swerve wrote:Peter, Paul, and Mary each received a passing score on his/her history midterm. The average (arithmetic mean) of the three scores was 78. What was the median of the three scores?

1) Peter scored a 73 on his exam.
2) Mary scored a 78 on her exam.

The OA is B

Source: Manhattan Prep

Given: Average (arithmetic mean) of 3 scores = 78

We have to get the value of Median.

1) Peter scored a 73 on his exam.

Case 1: Say the scores are 73, 78 and 83. This makes the average (arithmetic mean) of 3 scores = 78. Thus, median = 78.
Case 2: Say the scores are 73, 80 and 81. This makes the average (arithmetic mean) of 3 scores = 78. Thus, median = 80. No unique value. Insufficient.

2) Mary scored a 78 on her exam.

With average = 78 and one of the scores 78, there are two possibilities: (1) All the three scores are 78; thus, average = median = 78, and (2) With one score being 78, and to maintain the average = 78, if one of the remaining two scores is less than 78, the other one must be greater than 78; thus, average = median = 78. Sufficient.

The correct answer: B

Hope this helps!

-Jay
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swerve wrote:Peter, Paul, and Mary each received a passing score on his/her history midterm. The average (arithmetic mean) of the three scores was 78. What was the median of the three scores?

1) Peter scored a 73 on his exam.
2) Mary scored a 78 on her exam.

The OA is B

Source: Manhattan Prep

Target question: What was the median of the three scores?

Since there are 3 values, the median will be the middle-most value (when the values are arranged in ascending order).

We also know that: Total of all values = (median)(# of values)
So, the sum of all 3 scores = (78)(3) = 234

Statement 1: Peter scored a 73 on his exam.
There are several sets of scores that meet this condition. Here are two:
Case a: Peter:73, Paul:74, Mary:87, in which case the median is 74
Case b: Peter:73, Paul:75, Mary:86, in which case the median is 75
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Mary scored a 78 on her exam
NOTE: For scores above 78, I'll use the notation 78+ and for scores below 78, I'll use the notation 78-
If the mean is 78 and Mary scored a 78, then there are only 3 scenarios possible:
scenario 1: Peter:78, Mary:78, Paul:78, in which case the median is 78
scenario 2: Peter:78-, Mary:78, Paul:78+, in which case the median is 78
scenario 3: Peter:78+, Mary:78, Paul:78-, in which case the median is 78

Notice that no other scenarios are possible. For example, consider this scenario:
Peter:78+, Mary:78, Paul:78+
This scenario is impossible, because the sum of all three values must be 234, and we know that 78+78+78=234.
So, it is impossible for (78)+(78+)+(78+) to equal 234

Using similar logic and notation we can show that other scenarios are impossible.
As you can see, statement 2 consistently yields the same answer to the target question.
So, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent