Difference between revisions of "2019 AMC 8 Problems/Problem 11"
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| − | ==Problem | + | ==Problem== |
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? | 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) } | + | <math>\textbf{(A) }16\qquad\textbf{(B) }53\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70</math> |
==Solution 1== | ==Solution 1== | ||
| Line 11: | Line 11: | ||
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 | + | But we want the number of students taking only a math class, |
| − | + | which is <math>70 - 31 = 39</math>. | |
<math>\boxed{\textbf{(D)}\ 39}</math> | <math>\boxed{\textbf{(D)}\ 39}</math> | ||
| Line 20: | Line 20: | ||
==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 we subtract the people who didn't take only one math class, so we get <math>70 - 31 = \boxed{\textbf{D} \, 39}</math> -fath2012 | + | 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>. |
| + | |||
| + | -fath2012 | ||
==Solution 3== | ==Solution 3== | ||
| Line 42: | Line 44: | ||
-mathnerdnair | -mathnerdnair | ||
| + | |||
| + | ==Solution 4== | ||
| + | We are looking for students in math only, which is the complement (exactly the rest of the students) compared to those taking a language class. Since <math>54</math> students take a language (with or without math), we subtract that from the total number of students. Since <math>93-54= 39,</math> our answer is <math>\boxed{(D) 39}.</math> (It's not necessary to know that <math>70</math> students take math.) | ||
| + | ~hailstone | ||
| + | |||
| + | ==Video Solution by Math-X (First fully understand the problem!!!)== | ||
| + | https://youtu.be/IgpayYB48C4?si=hwYx8EF3iCm50Ms8&t=3557 | ||
| + | |||
| + | ~Math-X | ||
==Solution Explained== | ==Solution Explained== | ||
https://youtu.be/gOZOCFNXMhE ~ The Learning Royal | https://youtu.be/gOZOCFNXMhE ~ The Learning Royal | ||
| − | ==Solution | + | ==Solution 5== |
Associated video - https://www.youtube.com/watch?v=onPaMTO3dSA | Associated video - https://www.youtube.com/watch?v=onPaMTO3dSA | ||
| Line 52: | Line 63: | ||
Solution detailing how to solve the problem: https://www.youtube.com/watch?v=Kanl4ni2y0o&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=12 | 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 | ||
| + | |||
| + | ==Video Solution (CREATIVE ANALYSIS!!!)== | ||
| + | https://youtu.be/2K2ZUw-l1ek | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | ==Video Solution by The Power of Logic(1 to 25 Full Solution)== | ||
| + | https://youtu.be/Xm4ZGND9WoY | ||
| + | |||
| + | ~Hayabusa1 | ||
==See also== | ==See also== | ||
Latest revision as of 09:31, 9 November 2024
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 4
- 6 Video Solution by Math-X (First fully understand the problem!!!)
- 7 Solution Explained
- 8 Solution 5
- 9 Video Solution
- 10 Video Solution
- 11 Video Solution (CREATIVE ANALYSIS!!!)
- 12 Video Solution by The Power of Logic(1 to 25 Full Solution)
- 13 See also
Problem
The eighth grade class at Lincoln Middle School has 
students. Each student takes a math class or a foreign language class or both. There are 
eighth graders taking a math class, and there are 
eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?
Solution 1
Let 
be the number of students taking both a math and a foreign language class.
By P-I-E, we get 
= .
Solving gives us 
.
But we want the number of students taking only a math class,
which is 
.
~phoenixfire
Solution 2
We have 
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 
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 
.
-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]](http://latex.artofproblemsolving.com/b/2/f/b2f07041effb65f84f21bac99d7f4f0c292c18b7.png)
We know that the sum of all three areas is
So, we have:
![\[93 = 70-x+x+54-x\]](http://latex.artofproblemsolving.com/9/5/7/957e6020bc84fcd4dc822b45f5c0c0e9bbc10fb8.png)
![\[93 = 70+54-x\]](http://latex.artofproblemsolving.com/2/8/9/2892d379e250aab5337706f0020b1b6dc440370a.png)
![\[93 = 124 - x\]](http://latex.artofproblemsolving.com/5/e/5/5e5f58eb7dacfae6741cac7101024975dc4727f6.png)
![\[-31=-x\]](http://latex.artofproblemsolving.com/1/8/7/187c88af85602ec8c5e334c05cc4379424a4711f.png)
![\[x=31\]](http://latex.artofproblemsolving.com/7/c/0/7c0ebdc4866f572d386222d480abe1b80674c659.png)
We are looking for the number of students in only math. This is 
. Substituting with , our answer is 
.
-mathnerdnair
Solution 4
We are looking for students in math only, which is the complement (exactly the rest of the students) compared to those taking a language class. Since students take a language (with or without math), we subtract that from the total number of students. Since 
our answer is 
(It's not necessary to know that students take math.)
~hailstone
Video Solution by Math-X (First fully understand the problem!!!)
https://youtu.be/IgpayYB48C4?si=hwYx8EF3iCm50Ms8&t=3557
~Math-X
Solution Explained
https://youtu.be/gOZOCFNXMhE ~ The Learning Royal
Solution 5
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
~savannahsolver
Video Solution (CREATIVE ANALYSIS!!!)
~Education, the Study of Everything
Video Solution by The Power of Logic(1 to 25 Full Solution)
~Hayabusa1
See also
| 2019 AMC 8 (Problems • Answer Key • Resources) | ||
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
