# CAT Practice : Averages, Ratios, Mixtures

Finding the maximum and minimum possible values of the median given the average of a set of numbers.

## Mean, Median, Maximum

Q.17: The average of 5 distinct positive integers if 33. What are the maximum and minimum possible values of the median of the 5 numbers if the average of the three largest numbers within this set is 39?

23, 25, 38, 39, 40

## Detailed Solution

Let the numbers be a, b, c, d, e in ascending order. a + b + c + d + e = 165. Average of the three largest numbers is 39, so c + d + e = 117, or a + b = 48.

We need to find the maximum and minimum possible values of c.

For minimum value, a and b have to be minimum. a + b = 48. Let us assume a = 23, b = 25, ca can be as low as 26.
23, 25, 26, 45, 46 is a possible sequence that satisfies the conditions.

For maximum value, we need d + e to be minimum as c + d + e = 117. we can have c = 38, d = 39 and e = 40. Or, the maximum value c can take = 38.

23, 25, 38, 39, 40 is a possible sequence that satisfies the conditions specified.

Correct Answer: 23, 25, 38, 39, 40

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