ACT MATH
Triangle Inequality Theorem
The sum of the 2 shortest sides of a triangle is always greater than the length of the third side.
Equilateral
These triangles have three equal sides, and all three angles are 60◦
Pythagorean Triple
Three integers that, as side lengths of a triangle, form a right triangle. Ex. 3/4/5 or 5/12/13
Reverse FOIL
To solve a quadratic such as x 2+bx+c = 0, first factor the left side to get (x+a)(x+b) = 0, then set each part in parentheses equal to zero. E.g., x 2 + 4x + 3 = (x + 3)(x + 1) = 0 so that x = −3 or x = −1.
Reverse FOIL
To solve two linear equations in x and y: use the first equation to substitute for a variable in the second. E.g., suppose x + y = 3 and 4x − y = 2. The first equation gives y = 3 − x, so the second equation becomes 4x − (3 − x) = 2 ⇒ 5x − 3 = 2 ⇒ x = 1, y = 2.
Similar Triangles
Triangles that have the same angle measures but different side lengths. Solve by setting up a proportion.
Complimentary Angles
Two angles that add up to 90°.
Supplementary Angles
Two angles that form a line and add up to 180°.
Similar
Two or more triangles are similar if they have the same shape. The corresponding angles are equal, and the corresponding sides are in proportion. For example, the 3-4-5 triangle and the 6-8-10 triangle from before are similar since their sides are in a ratio of 2 to 1.
Point-Slope Formula
Use if you know the slope and a point on the line
Slope-Intercept Formula
Use if you know the slope and the y-intercept
Midpoint Formula
Used to find the midpoint of a line
Pythagorean Theorem
Used to find the missing side of a right triangle. "c" is always the length of the hypotenuse. a²+b²=c²
Translating word problems: Mathematical equivalent of the word "a number"
V ariable (typically x or y)
Volume of a Sphere
V=(4/3)πr³
Volume of a Prism
V=lwh
Volume of a Cube
V=side³ V=s³
Volume of a Cylinder
V=πr²h
Mode
Value(s) that occurs most frequently!
If You Miss a Question, Re-Solve It First
When you're doing practice questions, the first thing you probably do is read the answer explanation and at most reflect on it a little. This is a little too easy. I consider this passive learning - you're not actively engaging with the mistake you made. Instead, try something different - find the correct answer choice (A-E), but don't look at the explanation. Instead, try to resolve the question once over again and try to get the correct answer.
Interior Angle of a Polygon
Where n is the number of sides
If You Have Content Gaps, Be Ruthless About Filling Them
Within ACT Math, you have to master a lot of subjects. At the high level, you need to know number operations, algebra, geometry, trigonometry, probability, and more. Even within each subject, you have subskills to master. Within algebra, you need to know how to solve equations, how to deal with word problems, properties of functions, etc. Unless you're a math whiz and are already scoring a 34-36, it's unlikely that you've mastered all of these evenly. You probably have different strengths and weaknesses across these subjects.
Negative Exponent
X⁻ⁿ = 1/Xⁿ
Reverse FOIL
You can use Reverse FOIL to factor a polynomial by thinking about two numbers a and b which add to the number in front of the x, and which multiply to give the constant. For example, to factor x 2 + 5x + 6, the numbers add to 5 and multiply to 6, i.e., a = 2 and b = 3, so that x 2 + 5x + 6 = (x + 2)(x + 3)
Finish With Extra Time and Double Check
Your goal at the end of all this work is to get so good at ACT Math that you solve every question and have extra time left over at the end of the section to recheck your work. I'll admit, this is hard for the ACT. You have 60 minutes to solve 60 questions, which of course gives an average of a minute per question. But the last, most difficult questions might take you 2 or 3 minutes each.In high school and even now, I can finish a 60 minute section in 40 minutes or less. I then have 20 minutes left over to recheck my answers at least once.
