Problem Solving

Pros, could you please help?

List T consists of 30 +ve decimals, none of which is an integer, and the sum of the 30 decimals is S. The estimated sum of the 30 decimals, E, is defined as follows. Each decimal in T whose tenths digit is even is rounded up to the nearest integer and each decimal in T whose tenths digit is odd is rounded down to the nearest integer; E is the sum of the resulting integers. if 1/3 of the decimals in T have a tenths digit that is even, which of the following is a possible value of E-S?

I -16
II 6
III 10

A) I only
B) I & II only
C) I & III only
D) II & III only
E) I, II & III

Here's a short approach:

To MINIMIZE E - S, we'll make E as small as we can and S as large as we can. To do this, we want our even tenths digit to be big (8) and our odd tenths digit to be big (9).

E will be very small if our set consists of ten 1.8s and twenty 1.9s. In this case, E = 10*2 + 20*1 and S = 10*1.8 + 20*1.9 = -16.

To MAXIMIZE E - S, we'll make E as large as we can and S as small as we can. Now we want our even tenths digit to be small (0) and our odd tenths digit to be small (1).

A nice set here is ten 1.01s and twenty 1.1s. Now E = 10*2 + 20*1 and S = 10*1.01 + 20*1.1. Here E - S = 7.9.

We can use any decimals we like, so any number in this range is fine: -16 and 6 both check out. 10 is greater than our max E - S (which is 8), so we can't have that.