2024 AMC 8
Complete problem set with solutions and individual problem pages
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 . Then green frogs moved to the sunny side and yellow frogs moved to the shady side. Now the ratio is . What is the difference between the number of green frogs and the number of yellow frogs now?
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Solution 1
Let the initial number of green frogs be and the initial number of yellow frogs be . Since the ratio of the number of green frogs to yellow frogs is initially , . Now, green frogs move to the sunny side and yellow frogs move to the shade side, thus the new number of green frogs is and the new number of yellow frogs is . We are given that , so , since , we have , so and . Thus the answer is
Solution 2
Since the original ratio is and the new ratio is , the number of frogs must be a multiple of , the only solutions left are and .
Let's start with frogs:
We must have frogs in the shade and frogs in the sun. After the change, there would be frogs in the shade and frog in the sun, which is not a ratio.
Therefore the answer is: .
