10 questions similar to GRE quantitative reasoning questions, complete with detailed solutions and explanations, are presented below.
If \( w \) is the average of the numbers \( a, b, c \) and \( d \), then the average of \( m(a + k),\quad m(b + k),\quad m(c + k),\quad m(d + k) \) is given by
Let \( w \) be the mean of \( a, b, c \) and \( d \):
\[ w = \dfrac{a + b + c + d}{4} \]Let \( W \) be the average of \( m(a + k) \), \( m(b + k) \), \( m(c + k) \), and \( m(d + k) \):
\[ W = \dfrac{m(a + k) + m(b + k) + m(c + k) + m(d + k)}{4} \] \[ W = \dfrac{m(a + b + c + d + 4k)}{4} = \dfrac{m(a + b + c + d)}{4} + mk \] \[ W = m(w + k) \]Answer: B) \( m(w + k) \)
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?
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:
\[ R = 3r \]The areas \( A_1 \) of the smaller and \( A_2 \) of the larger circles are given by:
\[ A_1 = \pi r^2 \] \[ A_2 = \pi R^2 = \pi (3r)^2 = 9\pi r^2 \]The ratio of areas (larger / smaller) is:
\[ \text{Ratio} = \dfrac{9\pi r^2}{\pi r^2} = 9 \]Answer: E) 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?
Let \( S_1 \) be the total salary of the group of 20 employees:
\[ 35,000 = \dfrac{S_1}{20} \implies S_1 = 20 \times 35,000 = 700,000 \]Let \( S_2 \) be the total salary of the group of 30 employees:
\[ 40,000 = \dfrac{S_2}{30} \implies S_2 = 30 \times 40,000 = 1,200,000 \]The average of all 50 employees is given by:
\[ \dfrac{700,000 + 1,200,000}{50} = \dfrac{1,900,000}{50} = 38,000 \]Answer: B) $38,000
Which of the following is equal to \( \sqrt{48}\)?
Rewrite the given expression using the fact that \( 48 = 16 \times 3 \):
\[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \]Answer: C) \( 4\sqrt{3} \)
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?
Since the three angles are in the ratio 2:4:3, their sizes may be written as:
\[ \text{Size of A} = 2k, \quad \text{Size of B} = 4k, \quad \text{Size of C} = 3k \]where \( k \) is a constant. The sum of the angles in any triangle is \( 180^\circ \):
\[ 2k + 4k + 3k = 180 \implies 9k = 180 \implies k = 20 \]The smallest angle is A with size \( 2k \):
\[ 2k = 2 \times 20 = 40^\circ \]Answer: A) 40
If n is even and m is odd, then which of the following is true?
Let \( n = 2k \) and \( m = 2K + 1 \) where \( k \) and \( K \) are integers.
Thus, statement D is true.
Answer: D) \( n^2 + m^2 + 1 \) is even
Use the facts that \( 25 = 5^2 \) and \( 125 = 5^3 \):
\[ 5^{100} + (5^2)^{50} + 3\left( \dfrac{(5^3)^{34}}{5^2}\right) = 5^{100} + 5^{100} + 3\left( \dfrac{5^{102}}{5^2}\right) \] \[ = 5^{100} + 5^{100} + 3(5^{100}) = (1 + 1 + 3)5^{100} = 5 \cdot 5^{100} = 5^{101} \]Answer: E) \( 5^{101} \)
Factor the numerator and denominator:
\[ 6x^{10} - 2x^9 = 2x^9(3x - 1) \] \[ 9x^2 - 1 = (3x - 1)(3x + 1) \]Substitute and simplify:
\[ \dfrac{2x^9(3x - 1)}{(3x - 1)(3x + 1)} = \dfrac{2x^9}{3x + 1} \]Answer: B) \( \dfrac{2x^9}{3x + 1} \)
Expand using the binomial square identity \((a + b)^2 = a^2 + 2ab + b^2\):
\[ (-2x + 6)^2 = (-2x)^2 + 2(-2x)(6) + 6^2 = 4x^2 - 24x + 36 \]Answer: D) \( 4x^2 - 24x + 36 \)
The sum of all interior angles of a regular polygon is \( 1800^\circ \). How many sides does this polygon have?
The sum of all interior angles of an \( n \)-sided polygon is given by:
\[ (n - 2) \cdot 180^\circ \]Set this equal to \( 1800^\circ \):
\[ (n - 2) \cdot 180 = 1800 \implies n - 2 = 10 \implies n = 12 \]Answer: C) 12