30. Set B has three positive integers with a
median of 9. If the largest possible range of the
three numbers is 19, given a certain mean, what is
that mean?
(A) 22
(B) 10
(C) 9.6
(D) 9
(E) It cannot be determined with the information given
You have the set B {x,9,y} where all elements are positive integers.
We know x < 9 < y because if this were not true, 9 wouldn't be the median.
Now, the smallest possible value for x is 1, because 0 is not positive or negative. So the smallest possible value for y is then 1+19= 20.
The mean would then be:
(x+9+y)/3 =
(1+9+20)/3 =
30/3 = 10
10 is the minimum mean. So we can eliminate C and D because they are too low.
But wait, what's the largest possible value for x? That would be 9-1 = 8 and the largest possible value for y would then be 8+19 = 27.
Doing the math gives a mean of (8+9+27)/3 = 14 2/3. Since that is also the maximum mean, A (22) is eliminated because it's too high.
So we're left with B (10) and E (cannot be determined). The question contains the words "given a certain mean." The confusion is surrounding the word "certain" because people often think the word means unspecified. However, the word certain actually means:
i)definite but not specified or identified
or
ii)established beyond doubt or question; definitely known
Since the mean is definitely known, then it essentially translates into "the mean can be determined from the information in the question stem." E can be eliminated.
Finally, the correct answer is B.
