Question 236342: The average of 12 different counting numbers is 12. What is the greatest possible value of one of these numbers?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if it was an average of 1 number then the maximum or minimum value could be 1.
if it was an average of 2 numbers, then the maximum or minimum value could be 24 since 24 / 2 = 12
this assumes the numbers have to be positive or 0.
if the numbers could be negative, then the maximum number could be just about anything as long as one of the numbers was 24 more than the sum of both numbers.
example:
numbers are 1*10^24 + 24 and -1*10^24
add these numbers together and you get 24 because the 1*10^24 cancels the -1*10^24.
divide by 2 and the average is 12.
so i'll assume you mean that the numbers can't be negative.
using that assumption.
if 1 number, then the max is 12 because 12/1 = 1
if 2 numbers, then the max is 24 because 24 + 0 = 24/2 = 12
if 3 numbers, then the max is 36 - 1 = 35 because 35 + 0 + 1 = 36 / 3 = 12
if 4 numbers, then the max is 48 - 3 = 45 because 45 + 0 + 1 + 2 = 48/4 = 12
there's a pattern developing.
let n = number of numbers
max number = n*12 - sum(x) from x = 0 to x = n-2.
assuming n = 4, this becomes 4*12 = 48 - (0+1+2) = 48-3 = 45 which we know is good from above.
the 4 numbers are:
0
1
2
45
using this with n = 12, we get:
12*12 - sum(x) from x = 0 to x = 10.
this would be an arithmetic progression starting at 0 and ending at 10
the answer is 55 because it was easier to just add the numbers up.
if the numbers were higher, it might be easier to use the formula for the sum of an arithmetic progression.
that formula is:
sn = n(t1 + tn)/2
our first number was 0 and out last number was 10
we had 11 numbers.
formula becomes 11 * 10 / 2 = 110/2 = 55 (it works !!!!!)
with nj = 12, our highest number becomes:
144 - 55 = 89
89 should be the maximum number assuming none of the numbers can be negative.
let's see if this holds up;
we have:
0
1
2
3
4
5
6
7
8
9
10
89
sum is 144 / 12 = average of 12.
looks like the formula is good.
I would say your maximum number is 89 assuming all number have to be different and all numbers have to equal 0 or be greater than 0.
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