ScoreTop VIP - Median Problem

[wizardofwashington]

The average of five positive integers is 70 and the median value is 84. What is the greatest possible value of smallest integer?

Can someone help me understand how to solve this time sucker ..

[givemeanid]

Avg = 70
Sum = 70*5 = 350
To maximize the smallest integer, minimize the largest ones.

So, the sequence would look like a,b,84,84,84
So, a+b = 350-84*3 = 98
Max a will be when a = b. So, a = 49

[wizardofwashington]

Avg = 70
Sum = 70*5 = 350
To maximize the smallest integer, minimize the largest ones.

So, the sequence would look like a,b,84,84,84
So, a+b = 350-84*3 = 98
Max a will be when a = b. So, a = 49

wow.. That's one heck of a solution..Thanks for helping out..BTW, how is your application process going on..

[givemeanid]

Since I was slacking a week or two before my test, I am catching up at work right now. The apps are going slower than I thought... but I have plenty of time for Round 2 since I would not be able to put the best app together for Round 1.

Thanks for asking.

[camitava]

Givemeanid, I am having some doubt! Refer your approach below -

Avg = 70
Sum = 70*5 = 350
To maximize the smallest integer, minimize the largest ones.

So, the sequence would look like a,b,84,84,84
So, a+b = 350-84*3 = 98
Max a will be when a = b. So, a = 49

Now my question is why you took a and b as unknown? I mean that I can only take a as unknown and the equation becomes -
a = 350 - 84*4 = 14
OR
The sequence would look like a,b,C,84,84
a + b + c = 350 - 84 * 2 = 168
Max a will be when a = b = c. So, a = 84
By this I want to say that I really did not find the logic you implemented here. Can you please make me understand?

[Bharat]

Hi, let me try to explain.
Let the numbers be A, B, C, D & E. (in increasing numerical value order)
we know the average (mean) is 70 hence
A+B+C+D+E = 70*5 = 350
Median for an odd cardianlity set is the central element (middle value)
hence C = 84
& we have to identify the maximum value for A.
for this
1. B +D + E has to be minimum.
2. B is greater than or equal to A
3. D & E are greater than or equal to C (=84)
Hence for A to achieve maximum, B, D, E have to asssume minimum possible, i.e., B = A & D=E=C=84
Thus solve & get A = 49.

Let me know if there are still some doubts about this approach.

[camitava]

Ohhhhh! Now I understood. Thank u, Bharat. Its a really nice problem to solve then. Actually I have never faced this kind of prob earlier. A good learning for me.