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Difference between revisions of "2016 AMC 8 Problems/Problem 6"

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Why is this empty?
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== Problem ==
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The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?
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<asy>
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unitsize(0.9cm);
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draw((-0.5,0)--(10,0), linewidth(1.5));
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draw((-0.5,1)--(10,1));
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draw((-0.5,2)--(10,2));
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draw((-0.5,3)--(10,3));
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draw((-0.5,4)--(10,4));
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draw((-0.5,5)--(10,5));
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draw((-0.5,6)--(10,6));
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draw((-0.5,7)--(10,7));
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label("frequency",(-0.5,8));
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label("0", (-1, 0));
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label("1", (-1, 1));
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label("2", (-1, 2));
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label("3", (-1, 3));
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label("4", (-1, 4));
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label("5", (-1, 5));
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label("6", (-1, 6));
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label("7", (-1, 7));
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filldraw((0,0)--(0,7)--(1,7)--(1,0)--cycle, black);
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filldraw((2,0)--(2,3)--(3,3)--(3,0)--cycle, black);
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filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, black);
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filldraw((6,0)--(6,4)--(7,4)--(7,0)--cycle, black);
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filldraw((8,0)--(8,4)--(9,4)--(9,0)--cycle, black);
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label("3", (0.5, -0.5));
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label("4", (2.5, -0.5));
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label("5", (4.5, -0.5));
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label("6", (6.5, -0.5));
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label("7", (8.5, -0.5));
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label("name length", (4.5, -1));
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</asy>
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<math>\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7</math>
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== Solution 1 ==
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We first notice that the median name will be the <math>(19+1)/2=10^{\mbox{th}}</math> name. The <math>10^{\mbox{th}}</math> name is <math>\boxed{\textbf{(B)}\ 4}</math>.
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== Solution 2 ==
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To find the median length of a name from a bar graph, we must add up the number of names. Doing so gives us <math>7 + 3 + 1 + 4 + 4 = 19</math>. Thus the index of the median length would be the 10th name. Since there are <math>7</math> names with length <math>3</math>, and <math>3</math> names with length <math>4</math>, the <math>10</math>th name would have <math>4</math> letters. Thus our answer is <math>\boxed{\textbf{(B)}\ 4}</math>.
 +
 
 +
== Video Solution ==
 +
 
 +
https://youtu.be/M9Hooi5UwDg?si=4CPixqDwQ_9BCh6m
 +
 
 +
A solution so simple a 12-year-old made it!
 +
 
 +
~Elijahman~
 +
 
 +
== Video Solution (CREATIVE THINKING!!!) ==
 +
https://youtu.be/Xab3qcUUDRY
 +
 
 +
~Education, the Study of Everything
 +
 
 +
== Video Solution by OmegaLearn ==
 +
https://youtu.be/TkZvMa30Juo?t=1830
 +
 
 +
~ pi_is_3.14
 +
 
 +
== Video Solution ==
 +
https://youtu.be/800KF_3XSmM
 +
 
 +
~savannahsolver
 +
 
 +
== See Also ==
 +
 
 +
{{AMC8 box|year=2016|num-b=5|num-a=7}}
 +
{{MAA Notice}}

Latest revision as of 19:01, 21 August 2024

Problem

The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names? [asy] unitsize(0.9cm); draw((-0.5,0)--(10,0), linewidth(1.5)); draw((-0.5,1)--(10,1)); draw((-0.5,2)--(10,2)); draw((-0.5,3)--(10,3)); draw((-0.5,4)--(10,4)); draw((-0.5,5)--(10,5)); draw((-0.5,6)--(10,6)); draw((-0.5,7)--(10,7)); label("frequency",(-0.5,8)); label("0", (-1, 0)); label("1", (-1, 1)); label("2", (-1, 2)); label("3", (-1, 3)); label("4", (-1, 4)); label("5", (-1, 5)); label("6", (-1, 6)); label("7", (-1, 7)); filldraw((0,0)--(0,7)--(1,7)--(1,0)--cycle, black); filldraw((2,0)--(2,3)--(3,3)--(3,0)--cycle, black); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, black); filldraw((6,0)--(6,4)--(7,4)--(7,0)--cycle, black); filldraw((8,0)--(8,4)--(9,4)--(9,0)--cycle, black); label("3", (0.5, -0.5)); label("4", (2.5, -0.5)); label("5", (4.5, -0.5)); label("6", (6.5, -0.5)); label("7", (8.5, -0.5)); label("name length", (4.5, -1)); [/asy]

$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7$

Solution 1

We first notice that the median name will be the $(19+1)/2=10^{\mbox{th}}$ name. The $10^{\mbox{th}}$ name is $\boxed{\textbf{(B)}\ 4}$.

Solution 2

To find the median length of a name from a bar graph, we must add up the number of names. Doing so gives us $7 + 3 + 1 + 4 + 4 = 19$. Thus the index of the median length would be the 10th name. Since there are $7$ names with length $3$, and $3$ names with length $4$, the $10$th name would have $4$ letters. Thus our answer is $\boxed{\textbf{(B)}\ 4}$.

Video Solution

https://youtu.be/M9Hooi5UwDg?si=4CPixqDwQ_9BCh6m

A solution so simple a 12-year-old made it!

~Elijahman~

Video Solution (CREATIVE THINKING!!!)

https://youtu.be/Xab3qcUUDRY

~Education, the Study of Everything

Video Solution by OmegaLearn

https://youtu.be/TkZvMa30Juo?t=1830

~ pi_is_3.14

Video Solution

https://youtu.be/800KF_3XSmM

~savannahsolver

See Also

2016 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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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