AMC8 2024

AMC8 2024 · Q21

AMC8 2024 · Q21. It mainly tests Linear equations, Ratios & proportions.

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（称为一个军团）住在一棵树上。青蛙在阴凉处变绿，在阳光下变黄。最初，绿蛙与黄蛙的比例为 $3 : 1$。然后 $3$ 只绿蛙移动到阳光侧，$5$ 只黄蛙移动到阴凉侧。现在比例为 $4 : 1$。现在绿蛙与黄蛙的数量差是多少？

(A) 10 10

(B) 12 12

(D) 20 20

(E) 24 24

Answer

Correct choice: (E)

正确答案：（E）

Solution

Let the initial number of green frogs be $g$ and the initial number of yellow frogs be $y$. Since the ratio of the number of green frogs to yellow frogs is initially $3 : 1$, $g = 3y$. Now, $3$ green frogs move to the sunny side and $5$ yellow frogs move to the shade side, thus the new number of green frogs is $g + 2$ and the new number of yellow frogs is $y - 2$. We are given that $\frac{g + 2}{y - 2} = \frac{4}{1}$, so $g + 2 = 4y - 8$, since $g = 3y$, we have $3y + 2 = 4y - 8$, so $y = 10$ and $g = 30$. Thus the answer is $(g + 2) - (y - 2) = 32 - 8 = \boxed{(E) \hspace{1 mm} 24}.$

设最初绿蛙数量为 $g$，黄蛙数量为 $y$。由于绿蛙与黄蛙比例最初为 $3 : 1$，故 $g = 3y$。现在，$3$ 只绿蛙移到阳光侧，$5$ 只黄蛙移到阴凉侧，因此新绿蛙数量为 $g + 2$（因为 $3$ 只绿蛙离开阴凉但 $5$ 只黄蛙进入阴凉），新黄蛙数量为 $y - 2$。已知 $\frac{g + 2}{y - 2} = \frac{4}{1}$，所以 $g + 2 = 4y - 8$，代入 $g = 3y$ 得 $3y + 2 = 4y - 8$，解得 $y = 10$，$g = 30$。因此答案为 $(g + 2) - (y - 2) = 32 - 8 = \boxed{(E) \hspace{1 mm} 24}$。

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