Queations 1-3

1.
 X / |X| < X. Which of the following must be true about integer X? X is not equal to 0.
A. X > 1
B. X > -1
C. |X| < 1
D. |X| = 1
E.|X|^2 > 1

Solution:

We know that x doesn't equal 0. So let's break it down into the two other possible cases:  x>0 and x<0
If x is positive, then |x| = x and we can simplify:
x/x < x
 1 < x
 so, if x is positive, x must be greater than 1.
If x is negative, then |x| = -x (since a negative times a negative gives us a positive) and we can simplify:
 x/-x < x
 -1 < x
However, we know that x must be an integer (because that's what the question tells us). Since there are no negative integers greater than -1, there are no possible negative values for x.  Accordingly, we can ignore the x < 0 case.
 Since x must be positive, x must be greater than 1: choose A.


2.
 If each term in the sum a1+a2+a3...+an is either 7 or 77 and the sum is 350, which of the following could be n?
a)38
b)39
c)40
d)41
e)42

Solution:
We know that the units digit in each term in the sum is 7. We also know that the sum ends in 0. The only way to get a bunch of 7s to add up to a 0 is if the number of terms is a multiple of 10.  Only (c) is a multiple of 10


3.
Among 200 people 56% like strawberry jam, 44% like apple jam, and 40% like raspberry jam. If 30% of people like both strawberry and apple jam, what is the largest possible number of people who like raspberry jam but do not like either strawberry or apple jam?

Solution:
40% of the 200 like Raspberry (I'm assuming you got 80 by simply taking 40% of 200). However, there are people who like other jams as well.
 In general, when we're asked to maximize one thing in a GMAT question, we want to minimize everything else.
 In this question, to maximize the number of people who like just raspberry, we need to minimize the number of people who like strawberry and/or apple PLUS raspberry.
  Here's what we know about Strawberry/Apple:
  112 people like Strawberry
88 people like Apple.
60 people like both of them.
 Since only 60 people like both of them, this means that:
 52 people like only strawberry;
28 people like only apple;
 and  60 people like both.
That's already 140 people.
 We only started with 200, so the maximum possible number of people who could dislike both apple and strawberry is 60.