Problem:
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 yellow frogs now?
Answer Choices:
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Solution:
We can let be the number of green frogs and be the number of yellow frogs. Initially, we have . Then, after some frogs moved, we have the following proportion:
Substituting for will allow us to determine the number of yellow frogs originally :
This simplifies to:
Rearranging gives .
Hence, there were yellow frogs and green frogs initially. After some frogs moved, we now have yellow frogs and green frogs, giving us a difference of between the number of green and yellow frogs.
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions