Question

一群青蛙住在树上。青蛙在阴影中会变成绿色，在阳光下会变成黄色。最初绿色青蛙与黄色青蛙的比为 $3 : 1$ 。后来 $3$ 只绿色青蛙移到阳光处， $5$ 只黄色青蛙移到阴影处。现在比为 $4 : 1$ 。现在绿色青蛙和黄色青蛙的数量相差多少？

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?

$10$

$12$

$16$

$20$

$24$

答案：E

知识点：比与比例一次方程换元法

难度评级：1340

视频讲解：

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文字解答：

设最初绿色青蛙有 $g$ 只，黄色青蛙有 $y$ 只，则 $g=3y$ 。移动后有 $\dfrac{g+5-3}{y+3-5} = \dfrac{4}{1}.$

把 $3y$ 代入 $g$ ，即可求出原来的黄色青蛙数 $y$ ： $\dfrac{3y+2}{y-2} = 4$ $3y+2 = 4(y-2) = 4y-8$ $y = 10$

因此最初黄色有 $10$ 只，绿色有 $3 \times 10 = 30$ 只。移动后黄色有 $10+3-5 = 8$ 只，绿色有 $30+5-3=32$ 只，差为 $32-8=24$ 。

所以正确答案是 E。

We can let $g$ be the number of green frogs and $y$ be the number of yellow frogs. Initially, we have $g=3y$ . Then, after some frogs moved, we have the following proportion: $\dfrac{g+5-3}{y+3-5} = \dfrac{4}{1}.$

Substituting $3y$ for $g$ will allow us to determine the number of yellow frogs originally ( $y$ ): $\dfrac{3y+2}{y-2} = 4$ $3y+2 = 4(y-2) = 4y-8$ $y = 10$

Hence, there were $10$ yellow frogs and $3 \times 10 = 30$ green frogs initially. After some frogs moved, we now have $10+3-5 = 8$ yellow frogs and $30+5-3=32$ green frogs, giving us a difference of $32-8=24$ between the number of green and yellow frogs.

Thus, E is the correct answer.

2024 AMC 8 第 21 题

答案：E

难度评级：1340

视频讲解：

文字解答：

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