Math questions, with detailed solutions, are presented. The questions are those in ACT sample 2.

What is the greatest common factor (GCF) of 45, 135 and 270?
Solution
Use prime factorization to rewrite 45, 135 and 210 as follows
45 = 1 * 3^{2} * 5
135 = 1 * 3^{3} * 5
270 = 1 * 2 * 3^{3} * 5
The GCF = 5 * 3^{2} = 45

What is the value of 2x + 1/2 when 5x  3 = 3x + 5
Solution
We first solve the equation 5x  3 = 3x + 5
5x + 3x = 5 + 3
8x = 8
x = 1
We now calculate 2x + 1/2
2x + 1/2 = 2(1) + 1/2 = 2 1/2

The width W of a rectangle is 2 inches less than half its length L. Express the perimeter P of the rectangle in terms of the length L.
Solution
We first express the width W in terms of the length L
W = L/2  2
We now express P in terms of L as follows
P = 2(W + L) = 2(L/2  2 + L) = L  4 + 2L = 3L 4

The total surface area of all six sides of the rectangular box below is equal to 80 square inches. What is the volume of the rectangular box in inches cubed?
.
Solution
Let us first find the total area of all six sides:
Area of left and right sides: 4x + 4x = 8x
Area of front and back sides: 4x + 4x = 8x
Area of top and bottom sides: 16 + 16 = 32
Total area = 16x + 32 = 80
Solve for x
16x = 80  32
16x = 48
x = 3 in
Volume of the box is
4 * 4 * 3 = 48 inches cubed

The points A(4 , 4), B(1 , 2) and C(x , 8) are the vertices of a right triangle with the right angle at (1 , 2). Find the value of x.
Solution
Since the right angle is at point B, BA and BC are the legs of the right triangle and AC is the hypotenuse. Find squares of distances AB, BC and AC.
AB^{2} = (1 + 4)^{2} + (2 + 4)^{2} = 13
BC^{2} = (x + 1)^{2} + (8 + 2)^{2} = (x + 1)^{2} + 36
AC^{2} = (x + 4)^{2} + (8 + 4)^{2} = (x + 4)^{2} + 16
Apply Pythagora's theorem.
AB^{2} + BC^{2} = AC^{2}
13 + (x + 1)^{2} + 36 = (x + 4)^{2} + 16
Expand and simplify.
13 + x^{2} + 2x + 1 + 36 = x^{2} + 8x + 16 + 16
6x = 18
x = 3

In the figure below, L1 and L2 are parallel lines. The correct relationship between angles y and x is
.
Solution
The angle supplementary to angle x on the right side of L3 is equal to y since they are corresponding angles. Hence
x + y = 180^{o}

If (x + y)^{2} = 144 and x^{2}  y^{2} = 24, then what is x if x and y are both positive?
Solution
Take the square root of both sides in (x + y)^{2} = 144 to obtain
x + y = 12
Factor the left side of x^{2}  y^{2} = 24 and write
(x + y)(x  y) = 24
Use x + y = 12 obtained above to write
12(x  y) = 24
Simplify
x  y = 2
Solve for x the system of equation: x + y = 12 and x  y = 2. Add the right and left sides of the two equations top obtain
2x = 14
x = 7

How many 3 digit numbers can we make using the digits 4, 5, 7 and 9 and where repetition is allowed?
Solution
We have four digits. The numbers we need to make have 3 digits. There are 4 choices for the first digit, 4 choices for the second digit and 4 choices for the third digit since repetition is allowed. Hence the total number of 3 digit numbers that can be made is equal to
4 * 4 * 4 = 4^{3}

A boat travels 10 miles East and then 24 miles South to an island. How many miles are there from the point of departure of the boat to the island?
Solution
In traveling East and then South, the boat is moving along the legs of 10 and 24 miles of a right triangle. The distance d then would be the hypotenuse which can be found using Pythagora's theorem.
d^{2} = 10^{2} + 24^{2} = 676
d = 26 miles.

What is the slope of any line perpendicular to the line 5x + 3y = 9?
Solution
Write the equation of the given line and find the slope.
5x + 3y = 9
3y = 5x + 9
y = (5/3)x + 3
slope = 5/3
slope m of perpendicular is such that (5/3) * m = 1. solve for m.
m = 3/5

Which of the following is a factor of the polynomial 2x^{2} + 7x  6?
A) 2x  3
B) 2x + 2
C) x  6
D) 2x  2
E) 2x + 3
Solution
Factor the given expression.
2x^{2} + 7x  6 = (2x + 3)(x  2)
2x + 3 is a factor of 2x^{2} + 7x  6.

