Solutions and detailed explanations to questions and problems similar to the questions in the GRE test. sample 2

If w is the average (arithmetic mean) of the numbers a, b, c and d, then the average of m(a + k), m(b + k), m(c + k) and m(d + k) is given by
Solution
w is the mean of a, b, c and d is written as
w = (a + b + c + d) / 4
w is the average W of m(a + k), m(b + k), m(c + k) and m(d + k) is given by
W = [ m(a + k) + m(b + k) + m(c + k) + m(d + k) ] / 4
W = m [ a + b + c + d + 4 k] / 4 = m [a + b + c + d ] / 4 + m k
= m (w + k)
If w is the average of a, b, c and d, then the average W of m(a + k), m(b + k), m(c + k) and m(d + k) is given by
W = m (w + k)

What is the ratio of the area of the larger circle to the area of the smaller circle such that the radius of the larger circle is three times the radius of the smaller circle?
Solution
Let r and R be the radii of the smaller and larger circles respectively. The radius of the larger circle is three times the radius of the smaller circle leads to
R = 3r
Areas A1 of smaller and A2 of larger circles are given by
A1 = Pi r^{2}
A2 = Pi R^{2} = Pi (3r)^{2} = 9 Pi r^{2}
ratio R of areas larger / smaller is equal to
R = 9 Pi r^{2} / Pi r^{2} = 9

A group of 20 employees in a company have an average (arithmetic mean) salary of $35,000 while a second group of 30 employees in the same company have an average salary of $40,000. What is the average salary of the 50 employees making the two groups?
Solution
Let S1 be the total salary of the group of 20 employess. Hence
35,000 = S1 / 20
S1 = 20 * 35,000 = $700,000
Let S2 be the total salary of the group of 30 employess. Hence
40,000 = S2 / 30
S2 = $1,200,000
The average of all 50 employers is given by
(700,000 + 1,200,000) / 50 = $38,000

Which of the following is equal to √48
A) 16
B) 3√4
C) 4√3
D) 18√3
E) 24
Solution
Rewrite the given expression using the fact that 48 = 3 × 16
√48 = √(3 × 16)
Use the formula √(a × b) = √a × √a to rewite √(3 × 16) as
√48 = √(3 × 16) = √3 × √16
= 4 √3

The sizes of the interior angles A, B and C of a triangle are in the ratio 2:4:3. What is the measure, in degrees, of the smallest angle?
Solution
Since the three angles are in the ration 2:4:3, their sizes they may be written in the form
Size of A = 2 k , Size of B = 4 k and size of C = 3 k , where k is a constant.
The sum of the angles of a ny triangle is equal to 180°; hence
2 k + 4 k + 3 k = 180
Solve for k
9 k = 180 , k = 20
The smallest angle is A and its size is equal to 2 k
2 k = 2 × 20 = 40°

If n is even and m is odd, then which of the following is true?
A) n + m is even
B) n  m is even
C) n * m is odd
D) n^{2} + m^{2} + 1 is even
E) 2n + 3m + 1 is odd
Solution
If n is even, it can be written as follows
n = 2 k , where k is an integer
If m is odd, it can be written as follows
m = 2 K + 1 , where K is an integer
We now express n + m in terms of k and K
n + m = 2 k + 2 K + 1 = 2(k + K) + 1
n + m is odd
We now express n  m in terms of k and K
n  m = 2 k  (2 K + 1) = 2 k  2 K  1
n  m = 2 (k  K)  1
n  m is odd
We now express n * m in terms of k and K
n * m = (2 k)(2 K + 1) = 2( k(2K + 1) )
n * m is even
We now express n^{2} + m^{2} + 1 in terms of k and K
n^{2} + m^{2} + 1 = (2 k)^{2} + (2 K + 1)^{2} + 1
= 4 k^{2} + 4 K^{2} + 4 K + 1 + 1
= 2 ( 2 k^{2} + 2 K^{2} + 2 K + 1)
n^{2} + m^{2} + 1 is even
Statement D is true.

5^{100} + 25^{50} + 3(125^{34} / 25) =
Solution
Use the facts that 25 = 5^{2} and 125 = 5^{3} to rewrite the given expression as follows
5^{100} + 25^{50} + 3(125^{34} / 25) = 5^{100} + (5^{2})^{50} + 3( (5^{3})^{34} / (5^{2}))
Use formula for exponents to simplify
= 5^{100} + 5^{100} + 3( 5^{102} / 5^{2})
= 5^{100} + 5^{100} + 3( 5^{100})
= 5 * 5^{100}
= 5^{101}

[ 6x^{10}  2x^{9} ] / (9x^{2}  1) =
Solution
Factor numerator as follows
6x^{10}  2x^{9} = 2x^{9} (3x  1)
Factor denominator as follows
9x^{2}  1 = (3x  1)(3x + 1)
Substitute numerator and denominator by their factored forms and simplify the given expression
[ 6x^{10}  2x^{9} ] / (9x^{2}  1) = [2x^{9} (3x  1)
] / [(3x  1)(3x + 1)]
= 2x^{9} / (3x + 1)

( 2x + 6)^{2} =
Solution
Expand by multiplication or using the identity (x + y)^{2} = x^{2} + 2 x y + y^{2}.
( 2x + 6)^{2} = (2x)^{2} + 2 (2x)(6) + 6^{2}
= 4 x^{2}  24 x + 36

The sum of all interior angle of a regular polygon is 1800°. How many sides does this polygon have?
Solution
The sum of all interior angles of a polygon of n sides is given by.
(n  2) * 180
and is equal to 1800°. Hence
(n  2) * 180 = 1800
Solve for n
(n  2) = 10
n = 12
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