For statement # 1) imagine a set that includes only the numbers 10 and 15. The range would be 5. r = range so r = 5. Now you are supposed to add this to the value to the set. Set is {5, 10, 15} Even though all values are positive as statement 1 requires, you see that you do end up with an increased range. So this is a yes.
It is easy to get a No for statement 1 as well. Just have a set with number 5 and 14. The range is 10. So r = 10 and if you add 10 to the set you get {4, 10, 14} and the range does not increase. So this is a No.
Statement 1 is not sufficient.
Now look at Statement 2) alone. This is where we can try some negative numbers. If you have two numbers in your set {-30, 30} the range = 60. So add r to the set and you know have the terms {-30, 30, and 60}. The new mean is 20 which is smaller than r as required by statement 2. So this gives you a YES for statement 2.
Can we also get a NO? Recycle the numbers from statement 1. Specifically use the set with {4, 14}. Add r = 10 and get {4, 10, 14}. The mean of this set is just a little less than 10, it is 9.33. So that satisfies statement 2 and does not increase the range. So this is a NO.
Statement 2 is not sufficient alone.
What about together?
Well I have already tried one set of numbers that works with both statements. Specifically the set with {4, 14}. Add r = 10 and get {4, 10, 14}. This is the one that does not increase the range so this is a NO.
Can you get a yes? You cannot. Statement 2 does not allow the first set I used with statement 1 since r = 5 is less than the mean of the new set. Statement 1 does not allow negatives, which is what made statement 2 not sufficient.
So the answer is C.