√(9)(4) + √(4) = ?
Solution
Simplify both terms as follows.
√(9)(4) = √36 = 6
√(4) = √((1)4) = √(1) √4 = i *2 = 2i
Hence.
√(9)(4) + √(4) = 6 + 2 i , where i = √(1)

Find the linear function f such that f(2) = 5 and f(3) = 5.
Solution
A linear function f has the form.
f(x) = m x + b
Use f(2) = 5 and f(3) = 5 to write
f(2) = 2 m + b = 5 and f(3) = 3 m + b = 5
Solve the system of equations: 2 m + b = 5 and 3 m + b = 5. Subtract right sides an left sides to eliminate b as follows
(3 m + b)  (2 m + b) =  5  5
m = 10
b = 5  2 m = 5  2 (10) = 5 + 20 = 25
f(x) =  10 x + 25

A circular garden has an area of 100π feet squared. What is the circumference of the garden in feet?
Solution
The area of the circular garden of radius R is equal to
π R^{2} = 100π
Solve for radius R
R = 10 feet
The circumference of the circular garden is equal to
2 π R = 20 π

If 5/x = 10 and 2/y = 6 then x/y = ?
Solution
If 5/x = 10, then
x/5 = 1/10
We now multiply the left sides and right sides of x/5 = 1/10 and 2/y = 6 to obtain
(x/5)(2/y) = (1/10)(6)
(x/y)(2/5) = 6/10
x/y = (6/10)(5/2) = 3/2

If 2^{m3} / 4^{2m} = 8, then 2m  1 = ?
Solution
Write 4^{2m} and 8 in exponential form with exponent 2
4^{2m} = (2^{2})^{2m} = 2^{4m}
8 = 2^{3}
Use the above in the given equation
2^{m  3} / 2^{4m} = 2^{3}
2^{m  3  4m} = 2^{3}
Which gives
m  3  4 m = 3
3m = 6
m =  2
2m  1 = 2( 2)  1 =  5

In the figure below, ABC is a right triangle. Points B, C and D are collinear; points D, E and F are also collinear and so are points B, A and F. The length of segments DC and DE are equal. What is the size, in degrees, of angle AFE?
.
Solution
Since ABC is a right triangle, then
angle BCA = 90°  50° = 40°
Since DE and DC have equal lengths, triangle DCE is isosceles and therefore
angle DEC = 40°
Angles AEF and DEC are vertical and therefore equal. Hence
angle AEF = 40°
Since angle BAE is equal to 90° then angle FAE is also equal to 90° and therefore
angle AFE = 90°  40° = 50°

Which of the following is an equation of a line perpendicular to the line with equation 3x  6y = 9?
A) y = 2
B) 3x + 6y = 9
C) x  2y = 3
D) 2x + 2y = 3
E) 2x + y = 7
Solution
Find slope of line 3x  6y = 9.
3x  6y = 9
 6y =  3x + 9
y = (1/2) x  3/2
slope = 1/2
Find slope of the given lines.
A) y = 2 , slope = 0
B) 3x + 6y = 9 , 6y =  3x + 9 , y = (1/2)x + 3/2 , slope = 1/2
C) x  2y = 3 , 2y =  x + 2 , y = (1/2) x  1 , slope = 1/2
D) 2x + 2y = 3 , 2y =  2x + 3 , y =  x + 3/2 , slope = 1
E) 2x + y = 7 , y = 2x + 7 , slope = 2
If we multiply the slope of the given line which is 1/2 by the slope in E) which 2 the answer is 1 and therefore the line in E) is perpendicular to the given line.

If a and b are any real numbers, then which of the following expressions is always positive?
A) a
B) a + b
C) a  b + 1/2
D) a^{2} + b^{2}
E) (a + b)^{2}

The geometric figure below consists of a right triangle and 2 semicircles. The diameters of the semicircles are the sides of the triangle. What is the area of the shaded region in square centimeters if the length of the hypotenuse of the triangle is 8 centimeters?
.
A) 64
B) 8π
C) 64π
D) 10π
E) 16

The mean of the numbers a, b, c, d and e is 23. The mean of the numbers a, b, c, d, e and f is 22. What is the value of f?
A) 23
B) 18
C) 22
D) 22.5
E) 20

Which of the following equations corresponds to the graph shown below?
.
A) y = (x  3)^{2}  1
B) y = (x  3)^{2} + 1
C) y = (x  3)^{2} + 1
D) y = x  3
E) y = (x  3)^{2}  1

Functions f and g are defined by f(x) = x^{2} + x and g(x) = √(x + 6). What is the value of g(f(2))?
A) 3
B) 3
C) 7
D) 6
E) 6

The sum of three consecutive integers is equal to 192. What is the product of these numbers?
A) 216000
B) 7077888
C) 576
D) 110592
E) 262080

What are the xcoordinates of the points intersection of the line with equation y = x + 1 and the circle with equation x^{2} + y^{2} = 5
A) 2 , 0
B) 1 , 2
C) 2 , 1
D) 2 , 1
E) 1 , 3

(1/2)sin(2x)(1 + cot^{2}(x)) =
A) tan(x)
B) sin(x)
C) cos(x)
D) cot(x)
E) sec(x)

Find the area of the rectangle ABCD shown in the figure below.
.
A) 2500 / √2
B) 2500
C) 2500 / √3
D) 1250
E) 5000

Solve for x: log_{x}(1024) = 5
A) 1/4
B) 4
C) 1/2
D) 1/8
E) 2

Simplify: 6 ^{3}√32 + 2 ^{3}√108
A) 32
B) 18 ^{3}√2
C) 36 ^{3}√2
D) 18 ^{3}√4
E) 36 ^{3}√4

Evaluate: 1 / (5)^{2}
A)  1 / 25
B) 1 / 25
C) 25
D) 25
E) 1 / 10

