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805+ (Hard)|   Min-Max Problems|                                 
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Ulki201
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List T consist of 30 positive decimals, none of which is an integer 07 Aug 2024, 09:39
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­It surprises me how everyone here forgets to consider 0 as an even integer. Values such as 1.09 or 1.000003 can also be among the 30 decimals and should be equally rounded up. The results will be the same but it's a minor difference to pay attention to.
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List T consist of 30 positive decimals, none of which is an integer 08 Jan 2025, 04:00
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Hi Kinshook, I wasa wondering how you got to the values of "At most increase/decrease in E = 10 , 20"?

Thank you.

Kinshook
Given: 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.

Asked: 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 ?

Let \(T = {t_1, t_2, .... , t_{30}}\)
\(t_1 + t_2 + ... + t_{30} = S\)

10 decimals in T have even tenth digit and 20 decimals in T have odd tenth digit

At most increase in E = 10
At most decrease in E = 20

Change in E may vary from -20 to 10 with extremes not possible since the numbers are not integers.

I. -16 : Possible
II. 6 : Possible
III. 10 : Not possible

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

IMO B
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thunderbolt27
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List T consist of 30 positive decimals, none of which is an integer 18 Jan 2025, 07:14
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* Assuming numbers from 1 to 10 are those with even digits at tenths place, and assuming numbers from 11 to 30 are those with odd digits at the tenths place, somehow get to the following equation:

* E - S = 10 - {[Sum of decimals of numbers from 1 to 10] + [Sum of decimals of numbers from 11 to 30]} = 10 - {a + b}

* Max value of (E - S) --> Min value of {a + b}

* Min value of (E - S) --> Max value of {a + b}

* Min value of {a + b} = 4

* Max value of {a + b} = 26
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List T consist of 30 positive decimals, none of which is an integer 18 Jan 2025, 08:01
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They approximated, increase of 10 is not possible - i assume they supposedly meant less than 10

To maximize the increase in E, we will assume even decimals to be 1.000001 and odd decimals to be 1.1
Each even decimal will add 0.99999 to it in rounding process, there are 10 such even decimals => 0.99999*10 = ~ 9.9999
Each odd decimal will subtract 0.1 from it in rounding process, there are 20 such odd decimals => 0.1*20 = 2 => -2 (as it is subtracted)
So maximum increase in E is ~ 9.9999 - 2 = ~ 7.99999

To maximize the decrease in E, we will assume even decimals to be 1.89999 and odd decimals to be 1.99999
Each even decimal will add 0.11111 to it in rounding process, there are 10 such even decimals => 0.11111*10 = ~ 1.1111
Each odd decimal will subtract 0.99999 from it in rounding process, there are 20 such odd decimals => 0.99999*20 = ~ 19.9998 => -19.9998 (as it is subtracted)
So maximum decrease in E is ~ 1.1111 - 19.9998 = ~ 18.8887

-19 < E - S < 8


BelisariusTirto
Hi Kinshook, I wasa wondering how you got to the values of "At most increase/decrease in E = 10 , 20"?

Thank you.

Kinshook
Given: 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.

Asked: 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 ?

Let \(T = {t_1, t_2, .... , t_{30}}\)
\(t_1 + t_2 + ... + t_{30} = S\)

10 decimals in T have even tenth digit and 20 decimals in T have odd tenth digit

At most increase in E = 10
At most decrease in E = 20

Change in E may vary from -20 to 10 with extremes not possible since the numbers are not integers.

I. -16 : Possible
II. 6 : Possible
III. 10 : Not possible

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

IMO B
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List T consist of 30 positive decimals, none of which is an integer 19 Jan 2025, 04:28
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This was very insightful, thank you Krunaal. Is this how you would approach this question if you had to tackle it in the test?

