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1. Bob and Carol and Ted and Alice either hate or love each of the other three
people. The following is true:
Bob loves Ted and hates Carol.
Carol loves two of the other three people.
Carol hates everyone who hates her.
Only one person loves Alice.
Therefore, which must be true:
I. Ted hates Alice.
II. Ted loves Carol.
III. Ted loves Bob.
We know that Carol hates Bob right back (Carol is like that) so the two people
she loves are Ted and Alice who love her back (otherwise, Carol would hate them).
Carol is the one person who loves Alice which means Bob and Ted both
hate Alice. So Ted hates Alice (I) is true and Ted loves Carol (II) is also true.
We don't know how Ted feels about Bob so III is not true. The answer is (D).
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2. Bob and Carol and Ted and Alice are backpacking in the wilderness. Each person
carries x pounds of equipment at the start of the trip. "Equipment" does not
include food so the total weight of equipment (4x) is constant.
During the first night, Bob sneaks 10 pounds of equipment from his pack
into Carol's.
Ted and Alice each sneak 7 pounds of equipment from their packs into Carol's.
Later, when the other three are picking berries, Carol divides all the equipment
in her pack evenly amongst the packs of her
three pals. If Bob's pack now contains 54 pounds of equipment,
how much did it contain originally?
Bob starts with x pounds of equipment, gives some away and is left with
(x–10) pounds. Meanwhile, Carol winds up with (x+10+7+7) = (x+24) pounds of equipment due
to the "generosity" of her friends. Then Carol gives Bob (and Ted and Alice) one third
of the equipment in her pack or (x/3 + 8) pounds. So Bob now has (x–10+x/3+8) pounds
of equipment and we are told this is 54 pounds. Solving for x, you get 56 = (4/3)x, so
x is 42 and the answer is (D).
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3. Bob and Carol and Ted and Alice smuggle treasure from a Caribbean island to the
United States. The FBI is onto them. There are four possible outcomes and each
of our four heroes will experience one outcome and each will experience a different
outcome.
Bob or Carol or Ted will go to jail.
Ted or Alice will get rich.
Carol or Alice will change her identity.
Bob or Ted or Alice will go into hiding.
Which of the following is true?
I. If Bob goes to jail, Alice gets rich.
II. If Ted goes to jail, Bob goes into hiding.
III. If Carol goes to jail, Ted gets rich.
Suppose Bob goes to jail. You could have Ted get rich, Carol change her
identity, and Alice go into hiding. So (I) isn't true. Suppose Ted goes to jail.
This means Alice has to get rich. If Alice gets rich, then Carol has to change
her identity. This means Bob goes into hiding. So (II) is true. Finally, suppose
Carol goes to jail. This means Alice has to change her identity. If Alice changes
her identity then Ted must get rich. So (III) is true. The answer is (D).
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4. If ax2 + bx + c = 0 for all values of x, then which of the following
is true?
I. a = b = c = 0
II. a > (b+c)
III. a2 > b2+c2
If a, b, and c are any non-zero numbers, you could solve for x using the quadratic
formula and get 0, 1, or 2 solutions. The only way to make the equation true
for ALL values of x (that's an infinite number of solutions including x = 0 and
x = 1 trillion) is if a, b, and c are all zero. So the answer is (B).
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