Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(→‎Solution)
(→‎Video Solution by OmegaLearn)
 
(27 intermediate revisions by 14 users not shown)
Line 1: Line 1:
 +
== Problem ==
 +
 
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?
 +
<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>
 +
 +
<math>\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7</math>
 +
 +
==Solutions==
 +
 +
=== Solution 1 ===
 +
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>.
 +
 +
=== 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 <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 (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
  
−
==Solution==
+
== See Also ==
−
If you have a solution, then please help us out by posting it.
 
  
 
{{AMC8 box|year=2016|num-b=5|num-a=7}}
 
{{AMC8 box|year=2016|num-b=5|num-a=7}}
 +
{{MAA Notice}}

Latest revision as of 14:11, 4 April 2023

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]

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

Solutions

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 (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

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_8_Problems/Problem_6&oldid=191780"