Difference between revisions of "2019 AMC 8 Problems/Problem 7"

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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 $81-33 = \boxed{48}$ .

@@ Line 10: / Line 10: @@
 ==Solution 2==
-Right now, she scored <math>76, 94,</math> and <math>87</math> points, with a total of <math>257</math> points. She wants her average to be <math>81</math> for her <math>5</math> tests so she needs to score <math>405</math> points in total. She needs to score a total of <math>(405-257)
+Right now, she scored <math>76, 94,</math> and <math>87</math> points, for a total of <math>257</math> points. She wants her average to be <math>81</math> for her <math>5</math> tests, so she needs to score <math>405</math> points in total. This means she needs to score a total of <math>405-257=
-</math> points in her <math>2</math> tests. So the maximum score she can get is when one of her <math>2</math> scores is <math>100</math>. So the least possible score she can get is <math>\boxed{\textbf{(A)}\ 48}</math>. <math>3.1415926</math>
+</math> points in her next <math>2</math> tests. Since the maximum score she can get on one of her <math>2</math> tests is <math>100</math>, the least possible score she can get is <math>\boxed{\textbf{(A)}\ 48}</math>.
 Note: You can verify that <math>\boxed{48}</math> 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.

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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