Problema Solution
A set of plastic discs is thrown into a box. The discs have a positive integer written on one side and the second side is blank. The average of all the integers is 56. If the disc with 68 written on it is taken out of the box the average of the rest of the integers is 55. If it is possible to have the same integer written on more than one disc, what is the greatest integer that could be written on one of the discs? Please explain your work.
Answer provided by our tutors
let 'n' be the number of discs and x1, x2, x3, ....,xn the values written on the discs.
the average is 56 thus
(x1 + x2 + ...+xn)/n = 56 => x1 + x2 + ...+xn = 56n
without losing generality we can assume that xn = 68 is taken out of the box and the average of the rest x1,....,Xn-1 is 55
(x1 + x2 + ...+xn-1)/(n-1) = 55 => x1 + x2 + ...+xn-1 = 55(n-1)
since x1 + x2 + ...+xn-1 = x1 + x2 + ...+xn - xn = 56n - 68 thus
56n - 68 = 55(n-1)
n = 13
this means we have 13 discs and
(x1 + x2 + ....+ x12)/12 = 55
x1 + x2 + ....+ x12 = 12*55
x1 + x2 + ....+ x12 = 660
we need to find the greatest integer that could be written on one of the discs
since x1,...,x12 are positive integers we have xi > 0, i=1,..,12
without losing of generality we can assume that x1 is the greatest value then
x1 = 660 - (x2+...+x12) <= 660 - 11 = 649
thus x1 = 649 for x2=x3=...=x12=1 is the greatest value that could be written in one of the discs.