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Difference between revisions of "2024 AMC 8 Problems/Problem 9"

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==Problem==
 
==Problem==
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[quote=ChuMath]All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?
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All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?
  
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<math>\textbf{(A) } 24\qquad\textbf{(B) } 25\qquad\textbf{(C) } 26\qquad\textbf{(D) } 27\qquad\textbf{(E) } 28</math>[/quote]
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<math>\textbf{(A) } 24\qquad\textbf{(B) } 25\qquad\textbf{(C) } 26\qquad\textbf{(D) } 27\qquad\textbf{(E) } 28</math>
  
 
==Solution 1==
 
==Solution 1==
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Since she has half as many red marbles as green, we can call the number of red marbles <math>x</math>, and the number of green marbles <math>2x</math>.
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Since she has half as many green marbles as blue, we can call the number of blue marbles <math>4x</math>.
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Adding them up, we have <math>7x</math> marbles. The number of marbles therefore must be a multiple of <math>7</math>. The only possible answer is <math>\boxed{\textbf{(E) 28}}.</math>
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==Solution 2==
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Suppose Maria has <math>g</math> green marbles and let <math>t</math> be the total number of marbles. She then has <math>\frac{g}{2}</math> red marbles and <math>2g</math> blue marbles. Altogether, Maria has
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<cmath>g + \frac{g}{2} + 2g = \frac{7g}{2} = t</cmath>
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marbles, implying that <math>g = \dfrac{2t}{7},</math> so <math>t</math> must be a multiple of <math>7</math>. The only multiple of <math>7</math> is <math>\boxed{\textbf{(E) 28}}.</math>
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-Benedict T (countmath1)
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==Video Solution 1 (easy to digest) by Power Solve==
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https://youtu.be/16YYti_pDUg?si=FAyIwmaHxE6UdSwP&t=238
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==Video Solution by Math-X (First fully understand the problem!!!)==
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https://youtu.be/BaE00H2SHQM?si=ORnDevWxiB6tv1jO&t=1912
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~Math-X
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==Video Solution by NiuniuMaths (Easy to understand!)==
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https://www.youtube.com/watch?v=V-xN8Njd_Lc
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~NiuniuMaths
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==Video Solution 2 by SpreadTheMathLove==
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https://www.youtube.com/watch?v=L83DxusGkSY
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== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
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https://www.youtube.com/watch?v=zqQTfBWr9T0
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==Video Solution by Interstigation==
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https://youtu.be/ktzijuZtDas&t=890
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==See Also==
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{{AMC8 box|year=2024|num-b=8|num-a=10}}
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{{MAA Notice}}

Revision as of 02:30, 6 February 2024

Problem

All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?

$\textbf{(A) } 24\qquad\textbf{(B) } 25\qquad\textbf{(C) } 26\qquad\textbf{(D) } 27\qquad\textbf{(E) } 28$

Solution 1

Since she has half as many red marbles as green, we can call the number of red marbles $x$, and the number of green marbles $2x$. Since she has half as many green marbles as blue, we can call the number of blue marbles $4x$. Adding them up, we have $7x$ marbles. The number of marbles therefore must be a multiple of $7$. The only possible answer is $\boxed{\textbf{(E) 28}}.$

Solution 2

Suppose Maria has $g$ green marbles and let $t$ be the total number of marbles. She then has $\frac{g}{2}$ red marbles and $2g$ blue marbles. Altogether, Maria has \[g + \frac{g}{2} + 2g = \frac{7g}{2} = t\] marbles, implying that $g = \dfrac{2t}{7},$ so $t$ must be a multiple of $7$. The only multiple of $7$ is $\boxed{\textbf{(E) 28}}.$

-Benedict T (countmath1)

Video Solution 1 (easy to digest) by Power Solve

https://youtu.be/16YYti_pDUg?si=FAyIwmaHxE6UdSwP&t=238

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

https://youtu.be/BaE00H2SHQM?si=ORnDevWxiB6tv1jO&t=1912

~Math-X

Video Solution by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=V-xN8Njd_Lc

~NiuniuMaths

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=L83DxusGkSY

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=zqQTfBWr9T0

Video Solution by Interstigation

https://youtu.be/ktzijuZtDas&t=890

See Also

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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

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