Difference Of Squares Ex. 16x-49
a^2 − b^2= (a + b)(a − b) a^2 + 2ab + b^2 = (a + b)(a + b) a^2 − 2ab + b^2 = (a − b)(a − b)
Multiply Power Exponents
add exponents and keep the same base EX: X₃ · X₂ = X₅
Area of square
area = (side)²
Circumference of a circle
area = 2πr
Area of parallelogram
area= base x height
Area of rectangle
area= length x width
Area of a Triangle
area= ½ (base)(height)
Quadratic Equation
ax²+bx+c= 0 EX: x²+ 9x+ 18=0
Factoring Difference of squares
a²-b² = (a-b) (a+b)
Distance Formula
d=√(x₁-x₂)² + (y₁-y₂)² (EX: Find distance between (5,6) and (1,3) √(5-1)² + (6-3)²= √(4)² + (3)²= √16 + 9= √25= 5=Distance
Arithmetic Sequences Ex. Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ..
each term is equal to the previous term plus d Sequence: t1, t1 + d, t1 + 2d, Example: d = 4 and t1 = 3 gives the sequence 3, 7, 11, 15
Geometric Sequence
each term is equal to the previous term times r Sequence: t1, t1 · r, t1 · r Example: r = 2 and t1 = 3 gives the sequence 3, 6, 12, 24, . . .
Pythagorean Theorem
for all right triangles (leg₁)² + (leg₂)² = (hypotenuse)² (a²+b²=c²) Special Triangles 3-4-5 30-60-90 5-12-13 45-45-90
Area of Parallelogram
lh (length times height)
Area of Rectangle Ex. What is the length if the area is 336 and the width is 21?
lw (length times width)
Rectangle Solid Volume
lwh (length times width times height)
Raising Power to Powers
multiply the exponents EX: X² · X³ = X⁶
Horizontal Lines
slope = 0; Defined by x=a, where a is a constant
Vertical Lines
slope = undefined; Defined by y=a, where a is a constant
Dividing Power Exponents
subtract exponents and keep same base EX: X₄ ÷ X₂ = X₂
Factors Ex. What are factors of 52?
the factors of a number divide into that number without a remainder Example: the factors of 52 are 1, 2, 4, 13, 26, and 52
Multiples Ex. What are the 4 lowest multiples of 20?
the multiples of a number are divisible by that number without a remainder Example: the positive multiples of 20 are 20, 40, 60, 80
Percents Ex. Maria paid $28.00 for a jacket that was discounted by 30%. What was the original price of the jacket? $36.00 $47.60 $40.00 $42.50
use the following formula to find part, whole, or percent part =percent/100 × whole Example: 75% of 300 is what? Solve x = (75/100) × 300 to get 225 Example: 45 is what percent of 60? Solve 45 = (x/100) × 60 to get 75% Example: 30 is 20% of what? Solve 30 = (20/100) × x to get 150
Reverse FOIL
x^2 + (b + a)x + ab = (x + a)(x + b)
Powers, Exponents, Roots
x^a· x^b = x^a+b (x^a)^b = x^a·b x^0 = 1 x^a/x^b = x^a−b (xy)^a = x^a· y^a √xy =√x ·√y 1/x^b = x^−b (−1)^n = +1, if n is even; −1, if n is odd.
Graphing Trigonometric Functions
y = Asin(Bx - C) + D
Vertical Shift up (ie: shift 2 units up)
y = f(x) + 2
Vertical Shift Down (ie: shift 2 units down)
y = f(x) - 2
Horizontal Shift Left (ie: 3 units left)
y = f(x+3)
Horizontal Shift Right (ie: 3 units right)
y = f(x-3)
Slope Intercept
y=mx+b Ex: y=2/3x + 5
Absolute Value
|-3| = 3
Absolute Value Ex. 3∣x−3∣+5=−2∣x−3∣+9
|x| = +x, if x ≥ 0; −x, if x < 0.
Quadratic Formula
²x = −b ± √b²-4ac/2a EX: Solve using quadratic formula x²+ 9x+ 18=0: x= −9±√(9)² − 4(1)(18)/2(1) −9 ± √81 − 72/2 −9 ± √9/2 −9 ± √3/2 = -9 + 3/2 and -9 -3/2 -9+3/2= -3 and -9-3/2= -6
Area of a Circle Ex.Suppose the diameter of a circle is 2. What is its area?
πr^2
Right Cylinder Volume
πr^2h
Area of a circle
πr²
Rational Exponent
ⁿ√x
Area of Trapezoid
(b1+b2/2)xh (base 1 plus base 2 divided by 2) times height
Arc Length of a Sector (Circle)
(n/360)(2πr), where n is the central angle.
Area of a Sector (Circle)
(n/360)(πr²), where n is the central angle.
