[spoiler]IMO:B[/spoiler]

2)let the original set be {20,25,27,28} (sum=100)
then the set formed will be (21,27,30,32) (sum=110)

Invariably sum of the new set will be 110 (as we add 1+2+3+4 to any number present in the respective positions), Hence B is sufficient

I chose B too. But the OA is A.

Explanation:

Statement 1:
The mean of Set S = (Sum of elements of S)/n
The mean of new Set = (Sum of elements of S + 1 + 2+ 3 + 4 +.... n)/n
= (Sum of elements of S)/n + (1+2+3+4+....n)/n
= Mean of set S + (1+2+3+4+...n)/n
If we know n , this statement is sufficient. From Statement 1, we know n =10. Therefore, this statement is sufficient.

Statement 2.
Consider Set S1 {0,100} (n elements - sum adds up to 100); Mean = 50
Corresponding new set {1,102} (as the lowest number is increased by 1 and the second lowest is increased by 2) Mean = 51.5
Difference in mean = 1.5

Consider set S2 {100} (set has only one element - hence sum is 100) ; Mean = 100
Corresponding new set {101} Mean = 100
Difference in mean = 1
Hence B alone is insufficient.

A is the OA. This question is from one of the 800score tests. I am still confused how a set containing only one element can be considered in this case.

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It is never too late to do anything

I will try to explain it with less complexity:

From the question:
The amount of the increase will be 1 + 2 + 3 + .... + n
Up to whatever number n is

So, the increase is a sequence of consecutive numbers from 1 to n.
We can find the total amount of increase by using the formula for sum of consecutive numbers:
(first + last)/2 * number of terms
--> the first part gives us the average of the terms in the sequence

first = 1
last = n
number of terms = n

Increase from one set to the other: (n+1)*n/2

That's how much the total goes up by. To find out how much the average goes up by, we need to divide the total increase by the number of terms (n)

[(n+1)*n/2] / n = (n+1)/2

So, the increase depends on the number of terms and has nothing to do with the actual terms themselves.

That is why St 1 is sufficient and St 2 is not.

The error in your example for B is that you have assumed there are 4 terms. If the set was:
18, 19, 20, 21, 22 (sums to 100)
The new set would be
19, 21, 23, 25, 27 (sums to 115)

So the increase depends on the number of terms, not the terms.
In your example, n=4 so (n)(n+1)/2 = 10
In the example above, n=5 so (n)(n+1)/2 = 15