List T consists of 30 positive 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
B) I and II
C) I and III
D) II and III only
E) I, II and III
Make the problem CONCRETE by plugging in easy values.
10 of the values must have an EVEN tenths digit.
20 of the values must have an ODD tenths digit.
To make the math easy, let's not consider decimals beyond the tenths place.
Let each of the 10 values with an even tenths digit be taken from the following list:
0.2, 0.4, 0.6, 0.8.
Let each of the 20 values with an ODD tenths digit be taken from the following list:
0.1, 0.3, 0.5, 0.7, 0.9.
E is the ESTIMATED sum: the sum yielded when each of the 10 values with an EVEN tenths digit is rounded UP to 1 (the next greatest integer), while each of the 20 values with an ODD tenths digit is rounded DOWN to 0 (the next smallest integer).
Thus, E = 10(1) + 20(0) = 10.
S is the ACTUAL sum: the sum of the ACTUAL decimals taken from the two lists above.
Since the smallest possible odd decimal is 0.1, S>0.
Since E=10 and S>0, it is not possible that E-S = 10, as indicated in statement III.
Eliminate C, D and E.
To determine whether it's possible that E-S = 6 -- the value indicated in statement II -- try to MAXIMIZE the value of E-S.
To MAXIMIZE the value of E-S, we must MINIMIZE the value of S.
To minimize the value of S -- the sum of the ACTUAL decimals -- the 30 actual decimals must be as small as possible.
The least possible value for each of the 10 even decimals is 0.2.
The least possible value for each of the 20 odd decimals is 0.1.
Thus, the least possible value of S = 10(0.2) + 20(0.1) = 4.
Since the least possible value of S = 4, the maximum possible value of E-S = 10-4 = 6.
Since statement II is possible, eliminate A.
The correct answer is
B.
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