Ujaswin
IanStewart
enigma123
The GMAT is scored on a scale of 200 to 800 in 10 point increments. (Thus 410 and 760 are real GMAT scores but 412 and 765 are not). A first-year class at a certain business school consists of 478 students. Did any students of the same gender in the first-year class who were born in the same-named month have the same GMAT score?
(1) The range of GMAT scores in the first-year class is 600 to 780.
(2) 60% of the students in the first-year class are male.
Bowtie's solution above is perfect, but I'll go into a bit more detail in case it's unclear:
This question (and another I responded to yesterday) is based on something called the 'pigeonhole principle'. That principle is usually explained as follows: if a postal worker has 4 envelopes which she will place in 3 pigeonholes, then it must be true that (at least) 2 envelopes will end up in the same pigeonhole, since even if the first 3 envelopes go into different pigeonholes, the 4th will need to go in a pigeonhole already containing an envelope. Changing the numbers to mimic the question above, say you have 20 people who take a test consisting of 19 questions. Then it must be true that two people answered the same number of questions correctly, since even if the first 19 people all had a different number of correct answers, that uses up all of the possibilities from 1 to 19, so the last person would need to have answered the same number of questions correctly as one of the first nineteen people.
We have a similar situation here. From Statement 1, there are only 19 possible GMAT scores in the class. So if we know we have 20 students of the same gender who were born in the same month, we could then be sure that (at least) two of them must have the same GMAT score. We have 478 students in total. At least half of these, or 239 students, must be of the same gender (you couldn't have less than 239 men *and* less than 239 women in the class). Further, it is impossible that there are only 19 students of the same gender born in *every* month of the year (since then you'd only have at most 12*19 = 228 students of that gender), so we must have at least 20 students of the same gender born in the same month. So two of these students must have the same GMAT score, and the answer is A.
How do we know the no.of men or women? The question doesn't give us any info about the number.
half of 478= 239.
Say no. of male =1, then no. of female= 478-1=477, so more than half of 478 are of same gender (477 females)
if no. of male = 200, no. of females= 278, still more than half are of same gender.. (278 females)
if no. of males = 300, no. of females= 178, again, more than half are of same gender.. (300 males)
if no. of males= 239, no. of female= 239, same no. of male and female.. again exactly half are of same gender.. (239 males, 239 females)
so if no. of males is greater than no. of females , then more than half will be male (of same gender)
and if no. of males is fewer than no. of females, then more than half will be female (again of same gender)
so in any case, exactly half or more than half will be of same gender..
Hope its clear...