Difference between revisions of "2019 AMC 8 Problems/Problem 11"

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==Problem 11==
 
==Problem 11==
The eighth grade class at Lincoln Middle School has <math>93</math> students. Each student takes a math class or a foreign language class or both. There are <math>70</math> eighth graders taking a math class, and there are <math>54</math> eight graders taking a foreign language class. How many eighth graders take ''only'' a math class and ''not'' a foreign language class?
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The eighth grade class at Lincoln Middle School has <math>93</math> students. Each student takes a math class or a foreign language class or both. There are <math>70</math> eighth graders taking a math class, and there are <math>54</math> eighth graders taking a foreign language class. How many eighth graders take ''only'' a math class and ''not'' a foreign language class?
  
 
<math>\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70</math>
 
<math>\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70</math>
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Solving gives us <math>x = 31</math>.
 
Solving gives us <math>x = 31</math>.
  
But we want the number of students taking only a math class.
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But we want the number of students taking only a math class,
  
Which is <math>70 - 31 = 39</math>.
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which is <math>70 - 31 = 39</math>.
  
 
<math>\boxed{\textbf{(D)}\ 39}</math>
 
<math>\boxed{\textbf{(D)}\ 39}</math>
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==Solution 2==
 
==Solution 2==
We have <math>70 + 54 = 124</math> people taking classes. However we over-counted the number of people who take both classes. If we subtract the original amount of people who take classes we get that <math>31</math> people took the two classes. To find the amount of people who took only math class web subtract the people who didn't take only one math class, so we get <math>70 - 31 = \boxed{\textbf{C} \, 39}</math> -fath2012
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We have <math>70 + 54 = 124</math> people taking classes. However, we over-counted the number of people who take both classes. If we subtract the original amount of people who take classes we get that <math>31</math> people took the two classes. To find the amount of people who took only math class, we subtract the people who didn't take only one math class, so we get <math>70 - 31 = \boxed{\textbf{D} \, 39}</math>.
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 +
-fath2012
  
 
==Solution 3==
 
==Solution 3==
We're basically just finding the number of students not taking a foreign language since all the rest would be taking only math. $93-54=
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<asy>
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draw(circle((-0.5,0),1));
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draw(circle((0.5,0),1));
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label("$\huge{x}$", (0, 0));
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label("$70-x$", (-1, 0));
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label("$54-x$", (1, 0));
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</asy>
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We know that the sum of all three areas is <math>93</math>
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So, we have:
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<cmath>93 = 70-x+x+54-x</cmath>
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<cmath>93 = 70+54-x</cmath>
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<cmath>93 = 124 - x</cmath>
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<cmath>-31=-x</cmath>
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<cmath>x=31</cmath>
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We are looking for the number of students in only math. This is <math>70-x</math>. Substituting <math>x</math> with <math>31</math>, our answer is <math>\boxed{39}</math>.
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 +
-mathnerdnair
  
==See Also==
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==Solution Explained==
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https://youtu.be/gOZOCFNXMhE ~ The Learning Royal
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 +
==Solution 4==
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Associated video - https://www.youtube.com/watch?v=onPaMTO3dSA
 +
 
 +
== Video Solution ==
 +
 
 +
Solution detailing how to solve the problem: https://www.youtube.com/watch?v=Kanl4ni2y0o&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=12
 +
 
 +
==Video Solution==
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https://youtu.be/4E8I-NJYAJY
 +
 
 +
~savannahsolver
 +
 
 +
==See also==
 
{{AMC8 box|year=2019|num-b=10|num-a=12}}
 
{{AMC8 box|year=2019|num-b=10|num-a=12}}
  
 
{{MAA Notice}}
 
{{MAA Notice}}
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 +
[[Category: Introductory Combinatorics Problems]]

Revision as of 02:04, 4 December 2022

Problem 11

The eighth grade class at Lincoln Middle School has $93$ students. Each student takes a math class or a foreign language class or both. There are $70$ eighth graders taking a math class, and there are $54$ eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?

$\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70$

Solution 1

Let $x$ be the number of students taking both a math and a foreign language class.

By P-I-E, we get $70 + 54 - x$ = $93$.

Solving gives us $x = 31$.

But we want the number of students taking only a math class,

which is $70 - 31 = 39$.

$\boxed{\textbf{(D)}\ 39}$

~phoenixfire

Solution 2

We have $70 + 54 = 124$ people taking classes. However, we over-counted the number of people who take both classes. If we subtract the original amount of people who take classes we get that $31$ people took the two classes. To find the amount of people who took only math class, we subtract the people who didn't take only one math class, so we get $70 - 31 = \boxed{\textbf{D} \, 39}$.

-fath2012

Solution 3

[asy] draw(circle((-0.5,0),1)); draw(circle((0.5,0),1)); label("$\huge{x}$", (0, 0)); label("$70-x$", (-1, 0)); label("$54-x$", (1, 0)); [/asy]

We know that the sum of all three areas is $93$ So, we have: \[93 = 70-x+x+54-x\] \[93 = 70+54-x\] \[93 = 124 - x\] \[-31=-x\] \[x=31\]

We are looking for the number of students in only math. This is $70-x$. Substituting $x$ with $31$, our answer is $\boxed{39}$.

-mathnerdnair

Solution Explained

https://youtu.be/gOZOCFNXMhE ~ The Learning Royal

Solution 4

Associated video - https://www.youtube.com/watch?v=onPaMTO3dSA

Video Solution

Solution detailing how to solve the problem: https://www.youtube.com/watch?v=Kanl4ni2y0o&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=12

Video Solution

https://youtu.be/4E8I-NJYAJY

~savannahsolver

See also

2019 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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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