is exactly the number we are asked for (guppies in tank 4). Each answer choice fully determines all four tanks by walking backwards (tank 3 = tank 4 - 3, tank 2 = tank 3 - 2, tank 1 = tank 2 - 1). We can test a single choice, add the four numbers, and see if it equals 90 — Tool #6. Combined with Tool #3 we can eliminate any choice whose total is not 90, and Tool #1 (a quick row of 4 boxes) keeps the four tanks straight.

1STEP 1

Draw 4 boxes for the tanks and mark the gaps +1, +2, +3 — knowing any one tank now fixes all four.

T₁ → T₂ → T₃ → T₄

2STEP 2

Guess the biggest choice for Tank 4, then step backward: Tank 3 = 23, Tank 2 = 21, Tank 1 = 20 — all whole numbers, a good sign.

T₄=26, T₃=23, T₂=21, T₁=20

3STEP 3

Add the four counts in friendly pairs: 20+21=41 and 23+26=49, so the total is 90 — the guess works.

20+21+23+26 = 41+49 = 90 ✓

4STEP 4

Because Tank 4 fixes all four tanks, only one choice can total 90 — e.g. (A) 20 gives sum 66, too small. So the biggest choice stands.

Choice (E)colon 20+21+23+26=90 → (E)

Answer

Check the magnitude: if all four tanks held the same amount, each would hold 90 ÷ 4 = 22.5. Tank 4 should be a bit bigger than 22.5 because it has the most extras (+1+2+3 over Tank 1). Our answer 26 is just a few above 22.5 — that fits perfectly. The four numbers (20, 21, 23, 26) are all positive whole numbers and add to 90, matching every condition in the problem.

💡Key takeaway

This AMC 8 problem only needs Grade 2 two-digit adding and subtracting you already know!

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