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Difference between revisions of "2025 AMC 8 Problems/Problem 7"

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==Problem==
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== Problem ==
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On the most recent exam on Prof. Xochi's class,  
 
On the most recent exam on Prof. Xochi's class,  
  
  
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<math>5</math> students earned a score of at least <math>95</math>%,  
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<math>5</math> students earned a score of at least <math>95\%</math>,  
  
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<math>13</math> students earned a score of at least <math>90</math>%,  
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<math>13</math> students earned a score of at least <math>90\%</math>,  
  
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<math>27</math> students earned a score of at least <math>85</math>%,  
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<math>27</math> students earned a score of at least <math>85\%</math>,  
  
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<math>50</math> students earned a score of at least <math>80</math>%,  
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<math>50</math> students earned a score of at least <math>80\%</math>,  
  
  
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How many students earned a score of at least 80% and less than 90%?
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How many students earned a score of at least <math>80\%</math> and less than <math>90\%</math>?
  
 
<math>\textbf{(A)}\ 8\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 37\qquad \textbf{(E)}\ 45 </math>
 
<math>\textbf{(A)}\ 8\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 37\qquad \textbf{(E)}\ 45 </math>
  
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==Solution==
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== Solution 1 ==
  
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<math>50</math> people scored at least <math>80\%</math>, and out of these <math>50</math> people, <math>13</math> of them earned at least <math>90\%</math>, so the people that scored in between <math>80\%</math> and <math>90\%</math> is <math>50-13 = \boxed{\text{(D)\ 37}}</math>.
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<math>50</math> people scored at least <math>80\%</math>, and out of these <math>50</math> people, <math>13</math> of them earned a score that was not less than <math>90\%</math>, so the number of people that scored in between at least <math>80\%</math> and less than <math>90\%</math> is <math>50-13 = \boxed{\text{(D)\ 37}}</math>.
  
 
~Soupboy0
 
~Soupboy0
  
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==Solution 2==
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== Solution 2 ==
  
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Let <math>b</math> denote the number of people who had a score of at least <math>85</math>, but less than <math>90</math>. Similarly, let <math>d</math> be the number of people who had a score of at least <math>80</math> but less than <math>85</math>. Now we can see the question is just asking for <math>c+d</math>, we find <math>c = 27 - 13 = 14</math>, while <math>d = 50 - 27 = 23</math>. Thus, the answer is <math>23 + 14 = \boxed{\text{(D)\ 37}}</math>.  
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Let <math>a</math> denote the number of people who had a score of at least <math>85</math>, but less than <math>90</math>, and let <math>b</math> denote the number of people who had a score of at least <math>80</math> but less than <math>85</math>. Our answer is equal to <math>a+b</math>. We find <math>a = 27 - 13 = 14</math>, while <math>b = 50 - 27 = 23</math>. Thus, the answer is <math>23 + 14 = \boxed{\text{(D)\ 37}}</math>.  
  
 
-vockey
 
-vockey
 +
 +
== Video Solution 1 by Cool Math Problems ==
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 +
https://youtu.be/BRnILzqVwHk?si=kOvMjPSxqVZR8Mmt&t=89
 
   
 
   
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==Vide Solution 1 by SpreadTheMathLove==
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==Video Solution 2 by SpreadTheMathLove==
 
https://www.youtube.com/watch?v=jTTcscvcQmI
 
https://www.youtube.com/watch?v=jTTcscvcQmI
  
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==Video Solution (A Clever Explanation You’ll Get Instantly)==
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== Video Solution 3 ==
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https://youtu.be/VP7g-s8akMY?si=P0Nar6jhTGl1yKZb&t=427
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[//youtu.be/VP7g-s8akMY?si=P0Nar6jhTGl1yKZb&t=427 ~hsnacademy]
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~hsnacademy
 
  
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==Video Solution by Thinking Feet==
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== Video Solution 4 by Thinking Feet ==
 
https://youtu.be/PKMpTS6b988
 
https://youtu.be/PKMpTS6b988
  
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==Video Solution by Daily Dose of Math==
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== Video Solution 5 by Daily Dose of Math ==
  
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https://youtu.be/nkpdskFVgdM
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[//youtu.be/nkpdskFVgdM ~Thesmartgreekmathdude]
  
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~Thesmartgreekmathdude
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== See Also ==
  
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==See Also==
 
 
{{AMC8 box|year=2025|num-b=6|num-a=8}}
 
{{AMC8 box|year=2025|num-b=6|num-a=8}}
 
{{MAA Notice}}
 
{{MAA Notice}}
 +
[[Category:Introductory Number Theory Problems]]

Latest revision as of 13:05, 4 February 2025

Problem

On the most recent exam on Prof. Xochi's class,


$5$ students earned a score of at least $95\%$,

$13$ students earned a score of at least $90\%$,

$27$ students earned a score of at least $85\%$,

$50$ students earned a score of at least $80\%$,


How many students earned a score of at least $80\%$ and less than $90\%$?

$\textbf{(A)}\ 8\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 37\qquad \textbf{(E)}\ 45$

Solution 1

$50$ people scored at least $80\%$, and out of these $50$ people, $13$ of them earned a score that was not less than $90\%$, so the number of people that scored in between at least $80\%$ and less than $90\%$ is $50-13 = \boxed{\text{(D)\ 37}}$.

~Soupboy0

Solution 2

Let $a$ denote the number of people who had a score of at least $85$, but less than $90$, and let $b$ denote the number of people who had a score of at least $80$ but less than $85$. Our answer is equal to $a+b$. We find $a = 27 - 13 = 14$, while $b = 50 - 27 = 23$. Thus, the answer is $23 + 14 = \boxed{\text{(D)\ 37}}$.

-vockey

Video Solution 1 by Cool Math Problems

https://youtu.be/BRnILzqVwHk?si=kOvMjPSxqVZR8Mmt&t=89

Video Solution 2 by SpreadTheMathLove

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

Video Solution 3

~hsnacademy

Video Solution 4 by Thinking Feet

https://youtu.be/PKMpTS6b988

Video Solution 5 by Daily Dose of Math

~Thesmartgreekmathdude

See Also

2025 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
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. AMC logo.png

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