Length of Arc
(n◦/360◦) · 2πr
Area of Sector
(n◦/360◦) · πr^2
Distance Formula
(think Pythagorean theorem)
FOIL (2 + 5x)(11 + 12x)
(x + a)(x + b) = x^2 + (b + a)x + ab Simplifying (2 + 5x)(11 + 12x) Multiply (2 + 5x) * (11 + 12x) (2(11 + 12x) + 5x * (11 + 12x)) ((11 * 2 + 12x * 2) + 5x * (11 + 12x)) ((22 + 24x) + 5x * (11 + 12x)) (22 + 24x + (11 * 5x + 12x * 5x)) (22 + 24x + (55x + 60x2)) Combine like terms: 24x + 55x = 79x (22 + 79x + 60x2)
Midpoint Formula
(x₁+x₂)/2, (y₁+y₂)/2
Function Transformations: Amplitude decrease of f(x) in which all values of y are multiplied by 0.5. A horizontal Stretch.
0.5(f(x))
Circumference of a Circle Ex. What is the circumference if the diameter was 50?
2πr
Function Transformations: Amplitude increase of f(x) in which all values of y are multiplied by 3. A vertical stretch.
3(f(x))
Complex Numbers (−5−4i)⋅(1+4i) (−5−4i)⋅(1+4i)= (−5⋅1)+(−5⋅4i)+(−4i⋅1)+(−4i⋅4i)
A complex number is of the form a + bi where i 2 = −1. When multiplying complex numbers, treat i just like any other variable (letter), except remember to replace powers of i with −1 or 1 as follows (the pattern repeats after the first four): i^0 = 1 i^4 = 1 i^1 = i i^5 = i i^2 = −1 i^6 = −1 i^3 = −i i^7 = −i
Function
A function is a rule to go from one number (x) to another number (y), usually written y = f(x). The set of possible values of x is called the domain of f(), and the corresponding set of possible values of y is called the range of f(). For any given value of x, there can only be one corresponding value y.
Prime Number
A positive number that can only be divided by 1 AND itself. ie: 2, 3, 5, 7, 11, 13, 17, 19, 23, 279, 31, 37...! (1 is NOT a prime number. 2 is the smallest prime number, and prime numbers cannot be negative or even)
Equilateral Triangle
A triangle in which all three sides are equal and all three interior angles are 60°.
Isosceles Triangle
A triangle with two equal sides. Base angles (angles across from the congruent sides) are also equal.
What is the smallest value of x that satisfies the equation x(x + 4) = -3
A) -1 B) 0 C) 1 D) 3 E) -3
If x + 4y = 5 and 5x + 6y = 7, then 3x + 5y = ?
A) 12 B) 6 C) 4 D) 2 E) 1
Which of the following is equal to √45
A) 15 B) 5√3 C) 9√5 D) 3√5 E) 3
What is the slope of the line 4x = -3y + 8
A) 4 B) -3/4 C) -4/3 D) 2 E) 8
If a = 3, then 2 / (1/7 + 1/a) = ?
A) 5 B) 21/10 C) 20 D) 10 E) 21/5
A group of 7 friends are having lunch together. Each person eats at least 3/4 of a pizza. What is the smallest number of whole pizzas needed for lunch?
A) 7 B) 5 C) 6 D) 28 E) 21
The measures of angles A, B and C of a triangle are in the ratio 3:4:5. What is the measure, in degrees, of the largest angle?
A) 75° B) 15° C) 12° D) 90° E) 60°
Collinear Points
A, B, and C are all collinear points
A car averages 27 miles per gallon. If gas costs $4.04 per gallon, which of the following is closest to how much the gas would cost for this car to travel 2,727 typical miles?
A. $ 44.44 B. $109.08 C. $118.80 D. $408.04 E. $444.40
What is the value of x when 2x + 3 = 3x - 4 ?
A. -7 B. negative one fifth C. 1 D. one fifth E. 7
A typical high school student consumes 67.5 pounds of sugar per year. As part of a new nutrition plan, each member of a track team plans to lower the sugar he or she consumes by at least 20% for the coming year. Assuming each track member had consumed sugar at the level of a typical high school student and will adhere to this plan for the coming year, what is the maximum number of pounds of sugar to be consumed by each track team member in the coming year?
