# The data items in a list are 75,86,87,91, and 93. What is the largest integer you can add to the list so that the mean of six items is less than their median?

Oct 1, 2016

Largest integer is $101$

#### Explanation:

There are 5 numbers in the list, but a sixth one is to be added. (as large as possible)

$75 \text{ "86" "87" "91" "93" } x$
$\textcolor{w h i t e}{\times \times \times \times \times} \uparrow$
The median will be $\frac{87 + 91}{2} = 89$

Mean will be: $\frac{75 + 86 + 87 + 91 + 93 + x}{6} < 89$

$432 + x < 6 \times 89$

$x < 534 - 432$

$x < 102$

The largest integer can be 101.

Check; If $x = 101$

Mean $= \frac{533}{6} = 88.83$

$88.83 < 89$