Let the last division send 'a' representatives.
Now the set becomes (3,4,5,5,a)
Note: We cant assume 'a' is the largest or smallest or in the middle for now, but what we can infer from the stem is this.
The median for the set can only be 4 or 5.
This is because the set has only integers. Divisons cant exactly send people in decimals. (Dark jokes aside).
So, why do I say we can only take 4 or 5 as medians?
Here is why.
The set can be as follows
(a 3 4 5 5) Median = 4 (a would have to be less than or equal to 3)
(3 a 4 5 5) Median = 4 (a would have to less than equal to 4 but greater than or equal to 3)
(3 4 a 5 5) Median = (a can be 4 or 5 only) a has to be less than or equal to 5 but greater than equal to 4)
(3 4 5 a 5) Median = 5 (a has to be 5)
(3 4 5 5 a) Median = 5 (a has to be at least 5 or more)
This is how you can be sure median is 4 or 5.
Now with S1.
We see that median for this set is greater than the mean.
Mean of the set = (17+a)/5
Because median has two possible values
We will consider both:
(17+a)/5<4
..a<3
or
(17+a)/5<5
..a<8.
For a<3, we can easily say that the range of the set is definitely greater than 2,
but for a<8, we can't say range is surely greater than 2. if a = 7, Range = 7-3 =4: YES
for a = 3, Range = 5-3 = 2, then NO.
Hence S1 is insufficient.
Now for S2)
Median is 4.
Set can now be as follows:
(a 3 4 5 5) Median = 4 (a would have to be less than or equal to 3) (Range is greater than or equal to two)
(3 a 4 5 5) Median = 4 (a would have to less than equal to 4 but greater than or equal to 3) (Range = 2)
(3 4 a 5 5) Median = a (can be 4 or 5 only) a has to be less than or equal to 5 but greater than equal to 4) Range =2.
Hence even with S2, we cant say for sure.
Combining S1 and S2, we can say median is 4 and median is greater than mean, so (17+ a)/5 is less than 4, implying a < 3. Hence 'a' can be 1 or 2, range can then be 4 or 3 only, which is greater than 2.
Hence C is the option.
KarishmaB request your tips on coming up with ways to reduce the time taken to solve this.