1.
In how many ways can the letters of the word MAXIMA be arranged such that all
vowels are
together?
(A)
12
(B)
18
(C)
30
(D)
36
(E)
72
The
base set can be formed as {{A, I, A}, M, X, M}. The unit {A, I, A} arranges
itself in 3P3/2!
ways.
The 4 units in the base set can be arranged in 4P4/2! ways. Hence, the
total number of ways of arranging the letters is 3P3/2! . 4 P4/2!
=3!/2! . 4!/2!
=
3.12 = 36
The
answer is (D).
2.
If x + z > y + z, then which of the following
must be true?
(I)
x – z > y – z
(II)
xz > yz
(III)
x/z > y/z
(A)
I only
(B)
II only
(C)
III only
(D)
I and II only
(E)
II and III only
Canceling z from both sides of the
inequality x + z > y + z yields x > y.
Adding –z to both sides yields x –z > y – z.
Hence, I is true.
If
z is negative, multiplying the inequality x > y by z
would flip the direction of the inequality resulting in the inequality xz
< yz. Hence, II may not be true.
If
z is negative, dividing the inequality x > y by z would
flip the direction of the inequality resulting in the inequality x/z <
y/z. Hence, III may not be true.
The
answer is (A) since we are asked for statements that MUST be true.
3.
If yz – zx = 3 and zx – xy = 4, then xy – yz
=
(A)
–7
(B)
1
(C)
3
(D)
4
(E)
7
Adding the two given equations yields
(yz
– zx) + (zx – xy) = 3 + 4
yz – zx + zx – xy
= 7
yz – xy = 7
xy – yz = –7
multiplying both sides by –1
The
answer is (A).
4.
The cost of production of a certain instrument is directly proportional to the
number of units produced. The cost of production for 300 units is $300. What is
the cost of production for 270 units?
(A)
270
(B)
300
(C)
325
(D)
370
(E)
395
The cost of production is proportional to
the number of units produced. Hence, we have the equation The Cost of
Production = k × Quantity, where k is a constant.
We
are given that 300 units cost 300 dollars. Putting this in the proportionality
equation yields 300 = k × 300. Solving the equation for k yields k
= 300/300 = 1. Hence, the Cost of Production of 270 units equals k
× 270 = 1 × 270 = 270. The answer is (A).
5.
The list price of a commodity is the price after a 20% discount on the
retail price. The festival discount price on the commodity is the price
after a 30% discount on the list price. Customers purchase commodities
from stores at a festival discount price. What is the effective discount offered
by the stores on the commodity on its retail price?
(A)
20%
(B)
30%
(C)
44%
(D)
50%
(E)
56%
Let r be the retail price. The list
price is the price after a 20% discount on the retail price. Hence, it equals r(1
– 20/100) = r(1 – 0.2) = 0.8r.
The
festival discount price is the price after a 30% discount on the list price.
Hence, the festival discount price equals (list price)(1 – 30/100) = (0.8r)(1
– 30/100) = (0.8r)(1 – 0.3) = (0.8r)(0.7) = 0.56r. Hence,
the total discount offered is (Original Price – Price after discount)/Original
Price × 100 =(r – 0.56r)/ r × 100 = 0.44 ×
100 = 44%.
The
answer is (C).
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