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Difference between revisions of "2022 AMC 8 Problems/Problem 12"

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(→‎Problem)
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==Problem==
 
==Problem==
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The arrows on the two spinners shown below are spun. Let the number <math>N</math> equal 10 times the
 
−
number on Spinner A, added to the number on Spinner B. What is the probability that N is a
 
−
perfect square number?
 
  
−
<math>\textbf{(A) } \dfrac{1}{16}\qquad\textbf{(B) } \dfrac{1}{8}\qquad\textbf{(C) } \dfrac{1}{4}\qquad\textbf{(D) } \dfrac{3}{8}\qquad\textbf{(E) } \dfrac{1}{2}</math>
+
The arrows on the two spinners shown below are spun. Let the number <math>N</math>  equal <math>10</math> times the number on Spinner <math>A</math>, added to the number on Spinner <math>B</math>. What is the probability that <math>N</math> is a perfect square number?
 +
 
 +
<math>\textbf{(A)} ~\dfrac{1}{16}\qquad\textbf{(B)} ~\dfrac{1}{8}\qquad\textbf{(C)} ~\dfrac{1}{4}\qquad\textbf{(D)} ~\dfrac{3}{8}\qquad\textbf{(E)} ~\dfrac{1}{2}</math>
 +
 
 +
<asy>
 +
//diagram by pog give me 1 billion dollars for this
 +
size(6cm);
 +
usepackage("mathptmx");
 +
filldraw(arc((0,0), r=4, angle1=0, angle2=90)--(0,0)--cycle,mediumgray*0.5+gray*0.5);
 +
filldraw(arc((0,0), r=4, angle1=90, angle2=180)--(0,0)--cycle,lightgray);
 +
filldraw(arc((0,0), r=4, angle1=180, angle2=270)--(0,0)--cycle,mediumgray);
 +
filldraw(arc((0,0), r=4, angle1=270, angle2=360)--(0,0)--cycle,lightgray*0.5+mediumgray*0.5);
 +
label("$5$", (-1.5,1.7));
 +
label("$6$", (1.5,1.7));
 +
label("$7$", (1.5,-1.7));
 +
label("$8$", (-1.5,-1.7));
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label("Spinner A", (0, -5.5));
 +
filldraw(arc((12,0), r=4, angle1=0, angle2=90)--(12,0)--cycle,mediumgray*0.5+gray*0.5);
 +
filldraw(arc((12,0), r=4, angle1=90, angle2=180)--(12,0)--cycle,lightgray);
 +
filldraw(arc((12,0), r=4, angle1=180, angle2=270)--(12,0)--cycle,mediumgray);
 +
filldraw(arc((12,0), r=4, angle1=270, angle2=360)--(12,0)--cycle,lightgray*0.5+mediumgray*0.5);
 +
label("$1$", (10.5,1.7));
 +
label("$2$", (13.5,1.7));
 +
label("$3$", (13.5,-1.7));
 +
label("$4$", (10.5,-1.7));
 +
label("Spinner B", (12, -5.5));
 +
</asy>
  
 
==Solution==
 
==Solution==

Revision as of 19:03, 28 January 2022

Problem

The arrows on the two spinners shown below are spun. Let the number $N$ equal $10$ times the number on Spinner $A$, added to the number on Spinner $B$. What is the probability that $N$ is a perfect square number?

$\textbf{(A)} ~\dfrac{1}{16}\qquad\textbf{(B)} ~\dfrac{1}{8}\qquad\textbf{(C)} ~\dfrac{1}{4}\qquad\textbf{(D)} ~\dfrac{3}{8}\qquad\textbf{(E)} ~\dfrac{1}{2}$

[asy] //diagram by pog give me 1 billion dollars for this size(6cm); usepackage("mathptmx"); filldraw(arc((0,0), r=4, angle1=0, angle2=90)--(0,0)--cycle,mediumgray*0.5+gray*0.5); filldraw(arc((0,0), r=4, angle1=90, angle2=180)--(0,0)--cycle,lightgray); filldraw(arc((0,0), r=4, angle1=180, angle2=270)--(0,0)--cycle,mediumgray); filldraw(arc((0,0), r=4, angle1=270, angle2=360)--(0,0)--cycle,lightgray*0.5+mediumgray*0.5); label("$5$", (-1.5,1.7)); label("$6$", (1.5,1.7)); label("$7$", (1.5,-1.7)); label("$8$", (-1.5,-1.7)); label("Spinner A", (0, -5.5)); filldraw(arc((12,0), r=4, angle1=0, angle2=90)--(12,0)--cycle,mediumgray*0.5+gray*0.5); filldraw(arc((12,0), r=4, angle1=90, angle2=180)--(12,0)--cycle,lightgray); filldraw(arc((12,0), r=4, angle1=180, angle2=270)--(12,0)--cycle,mediumgray); filldraw(arc((12,0), r=4, angle1=270, angle2=360)--(12,0)--cycle,lightgray*0.5+mediumgray*0.5); label("$1$", (10.5,1.7)); label("$2$", (13.5,1.7)); label("$3$", (13.5,-1.7)); label("$4$", (10.5,-1.7)); label("Spinner B", (12, -5.5)); [/asy]

Solution

First, we realize that there are a total of $16$ possibilities. Now, we list all of them that can be spun. This includes $64$ and $81$. Then, our answer is $\frac{2}{16}=\boxed{\textbf{(B) }\dfrac{1}{8}}$.

~MathFun1000

See Also

2022 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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

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

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