1. E
Try different possible values for x and y,
eliminating the incorrect answer choices. Since x is
multiplied by 3, let's begin with the smallest positive integer
value for x:
If 3(1) + y = 14, then y = 11, and x +
y = 12. So choice (A) is out.
If 3(2) + y = 14, then y = 8, and x +
y = 10. So choice (B) is out. If 3(3) + y = 14,
then y = 5, and x + y = 8. So choice (C) is
also out. If you're really clever, you'll see at this point that
answer choice (E) is impossible (which makes it the right
choice). After all, the next smallest possible value of x
is 4, and since x and y must both be positive
integers, neither one can equal 0. (Zero is not positive or
negative.) So the sum of x and y must be greater
than 4. (Sure enough, if x = 4, then y = 2, and
x + y = 6, eliminating choice (D) as well.)
2. E
Each of the five surfaces is 60 by 180 centimeters, so tiles
measuring 12 by 18 centimeters can be laid down in 5 rows of 10
to exactly cover one surface. There are 5 surfaces so 5 x 5 x 10
= 250 tiles are needed.
3. D
If a machine labels 150 bottles in 20 minutes, it labels 15
bottles every 2 minutes. To label 60, or 4 x 15, bottles would
take 4 x 2, or 8 minutes.
4. C
Pick variables for the two items and translate the given
information into algebraic equations. Let h = the price of
a hot dog and s = the price of a soda. The first statement
is translated as 2h + s = $3.25, and the second as
3h + s = $4.50. If you subtract the first equation
from the second, the s is eliminated so you can solve for
h:
3h + s = $4.50
2h + s = $3.25
h = $1.25
Plug this value for h into the first equation to solve
for s:
2($1.25) + s = $3.25
$2.50 + s = $3.25
s = $0.75
So two sodas would cost 2 x $0.75 = $1.50.
5. C
Since he traveled the same distance both ways, it's tempting
to think we can simply average the speeds to find the answer. But
rates can't be averaged in that way. It's necessary to compute
the total distance and the total time and use these numbers in
the average formula.
Total miles: He travels 90 miles there and 90 miles back, for
a total of 180 miles. To find the total time, we must calculate
the time for each part of the trip, and then add them together.
Going there, he travels 90 miles at 20 miles per hour.
Since rate x time = distance, time = (distance/rate). So it
takes him (90 miles/20 miles per hour) = (9/2) hours to travel
there. Coming back, he travels 90 miles at 40 miles per hour, so
it takes him (90/40) = (9/4) hours to return home.
The total time spent was therefore:
(9/2) = (9/4) = (18/4) + (9/4) = (27/4) or 6 (3/4) hours.
So his average speed for the whole trip was (total miles/
total hours) = (180 x (4/27) ) = 80/3 miles per hour.
6. D
Any number that is divisible by 8 must also be divisible by 4.
A number is divisible by 4 is its last two digits form a number
that is divisible by 4; the only answer choices for W that will
make 5W2 divisible by 4 are 3 and 5 (since 32 and 52 are both
divisible by 4). Dividing 532 and 552 by 8, you see that only 552
leaves no remainder, so W must be 5, answer choice (D).
7. A
Let x = the amount of the chemical two days ago. Since
it decays at a rate of 20% each day, the amount of the chemical
remaining yesterday is equal to 80% of x, or .8x.
So the amount of the chemical remaining today equals 80% of
.8x, or .64x. Therefore, the .64x = 48, and
x = 75.
8. B
Number of terms X average = sum of the terms. For the first
group, 3 X 40 = 120, so the sum of 20, 70, and x is 120.
For the second group, 4 X 50 = 200, so 20 + 70 + x +
y = 200. Since the sum of the first three terms is 120,
120 + y = 200, y = 80.
9. A
The sum of the lengths of any two sides of a triangle must be
greater that the length of the third side. So, AB +
AC must be greater that BC; 6 + x > 12.
If x = 6, 6 + 6 = 12 is not greater than 12, so x
cannot equal 6.
10. B
The ratio 5 goblins to 3 witches tells you that for every 5
goblins in the city there must be three witches in the city. So,
the total number of goblins and witches in the city must be a
multiple of 8, since the smallest possible total is 5 + 3, or 8.
Since 36 is not divisible by 8, 36 cannot be the total number of
witches and goblins.
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