# CAT Practice : Averages, Ratios, Mixtures

Given a set of numbers, mean of the overall set, and median of a subset, how does one find the possible range of the median of the overall set.

## Change in average

Q.19: Average of 6 distinct positive integers is 33. The median of the three largest numbers is 43. What is the difference between the highest and lowest possible median of the 6 numbers?

Highest and lowest possible median is 38

## Detailed Solution

Let a, b, c, d, e, f be the six numbers in ascending order.
e = 43; a + b + c + d + e + f = 198
Median of 6 integers = ${{{\rm{c + d}}} \over {\rm{2}}}$

c + d / 2
is maximum when c and d are maximum since e = 43, dmax = 42 and cmax = 41
c + d / 2
is minimum when c and d are minimum amin = 1, bmin = 2, cmin = 3, dmin = 4
Therefore, =
c + d / 2
min = 3.5
Max – min = 41.5 – 3.5 = 38

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