Difference between revisions of "2024 AMC 8 Problems/Problem 21"

Latest revision as of 13:29, 21 April 2024

1 Problem
2 Solution 1
3 Solution 2
4 Video Solution
5 Video Solution by Power Solve (crystal clear)
6 Video Solution 1 by Math-X (First fully understand the problem!!!)
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 $3 : 1$ . 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 the number of yellow frogs now?

$\textbf{(A) } 10\qquad\textbf{(B) } 12\qquad\textbf{(C) } 16\qquad\textbf{(D) } 20\qquad\textbf{(E) } 24$

Solution 1

Let the initial number of green frogs be $g$ and the initial number of yellow frogs be $y$ . Since the ratio of the number of green frogs to yellow frogs is initially $3 : 1$ , $g = 3y$ . Now, $3$ green frogs move to the sunny side and $5$ yellow frogs move to the shade side, thus the new number of green frogs is $g + 2$ and the new number of yellow frogs is $y - 2$ . We are given that $\frac{g + 2}{y - 2} = \frac{4}{1}$ , so $g + 2 = 4y - 8$ , since $g = 3y$ , we have $3y + 2 = 4y - 8$ , so $y = 10$ and $g = 30$ . Thus the answer is $(g + 2) - (y - 2) = 32 - 8 = \textbf{(E) } 24$ .

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 $12$ , the only solutions left are $(B)$ and $(E)$ .

Let's start with $12$ frogs:

We must have $9$ frogs in the shade and $3$ frogs in the sun. After the change, there would be $11$ frogs in the shade and $1$ frog in the sun, which is not a $4:1$ ratio.

Therefore the answer is: $\boxed{(E) \hspace{1 mm} 24}$ .

-ILoveMath31415926535

Video Solution

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 1 by Math-X (First fully understand the problem!!!)

https://youtu.be/BaE00H2SHQM?si=yTyYiS2S75HoSfmI&t=6313

~Math-X

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

@@ Line 1: / Line 1: @@
 ==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==
+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 <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 ==
+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 1 by Math-X (First fully understand the problem!!!)==
+https://youtu.be/BaE00H2SHQM?si=yTyYiS2S75HoSfmI&t=6313
+~Math-X
+==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}}

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

Page

Toolbox

Search