A. 14 B. 44 C. 48 D. 54 E. 66
Sales for a business were 3 million dollars more the second year than the first, and sales for the third year were double the sales for the second year. If sales for the third year were 38 million dollars, what were sales, in millions of dollars, for the first year?
A. 16 B. 17.5 C. 20.5 D. 22 E. 35
Area of a Parallelogram
A=(base)(height) or A=bh
Area of a Rectangle
A=(length)(width) A=lw
Area of a Square
A=(side)(side) A=s²
Area of a Triangle
A=½(Base)(Height) A=½bh
Area of a Trapezoid
A=½(h)(b₁+b₂)
Area of a Polygon
A=½aP, where a is the apothem and P is the perimeter.
Area of a Circle
A=πr²
45-45-90 Special Triangle
Always in the ratio 1:1:√2 Isosceles right triangle
30-60-90 Special Triangle
Always in the ratio 1:√3:2
Right Angle
An angle that measures 90°.
Isosceles
An isosceles triangle has two equal sides. The "base" angles (the ones opposite the two sides) are equal.
Angles
Angles on the inside of any triangle add up to 180◦. The length of one side of any triangle is always less than the sum and more than the difference of the lengths of the other two sides. An exterior angle of any triangle is equal to the sum of the two remote interior angles.
Even Integer
Any integer that can be divided by 2 without a remainder (including zero!) ie: 2, 20, -30
Odd Integer
Any integer that cannot be divided by 2 without a remainder. ie: −111, −57, −1, 1, 67!
Real Number
Any number that can be found on a number line. Excludes infinity and imaginary numbers. ie: All integers, rational numbers, and irrational numbers!
Rational Number
Any number that can be written as a fraction (ratio of integers). ie: 0 ,4,12 ,−5,.20!
Irrational Number
Any number that cannot be written as a fraction.!
Integers
Any number that is not a decimal or a fraction. ie: -30, 1, 2, 50
Whole Number
Any number that is not a negative or a fraction. ie: 0, 2, 37, 455
Circumference of a Circle
C=2πr or C=πd
Linear Functions
Consider the line that goes through points A(x1, y1) and B(x2, y2). Distance from A to B:(x2 − x1)^2 + (y2 − y1)^2 Mid-point of the segment AB: (x1 + x2/2), (y1 + y2 /2) Slope of the line: y2 − y1 /x2 − x1=rise/run
Diagonal of a Cube
Diagonal = side√3 D=s√3
Diagonal of a Square
Diagonal=side(√2) D=s√2
Translating word problems: Mathematical equivalent of the word "per"
Divide (÷)
Translating word problems: Mathematical equivalent of the word "percent"
Divided by 100
Parallel Lines
Eight angles are formed when a line crosses two parallel lines. The four big angles (a) are equal, and the four small angles (b) are equal.
Translating word problems: Mathematical equivalent of the word "is"
Equals (=)
Consecutive Even Integers
Even numbers that follow each other on a number line. ie: −6,−4,−2,0... variable form: n, n+2, n+4, n+6
Understand Your High Level Weakness: Content or Time Management
Every student has different flaws in ACT Math. Some people aren't comfortable with the underlying math material. Others know the math material well, but can't solve questions quickly enough in the harsh time limit.
Corresponding Angles (Parallel Lines)
Ex. 1 & 5 are congruent
Alternate Exterior Angles (Parallel Lines)
Ex. 1 & 8 are congruent
Same Side Interior Angles (Parallel Lines)
Ex. 3 & 5 add up to 180º
Alternate Interior Angles (Parallel Lines)
Ex. 3 & 6 are congruent
How many irrational numbers are there between 1 and 6 ?
F. 1 G. 3 H. 4 J. 10 K. Infinitely many
What is the greatest common factor of 42, 126, and 210 ?
F. 2 G. 6 H. 14 J. 21 K. 42
When x = 3 and y = 5, by how much does the value of 3x^2 - 2y exceed the value of 2x^2 - 3y ?
F. 4 G. 14 H. 16 J. 20 K. 50
FOIL
First, Outside, Inside and Last. EX: (x-3) + (x+5) = x² + 5x -3x -15= x² + 2x-15
Perpendicular Lines
Form 90 degree angles; Slopes are negative reciprocals
Vertical Angles
Formed by 2 intersecting lines or segments. Always congruent.
