Good question.

I believe the answer is (D) - I & II only. Here's my logic.

Out of 20 numbers, 1/4th are even.
>Even = 5 numbers
>Odd = 15 numbers

Now, according to the property of set R, the numbers are rounded off to the nearest 10s' - if odd then scaled up, if even then scaled down. This essentially means, if set P contained {1,11,13,30,44......} then corresponding R would contain {10,20,20,30,40....}
P = { 1 ,11,13,30,44......}
R = {10,20,20,30,40....}


The below quesitons are required to be understood in order to crack this question.

Now, let's calculate the extreme values possible for R - P.

Question: What is the maximum possible scaling of an odd number?
Answer 1: 9 is the max possible scaling. Example: An odd number like 11 will be scaled to 20. 20-11=9. Any other number like 13,15,17 would be scaled to 20 but the scaling will be less.


Question: What is the maximum possible scaling of an even number?
Answer 2: -8. Example: An even number like 28 will scale down to 20. (20-28=-8 .). Any other number like 24, 26 will be scaled to 20 but not as much.

Question: What is the minimum possible scaling of an odd number?
Answer 3: 1. Example: 39 will be scaled to 40. 40-39=1. Any other number like 33, 37 will have a scaling more than 1.

Question: What is the minimum possible scaling of an even number?
Answer 4: 0. Example: 60 will be scaled to 60. 60-60=0. Any other number like 62, 66 will have a scaling more than 0.


Now with the above understanding, calculate the max/min range of R-P.
We have,
5 even numbers
Max possible scaling difference = 5 * (-8 ) = -40 ... -8 comes from answer 2
Min possible scaling difference = 5 * 0 = 0 ... 0 comes from answer 4

15 odd numbers
Max possible scaling difference = 15 * 9 = 135 ... 9 comes from answer 1
Min possible scaling difference = 15 * 1 = 15 ... 1 comes from answer 3

Thus,
R-P = Max limit when (MAX difference from ODD + Min difference from EVEN)
= 135 + 0
= 135

= Min limit when (MIN difference From ODD + MAX difference from EVEN)
= 15 + (-40)
= -25


So anything from -25 to 135 is possible. From the answer choices, (a) and (b) fit this criteria.



Please let us know if the answer is correct and if I've missed out anything. Hope it helps.

-g

i know its not the perfect solution but lest try

Assume we have same even number x 5 times and same odd number y 15 times in set P

x,y correspond to X,Y in Set R

so as per qestion stem
when x =2,4,....8, X = 0
when x =10, 12,14,16......18, X=10
Similarly
when y = 1,3,5,....9, Y=10
when y=11,13,15.....19, Y=20

Sum of P = 5x+15y
Sum of R = 5X + 15Y

Difference = 5(X-x) + 15(Y-y)

X-x can take values of -8, -6, -4, -2, 0
Y-y can take value of 1, 3, 5, 7, 9

Min Diff = -40 + 15 = -25
Max Diff = 0 + 135 = 135

So that makes 145 out. Answer D

It took more tan 5 minutes......

Edited to make changes in Max value

Thanks guys! hats off to you ..Answer is D. I am wondering if there is any faster way to do such questions.

Ian, I dont find the question statement contradicting as -

'rounded to the nearest tens' --> it just says that number are being rounded up or down to nearest tens.

And then later statements just detailed it further with - which Integers wll be rouded up and down to nearest tens.

Don't you agree?