Thank you again.
Krunaal
They approximated, increase of 10 is not possible - i assume they supposedly meant less than 10

To maximize the increase in E, we will assume even decimals to be 1.000001 and odd decimals to be 1.1
Each even decimal will add 0.99999 to it in rounding process, there are 10 such even decimals => 0.99999*10 = ~ 9.9999
Each odd decimal will subtract 0.1 from it in rounding process, there are 20 such odd decimals => 0.1*20 = 2 => -2 (as it is subtracted)
So maximum increase in E is ~ 9.9999 - 2 = ~ 7.99999

To maximize the decrease in E, we will assume even decimals to be 1.89999 and odd decimals to be 1.99999
Each even decimal will add 0.11111 to it in rounding process, there are 10 such even decimals => 0.11111*10 = ~ 1.1111
Each odd decimal will subtract 0.99999 from it in rounding process, there are 20 such odd decimals => 0.99999*20 = ~ 19.9998 => -19.9998 (as it is subtracted)
So maximum decrease in E is ~ 1.1111 - 19.9998 = ~ 18.8887

-19 < E - S < 8


BelisariusTirto
Hi Kinshook, I wasa wondering how you got to the values of "At most increase/decrease in E = 10 , 20"?

Thank you.

Kinshook
Given: 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.

Asked: 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 ?

Let \(T = {t_1, t_2, .... , t_{30}}\)
\(t_1 + t_2 + ... + t_{30} = S\)

10 decimals in T have even tenth digit and 20 decimals in T have odd tenth digit

At most increase in E = 10
At most decrease in E = 20

Change in E may vary from -20 to 10 with extremes not possible since the numbers are not integers.

I. -16 : Possible
II. 6 : Possible
III. 10 : Not possible

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

IMO B
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List T consist of 30 positive decimals, none of which is an integer 19 Jan 2025, 05:36
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Yes, focusing on extremes is one way to approach these problems - Check these out Max-Min Strategies: Establishing Base Case, How to Deal with Maximizing/Minimizing Strategies on the GMAT and Min-Max: Focusing on Extremes

All DS Min/Max Problems to practice: https://gmatclub.com/forum/search.php?s ... &tag_id=42
All PS Min/Max Problems to practice: https://gmatclub.com/forum/search.php?s ... &tag_id=63
BelisariusTirto
This was very insightful, thank you Krunaal. Is this how you would approach this question if you had to tackle it in the test?

Thank you again.
Krunaal
They approximated, increase of 10 is not possible - i assume they supposedly meant less than 10

To maximize the increase in E, we will assume even decimals to be 1.000001 and odd decimals to be 1.1
Each even decimal will add 0.99999 to it in rounding process, there are 10 such even decimals => 0.99999*10 = ~ 9.9999
Each odd decimal will subtract 0.1 from it in rounding process, there are 20 such odd decimals => 0.1*20 = 2 => -2 (as it is subtracted)
So maximum increase in E is ~ 9.9999 - 2 = ~ 7.99999

To maximize the decrease in E, we will assume even decimals to be 1.89999 and odd decimals to be 1.99999
Each even decimal will add 0.11111 to it in rounding process, there are 10 such even decimals => 0.11111*10 = ~ 1.1111
Each odd decimal will subtract 0.99999 from it in rounding process, there are 20 such odd decimals => 0.99999*20 = ~ 19.9998 => -19.9998 (as it is subtracted)
So maximum decrease in E is ~ 1.1111 - 19.9998 = ~ 18.8887

-19 < E - S < 8
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List T consist of 30 positive decimals, none of which is an integer 12 Feb 2025, 13:20
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I looked at it very simply:

We have a list, one summation is accurate, the other is inaccurate, though, we know HOW it's inaccurate (even tenths? round up to nearest int.. odd tenths? round down to nearest in..)

So, to get a range, let's find the minimum and maximum of this list.

Ok, so, to make the sum of the list the highest:

- We must have the highest increase AND the lowest decrease.

From here, we can do the following:

To get the highest increase (since even tenths are rounded up), we'd need to start at the lowest even #, which is .2
If .2 is rounded up to the nearest integer, that's an increase of .8.

To get the lowest possible decrease (since odds are rounded down), we'd start at the lowest even #, which is .1
If .1 is rounded down, it's only a decrease of .1.

Since we know the even's make up 1/3rd of the list of 30, there are 10 evens and 20 odds.

Thus, (.8) * 10 will get you the increase from the evens, and (.1) * 20 will get you the decrease from the odds.

So, the highest possible increase is (.8) * 10 minus (.1) * 20.

This equals a difference of 6.

Great! 6 is an option.

