What we see: a yes or no Data Sufficiency question about a set. Five different whole numbers, each bigger than 2. Is the mean bigger than the median?
Step 1. Name the numbers.
What we see: five different numbers.
What it means: write them in order as \(a < b < c < d < e\). The smallest is \(a\), the median is the middle one, \(c\), and the mean is the sum divided by 5. Since every number is bigger than 2 and a whole number, \(a\) is at least 3.
What we do and why: we keep "\(a \ge 3\)" in mind. It is easy to miss, and it decides the question at the end.
Step 2. Statement 1 alone.
What we see: the sum is \(20a\).
What it means: the mean is \(\frac{20a}{5} = 4a\). The question becomes: is \(4a > c\)?
What we do and why: try \(a = 3\), so the mean is 12 and the sum is 60.
Set 3, 4, 5, 6, 42: median 5, and 12 > 5. Yes.
Set 3, 4, 16, 18, 19: median 16, and 12 < 16. No.
Two answers. Not sufficient.
Step 3. Statement 2 alone.
What we see: the median is 8 more than the smallest, so \(c = a + 8\).
What it means: this says nothing about the sum.
What we do and why: try 3, 4, 11, 12, 13: mean 8.6, median 11. No. Try 3, 4, 11, 12, 100: mean 26, median 11. Yes. Not sufficient.
Step 4. Both statements together.
What we see: mean \(= 4a\) and median \(= a + 8\).
What it means: the question is "is \(4a > a + 8\)?", which is "is \(3a > 8\)?", which is "is \(a > 2.67\)?"
What we do and why: we know \(a \ge 3\), so the answer is always yes. A real set exists too: 3, 4, 11, 12, 30 has sum 60 \(= 20 \times 3\) and median 11 \(= 3 + 8\). One answer, yes. Sufficient.
The trap: (E). It looks like we know nothing about \(b\), \(d\) and \(e\), so it feels like the answer must stay open. But the question only compares the mean and the median, and both are written in terms of \(a\) alone once the statements are joined.
Faster way: write the mean and the median in terms of the same letter as soon as you can. Then the yes or no question becomes one small inequality.
Takeaway: in a yes or no question, rewrite the question with what the statements give you; then the conditions in the question stem (here "greater than 2") often finish it.
Answer: C
