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

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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?
 
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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<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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<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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==Solutions==
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=== Solution 1 ===
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We first notice that the median name will be the <math>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>.
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== See Also ==
  
==Solution==
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{{AMC8 box|year=2016|num-b=5|num-a=7}}
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{{MAA Notice}}

Latest revision as of 01:50, 15 January 2021

Problem

The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?

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

[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]

Solutions

Solution 1

We first notice that the median name will be the $10^{\mbox{th}}$ 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 $\boxed{\textbf{(B)}\ 4}$.

See Also

2016 AMC 8 (ProblemsAnswer KeyResources)
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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