Difference between revisions of "2019 AMC 8 Problems/Problem 6"

(Created page with "Problem 7 Shauna takes 5 tests, each worth a maximum of a 100 points. Her scores on the first three tests were 76, 94, and 87. In order to average an 81 on all five tests, wha...")
 
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Problem 7
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== Problem 6 ==
Shauna takes 5 tests, each worth a maximum of a 100 points. Her scores on the first three tests
 
were 76, 94, and 87. In order to average an 81 on all five tests, what is the lowest score she could
 
earn on one of the two tests?
 
A(48)  B(52)    C(66)    D(70)    E(74)
 
  
SOLUTION
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There are <math>81</math> grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point <math>P</math> is in the center of the square. Given that point <math>Q</math> is randomly chosen among the other 80 points, what is the probability that the line <math>PQ</math> is a line of symmetry for the square?
So far, she has scored 76+94+87=257 points on her tests. She needs to have 81*5=405 points in total to achieve an average of 81 on her 5 tests. To find the lowest score, one of the remaining scores must be the highest it can be (100). She needs to score 405-275=148. For the minimum score , 148-100=48. So its A(48).~heeeeeeheeeeeee
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<asy>
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draw((0,0)--(0,8));
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draw((0,8)--(8,8));
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draw((8,8)--(8,0));
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draw((8,0)--(0,0));
 +
dot((0,0));
 +
dot((0,1));
 +
dot((0,2));
 +
dot((0,3));
 +
dot((0,4));
 +
dot((0,5));
 +
dot((0,6));
 +
dot((0,7));
 +
dot((0,8));
 +
 
 +
dot((1,0));
 +
dot((1,1));
 +
dot((1,2));
 +
dot((1,3));
 +
dot((1,4));
 +
dot((1,5));
 +
dot((1,6));
 +
dot((1,7));
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dot((1,8));
 +
 
 +
dot((2,0));
 +
dot((2,1));
 +
dot((2,2));
 +
dot((2,3));
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dot((2,4));
 +
dot((2,5));
 +
dot((2,6));
 +
dot((2,7));
 +
dot((2,8));
 +
 
 +
dot((3,0));
 +
dot((3,1));
 +
dot((3,2));
 +
dot((3,3));
 +
dot((3,4));
 +
dot((3,5));
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dot((3,6));
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dot((3,7));
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dot((3,8));
 +
 
 +
dot((4,0));
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dot((4,1));
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dot((4,2));
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dot((4,3));
 +
dot((4,4));
 +
dot((4,5));
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dot((4,6));
 +
dot((4,7));
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dot((4,8));
 +
 
 +
dot((5,0));
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dot((5,1));
 +
dot((5,2));
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dot((5,3));
 +
dot((5,4));
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dot((5,5));
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dot((5,6));
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dot((5,7));
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dot((5,8));
 +
 
 +
dot((6,0));
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dot((6,1));
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dot((6,2));
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dot((6,3));
 +
dot((6,4));
 +
dot((6,5));
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dot((6,6));
 +
dot((6,7));
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dot((6,8));
 +
 
 +
dot((7,0));
 +
dot((7,1));
 +
dot((7,2));
 +
dot((7,3));
 +
dot((7,4));
 +
dot((7,5));
 +
dot((7,6));
 +
dot((7,7));
 +
dot((7,8));
 +
 
 +
dot((8,0));
 +
dot((8,1));
 +
dot((8,2));
 +
dot((8,3));
 +
dot((8,4));
 +
dot((8,5));
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dot((8,6));
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dot((8,7));
 +
dot((8,8));
 +
label("P",(4,4),NE);
 +
</asy>
 +
 
 +
<math>\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}</math>
 +
 
 +
==Solution 1==
 +
 
 +
 
 +
==See Also==
 +
{{AMC8 box|year=2019|num-b=14|num-a=16}}
 +
 
 +
{{MAA Notice}}

Revision as of 12:59, 20 November 2019

Problem 6

There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other 80 points, what is the probability that the line $PQ$ is a line of symmetry for the square?

[asy] draw((0,0)--(0,8)); draw((0,8)--(8,8)); draw((8,8)--(8,0)); draw((8,0)--(0,0)); dot((0,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((0,5)); dot((0,6)); dot((0,7)); dot((0,8));  dot((1,0)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((1,5)); dot((1,6)); dot((1,7)); dot((1,8));  dot((2,0)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((2,5)); dot((2,6)); dot((2,7)); dot((2,8));  dot((3,0)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((3,5)); dot((3,6)); dot((3,7)); dot((3,8));  dot((4,0)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); dot((4,5)); dot((4,6)); dot((4,7)); dot((4,8));  dot((5,0)); dot((5,1)); dot((5,2)); dot((5,3)); dot((5,4)); dot((5,5)); dot((5,6)); dot((5,7)); dot((5,8));  dot((6,0)); dot((6,1)); dot((6,2)); dot((6,3)); dot((6,4)); dot((6,5)); dot((6,6)); dot((6,7)); dot((6,8));  dot((7,0)); dot((7,1)); dot((7,2)); dot((7,3)); dot((7,4)); dot((7,5)); dot((7,6)); dot((7,7)); dot((7,8));  dot((8,0)); dot((8,1)); dot((8,2)); dot((8,3)); dot((8,4)); dot((8,5)); dot((8,6)); dot((8,7)); dot((8,8)); label("P",(4,4),NE); [/asy]

$\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}$

Solution 1

See Also

2019 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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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