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