Problem Solving

Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

m

10m/7

10m/7 - 9/7

5m/7 + 3/7

5m

OA:C

Please explain.I got it later,Distinct is the key word here

skprocks wrote:Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

m

10m/7

10m/7 - 9/7

5m/7 + 3/7

5m

OA:C

In Order to maximize the individual members of the Set why can't we chose all values from first to fourth value as m and subsequently choose the rest as 2m.Please explain.

@ skprocks

No, you can't take more than one value equal to m or 2 m, as the Set S contains seven distinct integers.

However, in order to maximize the average (arithmetic mean) of all values in set S, we must consider the integers being consecutive uptil m, with m being the fourth in counting order and the seventh integer being equal to or less than 2 m; why not take 7th equal to 2 m, and then downcount as 6th is 2 m - 1, and 5th is 2 m - 2.

Scene could look like: m - 3, m - 2, m - 1, m, 2 m - 2, 2 m - 1, 2 m

Hence, the highest possible average (arithmetic mean) of all values in set S

= 1/7 × (m - 3 + m - 2 + m - 1 + m + 2 m - 2 + 2 m - 1 + 2 m) = [spoiler]1/7 × (10 m - 9)

C[/spoiler]

I know my approach probably wouldnt be the best one.....but it worked! So here it is:

Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

S {a,b,c,d,e,f,g}

d = median
g = 2d

SO I just came up with a set of values that would fit into these rules.

S = {0,1,2,3,4,5,6}

Median = 3
Highest value = 3x2=6
All other values are lower than 6

To get the mean, you just add up the values --> 21/7=3

The equation in option (C) gives me the same solution even if i plug in the value for M from the values i gave set S.

So (C).

Cool question..

So say the question is

median=mean=20
smallest number = (largest/2)-5

what is the largest number in this 7 number set ?

skprocks wrote:Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

m

10m/7

10m/7 - 9/7

5m/7 + 3/7

5m

OA:C

Please explain.I got it later,Distinct is the key word here

supposed x1,x2,x3,x4,x5,x6,x7 are all 7 number we need to find, and all these numbers are different
x4= m and x1,x2, x3, x5,x6,x7<or =2m
to find the highest average value, suppose x7=2m and m<x6 and x5<2m, supposed em : 2m-1, 2m-2
and x1,x2,x3<m, supposed are m-3,m-2,m-1
added all this number together we get the answer C