Problem 21 · 2024 AMC 8
Hard
Algebra & Patterns
ratioproportionsubstitution
A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was 3 : 1. Then 3 green frogs moved to the sunny side and 5 yellow frogs moved to the shady side. Now the ratio is 4 : 1. What is the difference between the number of green frogs and yellow frogs now?
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Answer: E — 24 frogs.
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Hint 1 of 2
A 3 : 1 ratio means green = 3 × yellow, so the whole army rides on one number. Call yellow y and write everything in terms of it.
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Hint 2 of 2
Technique: track the net change per color (frogs move both ways), then set the new ratio equal to 4. After the moves green = 3y + 2, yellow = y − 2 — so (3y+2)/(y−2) = 4.
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Approach: let y = initial yellow, then use both ratios
- The 3 : 1 ratio pins green to yellow, so use one variable: let y = initial yellow, then initial green = 3y.
- Net the movements per color. Green: 5 yellow-turned-green arrive, 3 leave for the sun → 3y + 5 − 3 = 3y + 2. Yellow: 3 sun-turned-yellow arrive, 5 leave for shade → y + 3 − 5 = y − 2.
- New ratio 4 : 1 means green is 4 times yellow: 3y + 2 = 4(y − 2) = 4y − 8 → y = 10. So now green = 32, yellow = 8.
- The question asks the DIFFERENCE now: 32 − 8 = 24. Watch the ask: it wants the current gap, not the original counts — easy to stop a step early.
Another way — the total army never changes:
- Every move just recolors a frog (sun↔shade), so the total is fixed. Initially green:yellow = 3:1, so the total is a multiple of 3+1 = 4; finally it's 4:1, a multiple of 4+1 = 5. The total is a multiple of both, so a multiple of 20.
- Now the difference. Finally green:yellow = 4:1, so the gap is 4−1 = 3 parts out of 5, i.e. 35 of the total. With the total = 40 (the value consistent with the 3 net-out yellow and 4:1 split), the difference is 35×40 = 24.
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