Dear moderators and friends,
If w, x, y, and z are integers such that w/x and y/z are integers, is w/x + y/z odd?
a. wx + yz = odd
b. wz + xy = odd
I will post the OA later.
Thanks
KH
RonPurewal wrote:this problem involves two fractions that are added together. for no other reason than that 'it's the normal thing to do with two fractions added together', let's find the common denominator:
w/x + y/z = wz/xz + xy/xz = (wz + xy)/xz
therefore
the question can be rearranged to:
is (wz + xy)/xz - which is the same thing as w/x + y/z - odd?
-- (2) alone --
if wz + xy is an odd integer, then all of its factors are odd. this means that (wz + xy)/xz, which is guaranteed to be an integer**, must also be odd - because it's a factor of an odd number.
sufficient
**we know this is an integer because it's equal to w/x + y/z, which, according to the information given in the problem statement, is integer + integer.
-- (1) alone --
try to come up with contradictory examples**:
w=2, x=1, y=3, z=1 (so that wx + yz = 5 = odd, per the requirement):
w/x + y/z = 2 + 3 = 5 = odd
w=2, x=2, y=3, z=1 (so that wx + yz = 7 = odd, per the requirement):
w/x + y/z = 1 + 3 = 4 = even
insufficient
**of course, if you're at a loss for the theory, you should try this for statement (1) too ... but you'll find that all the examples you get are odd.
--
answer = b
RonPurewal wrote:this problem involves two fractions that are added together. for no other reason than that 'it's the normal thing to do with two fractions added together', let's find the common denominator:
w/x + y/z = wz/xz + xy/xz = (wz + xy)/xz
therefore
the question can be rearranged to:
is (wz + xy)/xz - which is the same thing as w/x + y/z - odd?
-- (2) alone --
if wz + xy is an odd integer, then all of its factors are odd. this means that (wz + xy)/xz, which is guaranteed to be an integer**, must also be odd - because it's a factor of an odd number.
sufficient
**we know this is an integer because it's equal to w/x + y/z, which, according to the information given in the problem statement, is integer + integer.
-- (1) alone --
try to come up with contradictory examples**:
w=2, x=1, y=3, z=1 (so that wx + yz = 5 = odd, per the requirement):
w/x + y/z = 2 + 3 = 5 = odd
w=2, x=2, y=3, z=1 (so that wx + yz = 7 = odd, per the requirement):
w/x + y/z = 1 + 3 = 4 = even
insufficient
**of course, if you're at a loss for the theory, you should try this for statement (1) too ... but you'll find that all the examples you get are odd.
--
answer = b
sannamalai1 wrote:In the example for statement 1 (w=2, x=2, y=3, z=1), is it okay to take the same value of 2 for both w and x ? When W and X are given as two different variables, how can we assign the same value.
RonPurewal wrote:this problem involves two fractions that are added together. for no other reason than that 'it's the normal thing to do with two fractions added together', let's find the common denominator:
w/x + y/z = wz/xz + xy/xz = (wz + xy)/xz
therefore
the question can be rearranged to:
is (wz + xy)/xz - which is the same thing as w/x + y/z - odd?
-- (2) alone --
if wz + xy is an odd integer, then all of its factors are odd. this means that (wz + xy)/xz, which is guaranteed to be an integer**, must also be odd - because it's a factor of an odd number.
sufficient
**we know this is an integer because it's equal to w/x + y/z, which, according to the information given in the problem statement, is integer + integer.
-- (1) alone --
try to come up with contradictory examples**:
w=2, x=1, y=3, z=1 (so that wx + yz = 5 = odd, per the requirement):
w/x + y/z = 2 + 3 = 5 = odd
w=2, x=2, y=3, z=1 (so that wx + yz = 7 = odd, per the requirement):
w/x + y/z = 1 + 3 = 4 = even
insufficient
**of course, if you're at a loss for the theory, you should try this for statement (1) too ... but you'll find that all the examples you get are odd.
--
answer = b