2016 AMC 10A Problems/Problem 7: Difference between revisions

Latest revision as of 17:19, 27 September 2023

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 1

Since $x$ is the mean, $\begin{align*} x&=\frac{60+100+x+40+50+200+90}{7}\\ &=\frac{540+x}{7}. \end{align*}$

Therefore, $7x=540+x$ , so $x=\boxed{\textbf{(D) }90}.$

Note: if the mean of a set is in the set, it can be discarded and the mean of the remaining numbers will be the same. This means that if using the mean in your reasoning, you can just take the mean of the other 6 numbers, and it'll solve it marginally faster. -Integralarefun (talk) 18:19, 27 September 2023 (EDT)

Solution 2

Note that $x$ must be the median so it must equal either $60$ or $90$ . You can see that the mean is also $x$ , and by intuition $x$ should be the greater one. $x=\boxed{\textbf{(D) }90}.$ ~bjc

Check

Order the list: $\{40,50,60,90,100,200\}$ . $x$ must be either $60$ or $90$ because it is both the median and the mode of the set. Thus $90$ is correct.

Video Solution (CREATIVE THINKING)

https://youtu.be/qDqe8pztaJQ

~Education, the Study of Everything

Video Solution

https://youtu.be/XXX4_oBHuGk?t=163

~IceMatrix

https://youtu.be/joLWmbpvrCw

~savannahsolver

2016 AMC 10A (Problems • Answer Key • Resources)
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2016 AMC 12A (Problems • Answer Key • Resources)
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These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

@@ Line 14: / Line 14: @@
 Therefore, <math>7x=540+x</math>, so <math>x=\boxed{\textbf{(D) }90}.</math>
+Note: if the mean of a set is in the set, it can be discarded and the mean of the remaining numbers will be the same. This means that if using the mean in your reasoning, you can just take the mean of the other 6 numbers, and it'll solve it marginally faster. -[[User:Integralarefun|Integralarefun]] ([[User talk:Integralarefun|talk]]) 18:19, 27 September 2023 (EDT)
 ==Solution 2==

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