Difference between revisions of "2019 AMC 12B Problems/Problem 7"

Revision as of 12:49, 14 February 2019

4, 6, 8, 17, x

What is the sum of all values of x such that the mean is equal to the median?

The mean is $\frac{4+6+8+17+x}{5}=\frac{35+x}{5}$ .

There are 3 possibilities: either the median is 6, 8, or x.

Let's start with 6.

$\frac{35+x}{5}=6$ when $x=-5$ and the sequence is -5, 4, 6, 8, 17 which has 6 as the median so we're good.

Now let the mean=8

$\frac{35+x}{5}=6$ when $x=5$ and the sequence is 4, 5, 6, 8, 17 which has median 6 so no go.

Finally we let the mean=x

$\frac{35+x}{5}=x \implies 35+x=5x \implies x=\frac{35}{4}=8.75.$ and the sequence is 4, 6, 8, 8.75, 17 which has median 8 so no go.

So the only option for x is $\boxed{-5}.$

--mguempel

2019 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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 AMC 12 Problems and Solutions

@@ Line 5: / Line 5: @@
 ==Solution==
-<math>\boxed{-5}</math> if my memory isn't trash
+The mean is <math>\frac{4+6+8+17+x}{5}=\frac{35+x}{5}</math>.
+There are 3 possibilities: either the median is 6, 8, or x.
+Let's start with 6.
+<math>\frac{35+x}{5}=6</math> when <math>x=-5</math> and the sequence is -5, 4, 6, 8, 17 which has 6 as the median so we're good.
+Now let the mean=8
+<math>\frac{35+x}{5}=6</math> when <math>x=5</math> and the sequence is 4, 5, 6, 8, 17 which has median 6 so no go.
+Finally we let the mean=x
+<math>\frac{35+x}{5}=x \implies 35+x=5x \implies x=\frac{35}{4}=8.75.</math> and the sequence is 4, 6, 8, 8.75, 17 which has median 8 so no go.
+So the only option for x is <math>\boxed{-5}.</math>
+--mguempel
 ==See Also==
 {{AMC12 box|year=2019|ab=B|num-b=6|num-a=8}}
 {{MAA Notice}}