Now, let's move on to make the sum of the list the lowest.

- To do this, we must have the lowest increase AND the highest decrease.

Thus, we do the following:

To get the highest decrease (reminder: odd tenths are rounded down), we'd need to start at the highest odd #, which is .9
If .9 is rounded down, it's a decrease of .9.

Since we know there are 20 odd #'s, this will be (.9) * 20

To get the lowest decrease (reminder: even tenths are rounded up), we'd need to start at the highest even #, which is .8
If .8 is rounded up, it's a minimal increase of .2.

Since we know there are 10 even #'s, this will be (.2) * 10

So, identical to before, let's combine these two:

(.2) * 10 will get you the increase from the evens, and (.9) * 20 will get you the decrease from the odds.

So, the lowest possible decrease is (.2) * 10 minus (.9) * 20.

This equals a difference of -16.

Thus, we just found the minimum and maximum. These two numbers, 6 <--> -16, form a range.

Easy enough: 6 & -16 are both options, but, we can rule out 10 since it doesn't fall within the min/max range.

Hope this helps!
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List T consist of 30 positive decimals, none of which is an integer 11 Apr 2025, 09:01
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chetan2u
I have not gone through the entire thread and would explain you the best way I can...
Since I am not aware what has been explained earlier, the way and solution I am giving may be NEW or already discussed..

Lets first take the INFO from the Q--



1) there are 30 decimals, NONE of them is an INTEGER..

2) decimals with even TENTHS is moved to upper integer..
which means any decimal .2,.4,.6,.8 moves to 1..

3) decimals with odd TENTHS is moved to lower integer..
which means any decimal .1,.3,.5,.7,.9 moves down to 0..

4) We are NOT concerned with what is to LEFT of decimal as that will get cancelled out in E-S

5) 10 are moving up (1/3 rd even decimals), whereas the other 20 move down

SOLUTION-


In such Qs, the best way is to look at the least and max values..
so lets take the smallest and biggest even and odd decimals..
EVEN--
smallest-0.2
EFFECT= 10 is moving up But in actual .2*10=2 is already moved up in S..
so effect on E-S= 10-2= +8
Largest-0.8
EFFECT= 10 is moving up But in actual .8*10=8 is already moved up in S..
so effect on E-S= 10-8= +2

ODD--
smallest-0.1
EFFECT= .1*20= 2moved down= -2
Largest-0.9
EFFECT= 0.9*20= 18 moved down = -18..

so now we can take combinations of effect of even and effect of moving down..
even= +2 and +8
odd= -2 and -18..
least (opposite effect)= -18+2=-16
biggest effect= 8-2=6..

so E-S will lie between -16 and 6, both inclusive..
so 10 is not possible
ans B
Hope it helps
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hey there how have you taken combinations ? whats the meaning of taking combinations ?
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List T consist of 30 positive decimals, none of which is an integer 17 May 2025, 00:31
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Hello Team kindly confirm whether I am right or not ?
Roundup(1.1)=2
Roundup(1.9)=2
Roundup(1.2)=2
Roundup(1.8)=2
Rounddown(1.1)=1
Rounddown(1.9)=1
Rounddown(1.2)=1
Rounddown(1.8)=1
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List T consist of 30 positive decimals, none of which is an integer 01 Aug 2025, 04:26
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Here's How I did it...

30 Numbers...10 with even decimals and 20 odd decimals...

S is actual and E is after the change
So 10 even decimals can bring a max change i.e increase the sum by 0.8 X 10 = 8 and minimum Increase by 0.2 x 10 = 2.
20 odd decimals can bring the max change i.e decrease the sum by 0.9 X 20 = 18 and minimum decrease by 0.1 X 20 = 2
Let's take 1 by 1 ....
Max Increase - Max decrease = 8-18 = -10
Now this is the change in S...So S - 10 = E...E - S = -10...We can't get 10....

- Max Increase - Min Decrease = 8-2= 6.....So increase of 6 in total..Means S + 6 = E..E-S = 6

- Min Decrease - Min Increase ..This is 0..

-Min Increase - Max Decrease = 2 - 18 = -16 So - 16 decrease in total..S - 16 = E ..E-S = - 16
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