Bunuel wrote:
A, B and C have received their Math midterm scores today. They find that the arithmetic mean of the three scores is 78. What is the median of the three scores?
(1) A scored a 73 on her exam.
(2) C scored a 78 on her exam.
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:Recall from the
arithmetic mean post that the sum of deviations of all scores from the mean is 0.
i.e. if one score is less than mean, there has to be one score that is more than the mean.
e.g. If mean is 78, one of the following must be true:
All scores are equal to 78.
At least one score is less than 78 and at least one is greater than 78.
For example, if one score is 70 i.e. 8 less than 78, another score has to make up this deficit of 8. Therefore, there could be a score that is 86 (8 more than 78) or there could be two scores of 82 each etc.
Statement 1: A scored 73 on her exam.
For the mean to be 78, there must be at least one score higher than 78. But what exactly are the other two scores? We have no idea! Various cases are possible:
73, 78, 83 or
73, 74, 87 or
70, 73, 91 etc.
In each case, the median will be different. Hence this statement alone is not sufficient.
Statement 2: C scored 78 on her exam.
Now we know that one score is 78. Either the other two will also be 78 or one will be less than 78 and the other will be greater than 78. In either case, 78 will be the middle number and hence will be the median. This statement alone is sufficient.
Answer (B) _________________