AMC 8 2024 Problem 21
AMC8 2024 年第 21 题
比与比例★★★☆☆
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 the number of yellow frogs now?
一群青蛙（号称青蛙军团）生活在树上。它们在树荫中呈绿色，在阳光下又变成黄色。刚开始，绿色青蛙和黄色青蛙的数量之比是3：1。随后，3只绿色青蛙跳到阳光下，5只黄色青蛙跳到树荫下，此时数量之比变成4：1。请问现在绿色青蛙和黄色青蛙的数量相差多少？
- A.10
- B.12
- C.16
- D.20
- E.24
正确答案 · Correct Answer
E
解析 · Solution
解法1
常规解法 · 一元一次方程（只设 1 个未知数）
设黄蛙有 只，则绿蛙 只（绿:黄 =3:1）。3 只绿蛙到阳面变黄、5 只黄蛙到阴面变绿，所以绿蛙净增 5-3=2、黄蛙净减 2：现在绿蛙 3x+2 只、黄蛙 x-2 只。新比 给出 ，即 3x+2=4(x-2)=4x-8，解得 x=10。现在两者之差 =(3x+2)-(x-2)=2x+4=24。
另解 · 设两个未知数
设初始绿蛙 只、黄蛙 只，由 得 g=3y。变化后绿蛙 g+2、黄蛙 y-2，由 得 g+2=4y-8，代入 g=3y：3y+2=4y-8，解得 y=10、g=30。现差 =(g+2)-(y-2)=32-8=24。
Solution 1
Standard — one unknown (a single linear equation). Let the number of yellow frogs be , so the green frogs number (green:yellow =3:1). Then 3 green frogs go to the sun (turning yellow) and 5 yellow frogs go to the shade (turning green), so green gains 5-3=2 and yellow loses $2$: now green =3x+2 and yellow =x-2. The new ratio gives , so 3x+2=4(x-2)=4x-8 and x=10. The current difference is (3x+2)-(x-2)=2x+4=24.
Alternative — two unknowns. Let the initial greens be and yellows ; from , g=3y. After the move, greens =g+2 and yellows =y-2, and gives g+2=4y-8. Substituting g=3y: 3y+2=4y-8, so y=10 and g=30. The current difference is (g+2)-(y-2)=32-8=24.
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