# 2019 AMC 8 Problems/Problem 7

## Problem 7

Shauna takes five tests, each worth a maximum of $100$ points. Her scores on the first three tests are $76$ , $94$ , and $87$ . In order to average $81$ for all five tests, what is the lowest score she could earn on one of the other two tests? $\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74$

## Solution 1

We should notice that we can turn the information we are given into a linear equation and just solve for our set variables. I'll use the variables $x$ and $y$ for the scores on the last two tests. $$\frac{76+94+87+x+y}{5} = 81,$$ $$\frac{257+x+y}{5} = 81.$$ We can now cross multiply to get rid of the denominator. $$257+x+y = 405,$$ $$x+y = 148.$$ Now that we have this equation, we will assign $y$ as the lowest score of the two other tests, and so: $$x = 100,$$ $$y=48.$$ Now we know that the lowest score on the two other tests is $\boxed{48}$.

~ aopsav

## Solution 2

Right now, she scored $76, 94,$ and $87$ points, for a total of $257$ points. She wants her average to be $81$ for her $5$ tests, so she needs to score $405$ points in total. This means she needs to score a total of $405-257= 148$ points in her next $2$ tests. Since the maximum score she can get on one of her $2$ tests is $100$, the least possible score she can get is $\boxed{\textbf{(A)}\ 48}$.

Note: You can verify that $\boxed{48}$ is the right answer because it is the lowest answer out of the 5. Since it is possible to get 48, we are guaranteed that that is the right answer.

## Solution 3

We can compare each of the scores with the average of $81$: $76$ $\rightarrow$ $-5$, $94$ $\rightarrow$ $+13$, $87$ $\rightarrow$ $+6$, $100$ $\rightarrow$ $+19$;

So the last one has to be $-33$ (since all the differences have to sum to $0$), which corresponds to $81-33 = \boxed{48}$.

## Solution 4

We know that she scored $76, 94,$ and $87$ points on her first $3$ tests for a total of $257$ points and that she wants her average to be $81$ for her $5$ total tests. Therefore, she needs to score a total of $405$ points. In addition, one of the final $2$ tests needs to be the maximum of $100$ points, to make the final test score—the one that we are looking for—the lowest score possible for her to earn. We can now see here that the sum of $76+94+100$ has a units digit of $0$ and that the final test score must have a units digit ending with a $5$. Now, $87$ needs to be added to a number that makes the sum divisible by $5$. Among the answer choices of $48, 52, 66, 70,$ and $74$, only $\boxed{\textbf{(A)}\ 48}$ has a units digit that works. ( $48+87=135$, giving us a units digit of $5$.)

~ saxstreak

## Video Solution 1

The Learning Royal : https://youtu.be/8njQzoztDGc

## Video Solution 2

Solution detailing how to solve the problem: https://www.youtube.com/watch?v=mwHrUESo2_A&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=8

~savannahsolver

~ saxstreak

## Video Solution by OmegaLearn

~ pi_is_3.14

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 