Difference between revisions of "2024 AMC 8 Problems/Problem 21"
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==Problem== | ==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 <math>3 : 1</math>. Then <math>3</math> green frogs moved to the | ||
+ | sunny side and <math>5</math> yellow frogs moved to the shady side. Now the ratio is <math>4 : 1</math>. What is the difference | ||
+ | between the number of green frogs and the number of yellow frogs now? | ||
+ | |||
+ | <math>\textbf{(A) } 10\qquad\textbf{(B) } 12\qquad\textbf{(C) } 16\qquad\textbf{(D) } 20\qquad\textbf{(E) } 24</math> | ||
+ | |||
+ | |||
==Solution 1== | ==Solution 1== | ||
+ | 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 = 24</math> | ||
+ | |||
+ | -anonchalantdreadhead | ||
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | Since the original ratio is <math>3:1</math> and the new ratio is <math>4:1</math>, the number of frogs must be a multiple of <math>12</math>, the only solutions left are <math>(B)</math> and <math>(E)</math>. | ||
+ | |||
+ | Let's start with <math>12</math> frogs: | ||
+ | |||
+ | We must have <math>9</math> frogs in the shade and <math>3</math> frogs in the sun. After the change, there would be <math>11</math> frogs in the shade and <math>1</math> frog in the sun, which is not a <math>4:1</math> ratio. | ||
+ | |||
+ | Therefore the answer is: <math>\boxed{(E) \hspace{1 mm} 24}</math>. | ||
+ | |||
+ | -ILoveMath31415926535 | ||
+ | |||
+ | == Video Solution == | ||
+ | ==Video Solution 1 by Math-X (First fully understand the problem!!!)== | ||
+ | https://youtu.be/BaE00H2SHQM?si=yTyYiS2S75HoSfmI&t=6313 | ||
+ | |||
+ | ~Math-X | ||
+ | |||
+ | https://youtu.be/H7d8c_YnvqE | ||
+ | |||
+ | Please like and subscribe! | ||
+ | |||
+ | ==Video Solution by Power Solve (crystal clear)== | ||
+ | https://www.youtube.com/watch?v=HodW9H55ZsE | ||
+ | |||
+ | ==Video Solution 2 by OmegaLearn.org== | ||
+ | https://youtu.be/Ah1WTdk8nuA | ||
+ | |||
+ | ==Video Solution 3 by SpreadTheMathLove== | ||
+ | https://www.youtube.com/watch?v=3ItvjukLqK0 | ||
+ | |||
+ | == Video Solution by NiuniuMaths (Easy to understand!) == | ||
+ | https://www.youtube.com/watch?v=looAMewBACY | ||
+ | |||
+ | ~NiuniuMaths | ||
+ | |||
+ | == Video Solution by CosineMethod [🔥Fast and Easy🔥]== | ||
+ | |||
+ | https://www.youtube.com/watch?v=3SUTUr1My7c&t=1s | ||
+ | ==Video Solution by Interstigation== | ||
+ | https://youtu.be/ktzijuZtDas&t=2562 | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2024|num-b=20|num-a=22}} | ||
+ | {{MAA Notice}} |
Latest revision as of 15:55, 17 June 2024
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Video Solution
- 5 Video Solution 1 by Math-X (First fully understand the problem!!!)
- 6 Video Solution by Power Solve (crystal clear)
- 7 Video Solution 2 by OmegaLearn.org
- 8 Video Solution 3 by SpreadTheMathLove
- 9 Video Solution by NiuniuMaths (Easy to understand!)
- 10 Video Solution by CosineMethod [🔥Fast and Easy🔥]
- 11 Video Solution by Interstigation
- 12 See Also
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
-anonchalantdreadhead
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: .
-ILoveMath31415926535
Video Solution
Video Solution 1 by Math-X (First fully understand the problem!!!)
https://youtu.be/BaE00H2SHQM?si=yTyYiS2S75HoSfmI&t=6313
~Math-X
Please like and subscribe!
Video Solution by Power Solve (crystal clear)
https://www.youtube.com/watch?v=HodW9H55ZsE
Video Solution 2 by OmegaLearn.org
Video Solution 3 by SpreadTheMathLove
https://www.youtube.com/watch?v=3ItvjukLqK0
Video Solution by NiuniuMaths (Easy to understand!)
https://www.youtube.com/watch?v=looAMewBACY
~NiuniuMaths
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://www.youtube.com/watch?v=3SUTUr1My7c&t=1s
Video Solution by Interstigation
https://youtu.be/ktzijuZtDas&t=2562
See Also
2024 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.