Fundamental Counting Principle Oula is going on an outdoor expedition with his family. The expedition will include a hunting event, a fishing event, a hiking event, and a camping event. There are 4 hunting, 7 fishing, 6 hiking, and 3 camping events for Oula's family to choose from.
If an event can happen in N ways, and another, independent event can happen in M ways, then both events together can happen in N × M ways. (Extend this for three or more: N1 × N2 × N3 . . .)
Fundamental Counting Principle
If an event can happen m ways and another, independent event can happen n ways, then both events can happen in m ∗ n ways.
Proportionality in Triangles
In every triangle, the longest side is opposite the largest angle and the smallest side is opposite the smallest angle.
Percent
Means "divided by 100"
Median
Middle point of an ordered list!
Translating word problems: Mathematical equivalent of the word "of"
Multiply (×)
Consecutive Integers
Numbers that directly follow each other on a number line. ie: −4,−3,−2,−1... or 3,4,5,6... variable form: n, n+1, n+2, n+3
Consecutive Odd Integers
Odd numbers that follow each other on a number line. ie: −5,−3,−1,1... variable form: n, n+2, n+4
Do a Ton of Practice, and Understand Every Single Mistake
On the path to perfection, you need to make sure every single one of your weak points is covered. Even one mistake on all of ACT Math will knock you down from an 36.
Angles in a Parallelogram
Opposite angles are equal.
Intersecting Lines
Opposite angles are equal. Also, each pair of angles along the same line add to 180◦ . In the figure above, a + b = 180◦ .
Perimeter of a Rectangle
P=2(length)+2(width) P=2l+2w
Order of Operations
Parentheses! Exponents! Multiplication! Division! Addition! Subtraction!
PEDMAS
Parenthesis, Exponents, Multiplication, Division, Add, Subtract. EX: 9-2 x (5-3)² + 6÷3= 9-2 x (2)² + 6÷3= 9-2 x (4) +6÷3= 9-8 +2= 9-6= 3
Percent Formula
Part=Percent x Whole
Linear Functions
Point-slope form: given the slope m and a point (x1, y1) on the line, the equation of the line is (y − y1) = m(x − x1). Slope-intercept form: given the slope m and the y-intercept b, then the equation of the line is y = mx + b. To find the equation of the line given two points A(x1, y1) and B(x2, y2), calculate the slope m = (y2 − y1)/(x2 − x1) and use the point-slope form. Parallel lines have equal slopes. Perpendicular lines (i.e., those that make a 90◦ angle where they intersect) have negative reciprocal slopes: m1 · m2 = −1
Slope
Rate of change of a line; rise over run; change in y /change in x
Surface Area of a Prism
SA=2(lw+lh+wh) SA=2B+Ph, where B is the area of the base, P is the perimeter of the base, and h is the height of the prism.
Surface Area of a Cylinder
SA=2πr²+2πrh
Surface Area of a Sphere
SA=4πr²
Parallel Lines
Same slope
Slope
Slope= Rise/Run=Change in Y/Change in X (y₂-y₁/x₁-x₂) (EX: Find slope of (5,6) and (1,3) 6-3/5-(-1)= 3/6= 1/2
Reverse FOIL
Solving two linear equations in x and y is geometrically the same as finding where two lines intersect. In the example above, the lines intersect at the point (1, 2). Two parallel lines will have no solution, and two overlapping lines will have an infinite number of solutions.
SOHCAHTOA
Some Old Hippy Caught Another Hippy Tripping On Acid Sin (opposite/hypotenuse) Cos (Adjacent/hypotenuse) Tan (opposite/adjacent)
Sum of Interior Angles of a Polygon
Sum=180(n-2), where n is the number of sides.
Remainder
The amount left over when a quantity is divided by another number.
Degree Measure of a Circle
The central angles of a circle add up to 360°.
Absolute Value
The distance from 0 (an absolute value takes any number and makes it positive)!
Degree Measure of a Triangle
The inside angles of a triangle always add up to 180°.
Degree Measure of a Quadrilateral
The interior angles of a quadrilateral add up to 360º.
Multiples
The numbers that divide evenly by a given number without a remainder. Multiples of 30 are 30, 60, 90, 120
Factors
The numbers that divide evenly into a given number without a remainder. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
