Peter, Paul, and Mary each received a passing score on his/

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swerve: Legendary Member; Posts: 2539; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

by swerve » Tue Jul 30, 2019 10:01 am

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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

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Source: Beat The GMAT — Data Sufficiency | Tue Jul 30, 2019 10:01 am

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Tue Aug 06, 2019 4:13 am

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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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Aug 06, 2019 5:35 am

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

Brent Hanneson - Creator of GMATPrepNow.com

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deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

by deloitte247 » Wed Aug 07, 2019 6:17 am

$$Mean=78$$
Let Peter's score = a,
Paul's score = b
Mary's score = c
$$So,\ \frac{\left(a+b+c\right)}{3}=78$$
$$Their\ total\ scores=78\cdot3=234$$
Question: What was the median of the three scores?
Statement 1: Paul scored 73 in his exams
$$So,\ \frac{\left(a+73+c\right)}{3}=78$$
$$a+73+c=234\ but\ \ note\ that\ the\ value\ of\ b\ and\ c\ are\ unknown.$$
Therefore, statement 1 is NOT SUFFICIENT.

Statement 2: Mary scored 78 in her exam.
$$So,\ \frac{\left(a+b+78\right)}{3}=78$$
Since 78 is the mean, that means one of the values of 'a' and 'b' must be less than or equal to 78, and the other must be greater than or equal to 78.
Whichever way it is, 78 will be in the middle when arranged in ascending order.
So, median = 78. Hence, statement 2 alone is SUFFICIENT.

Answer = option B.

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