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| − | ==Problem==
| + | #REDIRECT[[2019_AMC_10B_Problems/Problem_13]] |
| − | 4, 6, 8, 17, x
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| − | What is the sum of all values of x such that the mean is equal to the median?
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| − | ==Solution==
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| − | The mean is <math>\frac{4+6+8+17+x}{5}=\frac{35+x}{5}</math>.
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| − | There are 3 possibilities: either the median is 6, 8, or x.
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| − | Let's start with 6.
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| − | <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.
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| − | Now let the mean=8
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| − | <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.
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| − | Finally we let the mean=x
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| − | <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.
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| − | So the only option for x is <math>\boxed{-5}.</math>
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| − | --mguempel
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| − | ==See Also==
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| − | {{AMC12 box|year=2019|ab=B|num-b=6|num-a=8}}
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| − | {{MAA Notice}}
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