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

## 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 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 $54$ students take a language (with or without math), we subtract that from the total number of students. Since $93-54= 39,$ our answer is $\boxed{D 39}.$ (It's not necessary to know that $70$ students take math.) ~hailstone

## Solution Explained

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

## Video Solution

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

~savannahsolver

## Video Solution (CREATIVE ANALYSIS!!!)

~Education, the Study of Everything

~Hayabusa1