I am thinking of four counting numbers that are 100 or less....

Question

I am thinking of four counting numbers that are 100 or less. All of the numbers are different. The mean of my numbers is 82. What is the least number possible for one of my numbers?

Answer 1

Definitions:
  counting numbers = natural numbers (1, 2, 3, ...), although 0 is sometimes included
  mean = arithmetic average

 

mean = 82 = (a+b+c+d)/4

 

     If a, b, c, and d are all different and are each 100 or less, what is the smallest value possible for a?

 

     a+b+c+d = 328        [multiply both sides by 4]

     a = 328 - (b+c+d)       [subtract (b+c+d) from both sides]

  The least value of a is when (b+c+d) is the maximum allowed.  That is (100, 99, and 98).

     a = 328 - (100+99+98)

     a =  31

 

The four numbers could be 31, 100, 99, and 98.

Answer 2

If the mean of four numbers is 82, their sum must be 82 x 4 = 328.

 

To find the smallest possible number, we must assume the other three numbers are large. 97, 98 and 99 are the three largest possible numbers. 

 

328 - 99 - 98 - 97 = 34.  

 

34 is the smallest number possible.