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Median Problem 5

This topic has 4 expert replies and 1 member reply

Median Problem 5

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During a behavioral experiment in a psychology class, each student is asked to compute his or her lucky number by raising 7 to the power of the student's favorite day of the week. (Numbered 1 through 7 from monday through Sunday respectively), multiply the result by 3, and adding this to the doubled ages of students in years, rounded to the nearest year. If a class consists of 28 students, what is the probability that the median lucky number in the class will be a non-integer ?

A) 0%
B) 10%
C) 20%
D) 30%
E) 40%

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aditiniyer wrote:
During a behavioral experiment in a psychology class, each student is asked to compute his or her lucky number by raising 7 to the power of the student's favorite day of the week. (Numbered 1 through 7 from monday through Sunday respectively), multiply the result by 3, and adding this to the doubled ages of students in years, rounded to the nearest year. If a class consists of 28 students, what is the probability that the median lucky number in the class will be a non-integer ?

A) 0%
B) 10%
C) 20%
D) 30%
E) 40%
We need to know 2 things to answer this question.

First, each student's lucky number will ALWAYS be an ODD INTEGER.
We know this because...
lucky number = 7^(student's favorite day of the week - 1,2,3.. or 7) x 3 + (doubled ages of students in years)
In other words, lucky number = (ODD INTEGER x ODD INTEGER) + EVEN INTEGER
= ODD INTEGER + EVEN INTEGER
= ODD INTEGER

Second, when we have an even number of values (28 values), the MEDIAN equals the average (arithmetic mean) of the two middle-most integers (when all of the integers are arranged in ascending order).

Since all 28 values are guaranteed to be ODD (see point #1 above), then we know that the two middle-most integers will be ODD.
So, the median of the 28 values = (some ODD integer + some ODD integer)/2
= (an even integer)/2
= an integer.
In other words, the median of the 28 values is GUARANTEED to be an integer.

So, P(the median of the lucky numbers will be a non-integer) = 0%

Answer: A

Cheers,
Brent

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Hi aditiniyer,

Where did you find this question? I ask because it's at least 12 years old and practicing with such old material might not be in your best interest (since the GMAT has gone through some subtle and not-so-subtle changes over the years).

GMAT assassins aren't born, they're made,
Rich

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aditiniyer wrote:
During a behavioral experiment in a psychology class, each student is asked to compute his or her lucky number by raising 7 to the power of the student's favorite day of the week. (Numbered 1 through 7 from monday through Sunday respectively), multiply the result by 3, and adding this to the doubled ages of students in years, rounded to the nearest year. If a class consists of 28 students, what is the probability that the median lucky number in the class will be a non-integer ?

A) 0%
B) 10%
C) 20%
D) 30%
E) 40%
Say the favorite day = d and age of students = x

Thus, the lucky number = 3.7^d + 2x

We have to find out the median of 28 lucky numbers. Since 28 is an even number, the median lucky number would be the mean of 14th and 15th lucky number.

Say the 14th lucky number = 3.7^d + 2x and the 15th lucky number = 3.7^d' + 2x'

Thus, the median = (3.7^d + 2x + 3.7^d' + 2x')/2 = [3(7^d + 7^d') + 2(x + x')]/2 = 3(7^d + 7^d')/2 + (x + x')

= 3(7^d + 7^d')/2 + Integer; since (x + x') is an integer

Now the fate of the lucky median rests on the nature of 3(7^d + 7^d')/2

Since 7^d and 7^d' are essentially (Odd)^(integer) = Odd, 3(7^d + 7^d')/2 = Odd(Odd + Odd)/2 = (Odd x Even)/2 = Integer

Thus, median lucky number = Integer + Integer = Integer.

It means that whatever be the case, median lucky number is always an integer.

The correct answer: A

Hope this helps!

Relevant book: Manhattan Review GMAT Combinatorics and Probability Guide

-Jay
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Rich.C@EMPOWERgmat.com wrote:
Hi aditiniyer,

Where did you find this question? I ask because it's at least 12 years old and practicing with such old material might not be in your best interest (since the GMAT has gone through some subtle and not-so-subtle changes over the years).

GMAT assassins aren't born, they're made,
Rich
Found it in a bundle used by a popular coaching institute. Is there something wrong in practicing using them?

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Hi aditiniyer,

The value of any particular type of practice materials is ultimately measured in how well you score on the Official GMAT. For someone who was just beginning his studies, I wouldn't recommend that he use such old material. That having been said, how have you been scoring on your practice CATs? Have you taken the Official GMAT yet (and if so, then how did you score?)?

GMAT assassins aren't born, they're made,
Rich

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Contact Rich at Rich.C@empowergmat.com

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