Difference between revisions of "2016 AMC 12A Problems/Problem 4"

Revision as of 22:37, 3 February 2016

Problem

The mean, median, and mode of the 7 data values $60,100,x,40,50,200,90$ are all equal to $x$ . What is the value of $x$ ?

$\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100$

Solution

$x$ must be in the middle of the list. Order the known list first: $\{40,50,60,90,100,200\}$ . $x$ occurs the most times, and it is the average of the list.

$\#1:$ $x$ is either $60$ or $90$ because it is the median and the mode.

$\#2:$ $x$ is the average of the set values. $x=\frac{40+50+60+90+100+200+x}{7}$ $x=\frac{540+x}{7}$ $7x=540+x$ $6x=540$ $x=\boxed{\textbf{(D)}\text{ 90}}$

@@ Line 1: / Line 1: @@
+==Problem==
+The mean, median, and mode of the 7 data values <math>60,100,x,40,50,200,90</math> are all equal to <math>x</math>.
+What is the value of <math>x</math>?
+<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math>
 ==Solution==
 <math>x</math> must be in the middle of the list. Order the known list first: <math>\{40,50,60,90,100,200\}</math>.

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Difference between revisions of "2016 AMC 12A Problems/Problem 4"

Revision as of 22:37, 3 February 2016

Problem

Solution