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

Latest revision as of 09:30, 9 November 2024

1 Problem
2 Solution 1
3 Solution 2
4 Solution 3
5 Solution 4
6 Video Solution 1 by Math-X (First fully understand the problem!!!)
7 Video Solution 2
8 Video Solution 3
9 Video Solution 4
10 Video Solution 5
11 Video Solution by OmegaLearn
12 Video Solution (CREATIVE THINKING!!!)
13 Video Solution by The Power of Logic(1 to 25 Full Solution)
14 See also

Problem

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 by Math-X (First fully understand the problem!!!)

https://youtu.be/IgpayYB48C4?si=ESh8VTHfpwe7idKH&t=2030

~Math-X

Video Solution 2

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

Video Solution 3

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

Video Solution 4

https://youtu.be/CEUuZiHL8c8

~savannahsolver

Video Solution 5

https://youtu.be/3t6MQ2MK1fs

~ saxstreak

Video Solution by OmegaLearn

https://youtu.be/rQUwNC0gqdg?t=1399

~ pi_is_3.14

Video Solution (CREATIVE THINKING!!!)

https://youtu.be/s0O7YM2W3Bs

~Education, the Study of Everything

Video Solution by The Power of Logic(1 to 25 Full Solution)

https://youtu.be/Xm4ZGND9WoY

~Hayabusa1

@@ Line 1: / Line 1: @@
-==Problem 7==
+==Problem==
 Shauna takes five tests, each worth a maximum of <math>100</math> points. Her scores on the first three tests are <math>76</math> , <math>94</math> , and <math>87</math> . In order to average <math>81</math> for all five tests, what is the lowest score she could earn on one of the other two tests?
@@ Line 29: / Line 29: @@
 ~ saxstreak
-== Video Solution 1 ==
+==Video Solution 1 by Math-X (First fully understand the problem!!!)==
+https://youtu.be/IgpayYB48C4?si=ESh8VTHfpwe7idKH&t=2030
+~Math-X
+== Video Solution 2 ==
 The Learning Royal : https://youtu.be/8njQzoztDGc
-== Video Solution 2 ==
+== Video Solution 3 ==
 Solution detailing how to solve the problem: https://www.youtube.com/watch?v=mwHrUESo2_A&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=8
-==Video Solution 3==
+==Video Solution 4==
 https://youtu.be/CEUuZiHL8c8
 ~savannahsolver
-==Video Solution 4==
+==Video Solution 5==
 https://youtu.be/3t6MQ2MK1fs
 ~ saxstreak
+==Video Solution by OmegaLearn ==
+https://youtu.be/rQUwNC0gqdg?t=1399
+~ pi_is_3.14
+== Video Solution (CREATIVE THINKING!!!)==
+https://youtu.be/s0O7YM2W3Bs
+~Education, the Study of Everything
+==Video Solution by The Power of Logic(1 to 25 Full Solution)==
+https://youtu.be/Xm4ZGND9WoY
+~Hayabusa1
 ==See also==

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