shamanth25 wrote:
List T consist 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 digits is odd is rounded down to the nearest integer. 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 and II only
C. I and III only
D. II and III only
E. I, II, and III
OG 2019 PS01648
Official Explanation is HEREI made a blunder in this one even after spending a lot of time. Although i reached to get a figure of 18, 8 and 2, I subtracted 8 from 18 and rest, anyone can guess. Anyway, on retrospection, I figured out certain things that I'm sharing here.
First, that I was overwhelmed by the question for what it was about. I unnecessarily made my life miserable by overthinking as I was thought of 30 huge numbers - about the integer part and decimal part. I went into gimmicky approach as I thought integer part would change, impacting massively the result of E - S. And finally that I need the rough values of E and S.
However, the question talks only about the positive decimals. There are 10 decimals with even digit at tenths place that are rounded up and 20 with odd digit at tenths place that are rounded down. As we are looking for E - S, we know that the integer part would not get affected at all. Okay, that's not right perfectly since 0.2 becomes 1.0, changing the integer part.
But on a closer look we can calculate the total approximate change and not bother about the integer part at all. Whatever integer we get as result of either rounding up or rounding down, we are taking care of that by calculating - it not being carry forwarded.
Here's how it is:To know what possible values E - S can have, we need to first get the range of its values for which we need to calculate the maximum and minimum value of E-S.
\((E - S)_{max} = E_{max} - S = Δ_{E_{max}}\) AND
\((E - S)_{min} = E_{min} - S = Δ_{E_{min}}\)
(Note: E-S can have negative values depending on each value)
\(Δ_{E_{max}} = E_{Emax} + E_{Omin}\) - Eqn. 1 AND
\(Δ_{E_{min}} = E_{Emin} + E_{Omax}\) - Eqn. 2
Now, let the 30 decimals are(extreme possibilities)
0.00001, 0.00001 ....... (10 0.00001 values) and 0.9999, 0.9999 .... (20 0.9999 values)
0.00001 round up to 1.0, total Max change becomes ~10*0.9999 = 10 AND 0.9999 becomes 0.0, total Max change becomes ~0.9999*20 = 20
Precisely, the rounding up becomes max. a little < 10 and rounding down becomes max. a little < -20. For ease let's take 10 and -20('-' for rounding down).
0.8999, 0.8999 ..... (10 0.8999 values) and 0.1000, 0.1000 .... (20 0.1000 values)
0.8999 round up to 1.0, total Min change becomes ~10*0.1 = 1 AND 0.1 becomes 0.0, total Min change becomes 0.1*20 = 2
Precisely, the rounding up becomes min. a little > 1 and rounding down becomes min. a little = 2. For ease let's take 1 and -2('-' for rounding down).
Thus,
\(Δ_{E_{max}} = 10 - 2 = 8\) AND
\(Δ_{E_{min}} = 1 - 20 = -19\)
Hence only I and II are possible i.e. 6 and -16.
Subtlety in Question(or may be answer):
On the otherhand, GMAC would have been notoriously insane had it any one option with any of the possibilities viz. 7, 8, -17, -18 or -19. Here's how:
Under time pressure, normally people could take T as 0.2(smallest tenths digit) OR 0.8(largest tenths digit) for 10 decimals whose tenths digits is even and 0.1(smallest tenths digit) OR 0.9(largest tenths digit) for 20 decimals whose tenths digits is odd.
Maximum change: Rounding up becomes max. change = 0.8*10 = 8 and Rounding down becomes max. change = 0.9*20 = 18(actually '-18' for rounding down)
Minimum change: Rounding up becomes min. change = 0.2*10 = 2 and Rounding down becomes min. change = 0.1*20 = 2(actually '-2' for rounding down)
Thus,
\(Δ_{E_{max}} = 8 - 2 = 6\) AND
\(Δ_{E_{min}} = 2 - 18 = -16\)
Hope you got my point.
Finally, we can make our life easier if we consider all the 30 decimals as 0.xxxx OR at best each of the 10 as 0.01/0.89 and each of the 20 as 0.1/0.99.
Answer B.
HTH.