2019 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 13 Medium

What is the sum of all real numbers x for which the median of the numbers 4, 6, 8, 17, and x is equal to the mean of those five numbers? (2019 AMC 10B Problem, Question#13)

  • A.

    -5

  • B.

    0

  • C.

    5

  • D.

    \frac{15}{4}

  • E.

    \frac{35}{4}

Answer:A

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

There are three possibilties for the median: it is either 6. 8, or x.

Let's start with 6.

\frac{35+x}{5}=6 has solution x=-5, and the sequence is -5, 4, 6, 8, 17, which does have median 6, so this is a valid solution.

Now let the median be 8.

\frac{35+x}{5}=8 give x=5, so the  sequence is 4, 5, 6, 8, 17, which has median 6, so this is not valid.

Finally we let the median be x.

\frac{35+x}{5}=x \Rightarrow 35+x=5x \Rightarrow x= \frac{35}{4}=8.75, and the  sequene is 4, 6, 8, 8.75, 17，

which has median 8. This case is therefore again not valid.

Hence the only possible vaue of x is \text {(A)}-5.

Related Problems
Problem 11 2019 AMC 10 B
Problem 12 2019 AMC 10 B
Problem 14 2019 AMC 10 B
Problem 15 2019 AMC 10 B
Think Academy Powered by ChatGPT