Mean Problem 2

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aditiniyer: Senior | Next Rank: 100 Posts; Posts: 45; Joined: Fri Aug 19, 2016 1:42 am

by aditiniyer » Tue Feb 07, 2017 2:32 am

The average of x number of exams is y. When an additional exam of score z is added in, does the average score of the exams increase by 50% ?

A) 3x = y
B) 2z-3y = xy

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Source: Beat The GMAT — Data Sufficiency | Tue Feb 07, 2017 2:32 am

DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 amFollowed by: 128 members; Location: Boston, MA; Thanked: 1153 times; GMAT Score:770

by DavidG@VeritasPrep » Tue Feb 07, 2017 5:19 am

Let's rephrase the question. Our initial average was y, and our initial number of terms was x, so our initial sum was yx. If we add 'z', the new sum is yx + z, and the new number of terms is x + 1, so the new average would be (xy + z)/(x +1). We want to know if this is 50% more than the initial average of y, or 1.5y. In other words is (xy + z)/(x +1) = 1.5y? We can simplify

xy + z = 1.5y(x+1) --> xy + z = 1.5yx + 1.5y ---> z = .5yx + 1.5y. Rephrased question: Is z = .5yx + 1.5y?

S1: tells us nothing about z. Not sufficient.

S2: 2z - 3y = xy --> 2z = xy + 3y --> z = .5xy + 1.5y. Statement 2 alone is sufficient. Answer is B

Last edited by DavidG@VeritasPrep on Tue Feb 07, 2017 9:58 am, edited 2 times in total.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Feb 07, 2017 5:28 am

aditiniyer wrote:The average of x number of exams is y. When an additional exam of score z is added in, does the average score of the exams increase by 50% ?

A) 3x = y
B) 2z-3y = xy

Statement 1:
No information about z.
INSUFFICIENT.

Statement 2:
Case 1: x=1 and y=1, implying that 1 exam has an average score of 1
Sum of the scores = (number of exams)(average score) = 1*1 = 1.
Plugging x=1 and y=1 into 2z-3y = xy, we get:
2z - 3*1 = 1*1
2z = 4
z = 2.
When the additional exam with a score of 2 is included, the new average for the resulting 2 exams = (old sum + z)/2 = (1 + 2)/2 = 1.5.
Since the average increases from 1 to 1.5, it increases by exactly 50%.

Case 2: x=3 and y=2, implying that 3 exams have an average score of 2
Sum of the scores = (number of exams)(average score) = 3*2 = 6.
Plugging x=3 and y=2 into 2z-3y = xy, we get:
2z - 3*2 = 3*2
2z = 12
z = 6.
When the additional exam with a score of 6 is included, the new average for the resulting 4 exams = (old sum + z)/4 = (6 + 6)/4 = 3.
Since the average increases from 2 to 3, it increases by exactly 50%.

The two cases above illustrate that -- for any positive values x and y -- the average will increase by exactly 50%.
SUFFICIENT.

The correct answer is B.

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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 » Thu Feb 09, 2017 10:29 pm

aditiniyer wrote:The average of x number of exams is y. When an additional exam of score z is added in, does the average score of the exams increase by 50% ?

A) 3x = y
B) 2z-3y = xy

So, the question is: Is the new average = 3/2 times the old average (y)?

Current average = y

Sum of all exams = xy

New average = (xy+z)/(x+1)

We have to see if (xy+z)/(x+1) = (3/2)*(xy).

Let us take each statement one by one.

S1: 3x = y

We do not have any information about z. The answer may be YES or NO.

S2: 2z-3y = xy

=> z = (3y + xy)/2

By plugging in the value of z in new average = (xy+z)/(x+1)

We get,

New average = (xy+z)/(x+1) = (xy+(3y + xy)/2) / (x+1) = (2xy+xy+3y)/2(x+1) = (3xy+3y)/2(x+1) = 3y(x+1)/2(x+1) = 3y/2 = 50% more than old average

Sufficient.

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

The correct answer: B

Hope this helps!

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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 » Thu Feb 09, 2017 10:59 pm

aditiniyer wrote:The average of x number of exams is y. When an additional exam of score z is added in, does the average score of the exams increase by 50% ?

A) 3x = y
B) 2z-3y = xy

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