Question

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?

$10$

$12$

$16$

$20$

$24$

Answer: E

Concepts:ratio and proportion linear equation substitution

Difficulty rating: 1340

Video solution:

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Written solution:

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 Problem 21

Answer: E

Difficulty rating: 1340

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Problem 21 in Other Years