Difference between revisions of "2024 AMC 8 Problems/Problem 21"
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Let the initial number of green frogs be <math>g</math> and the initial number of yellow frogs be <math>y</math>. Since the ratio of the number of green frogs to yellow frogs is initially <math>3 : 1</math>, <math>g = 3y</math>. Now, <math>3</math> green frogs move to the sunny side and <math>5</math> yellow frogs move to the shade side, thus the new number of green frogs is <math>g + 2</math> and the new number of yellow frogs is <math>y - 2</math>. We are given that <math>\frac{g + 2}{y - 2} = \frac{4}{1}</math>, so <math>g + 2 = 4y - 8</math>, since <math>g = 3y</math>, we have <math>3y + 2 = 4y - 8</math>, so <math>y = 10</math> and <math>g = 30</math>. Thus the answer is <math>(g + 2) - (y - 2) = 32 - 8 = \textbf{(E) } 24</math>. | Let the initial number of green frogs be <math>g</math> and the initial number of yellow frogs be <math>y</math>. Since the ratio of the number of green frogs to yellow frogs is initially <math>3 : 1</math>, <math>g = 3y</math>. Now, <math>3</math> green frogs move to the sunny side and <math>5</math> yellow frogs move to the shade side, thus the new number of green frogs is <math>g + 2</math> and the new number of yellow frogs is <math>y - 2</math>. We are given that <math>\frac{g + 2}{y - 2} = \frac{4}{1}</math>, so <math>g + 2 = 4y - 8</math>, since <math>g = 3y</math>, we have <math>3y + 2 = 4y - 8</math>, so <math>y = 10</math> and <math>g = 30</math>. Thus the answer is <math>(g + 2) - (y - 2) = 32 - 8 = \textbf{(E) } 24</math>. | ||
+ | ==Solution 2== | ||
− | + | Since the original ratio is 3:1 and the new ratio is 4:1, the number of frogs must be a multiple of 12. Therefore, we can try all multiples of 12 and we get <math>(E) \boxed{24}</math>. | |
==Video Solution by Math-X (First fully understand the problem!!!)== | ==Video Solution by Math-X (First fully understand the problem!!!)== |
Revision as of 16:16, 25 January 2024
Contents
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 the number of yellow frogs now?
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 3:1 and the new ratio is 4:1, the number of frogs must be a multiple of 12. Therefore, we can try all multiples of 12 and we get .
Video Solution by Math-X (First fully understand the problem!!!)
https://www.youtube.com/watch?v=zBe5vrQbn2A
~Math-X