These notes are based upon the ETS Super Power Pack Textbooks
Analytical Writing
Two sections:
- Analyze an issue
- Analyze an argument
30 minutes each.
Issue looks for language, prose, and appropriate use of examples. Argument looks for reason and analysis. Combined to give one score out of 6.
Verbal Reasoning
Reading Comprehension
Understanding words, sentences, and paragraphs.
Inferring information and drawing conclusions.
Evaluating strengths and weaknesses.
Half of questions will be based on passages.
Quantitative Reasoning
- Odd numbers can be represented by the equation 2n + 1
- Squaring both sides is not reversible. Simple operations are reversible.
- Solve for the relationship imposed by the conditions. If no relationship is imposed then answer D
- The relationship flips on multiplying or dividing by a negative number.
- Distance between two points on the number line is equal to absolute difference in values
- 1 Mile = 5280 Feet
- 1 Hour = 3600 Seconds
- Evenly split numbers should always be rounded up if positive or down if negative
- Modulos and remainders are always positive
- Distances are always positive
- Quadrants are: 2 | 1 3 | 4
- Disjoint/mutually exclusive sets have no common members
- The empty set is a subset of every set
- Lists of numbers are 1-indexed
- Sets are declared inclusively by default. “From 1985 to 2005” refers to 21 years.
- Profit refers to Gross Profit (Revenue - COGS)
Math Review
Arithmetic
Multiplying odd integers always results in an odd number. Summing an odd number of odds results in an odd number and summing an even number of odds results in an even number.
Error in arithmetic exercise 1. g)
Algebra
The degree of a term is the sum of the exponents of the variables in a term. 5xy^2 is of degree 3.
Identities:
- \(ca + cb = c * (a+b)\)
- \(ca - cb = c * (a-b)\)
- \((a+b)^2 = a^2 + 2ab + b^2\)
- \((a-b)^2 = a^2 - 2ab + b^2\)
- \(a^2 - b^2 = (a+b)(a-b)\)
- \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
- \((a-b)^3 = a^3 + 3a^2b + 3ab^2 - b^3\)
Exponent Rules:
- \(x^a * x^b = x^{a+b}\)
- \(x^0 = 1\)
- \(x^a * y^a = (xy)^a\)
- \({x \over y}^a = {x^a \over y^a}\)
- \((x^a)^b = x^{a*b}\)
Linear equations are polynomial equations of degree 1 where each term is either a constant or a single variable multiplied by a coefficient.
Every linear equation with two or more variables has infinitely many solutions.
Systems of linear equations can be solved either by substitution or by elimination.
Quadratic equations can be solved by the quadratic formula: \[ x = {-b \pm \sqrt{b^2 - 4ac} \over 2a } \] or the simplified quadratic formula where a = 1: \[ {-b \over 2} \pm \sqrt{({b \over 2})^2 - c} \]
To solve an inequality find the set of numbers for which the inequality is true.
The direction of the inequality is reversed when multiplied or divided by a negative constant.
The domain of the function is the set of permissible inputs to that function.
Simple Interest: \[ V = P(1+{rt \over 100}) \]
Compound Interest (n times per year): \[ V =P(1+{r \over 100n})^{nt} \]
Directly proportional means \(a=kb\)
Inversely proportional means \(a=k \over b\)
Two lines are parallel if their slopes are equal. Two lines are perpendicular if their slopes are negative reciprocals of each other.
A circle is an equation of the form \((x-a)^2 + (y-b)^2 = r^2\) with its center at the point (a, b) and with radius r > 0.
Geometry
Congruent line segments are line segments with equal length.
When two lines intersect the opposite angles will have equal measure. Angles of equal measure are called congruent.
Two intersecting lines are perpendicular if each of their intersecting angles is 90\(\degree\). Perpendicular lines are denoted \(k \perp m\).
Acute angles are measures of \(\lt90 \degree\). An angle with measure between \(90 \degree\) and \(180 \degree\) is obtuse.
Two lines in the same plane that do not intersect are parallel. Denoted by \(k \parallel m.\)
A polygon is a closed figure formed by three or more line segments in the same plane.
A convex polygon has interior angles that are all less than 180 \(\degree\)
If a polygon has n sides it can be divided into n-2 triangles. The sum of the interior angles of an n-sided polygon is \((n-2)(180\degree)\)
The length of each side of a triangle must be less than the sum of the lengths of the other two sides.
An equilateral triangle has three equal interior angles of \(60 \degree\)
An isosceles triangle has two equal angles and accordingly two congruent sides.
A right triangle has an interior angle of \(90 \degree\)
The isosceles right triangle has two \(45\degree\) angles and one \(90\degree\) angle.
The \(30\degree-60\degree-90\degree\) triangle is half of an equilateral triangle. Its sides have ratios of \(1 : \sqrt{3} : 2\)
The height of a triangle is the length of the line segment that is perpendicular with the base and intersects the opposite vertex of the base.
Two triangles are congruent if they have the same angles and sides.
Two triangles are similar if they have the same angles but different sides.
Parallelograms have parallel opposite sides. All rectangles are parallelograms.
Trapezoids have at least one pair of opposite sides. The parallel sides are called the bases of the trapezoid.
For all parallelograms the area is the product of the base and height.
The area of a trapezoid is: \[ A = {1 \over 2} * (b_1 + b_2) * (h) \]
Two circles are congruent if they have the same radius.
A line segment joining two points on the circle is called a chord.
The ratio of circumference to diameter is \(\pi\). So: \[ {C \over d} = \pi \] \[ C = \pi * d \] \[ C = 2 \pi r \]
A central angle of a circle is an angle with its vertex at the center of the circle.
The area of a circle is \(\pi * r^2\)
A line is tangent to a circle if it intersects with a single point on the circle and is perpendicular to the radius at that point.
A polygon is inscribed in a circle if its vertices lie on the circle. The circle is said to be circumscribed by the polygon.
If a triangle is inscribed in a circle and one side passes through the center of the circle then the length of that line will be the diameter and the triangle will have a right angle.
A polygon is circumscribed about a circle if each side is tangent to the circle. The circle is said to be inscribed in the polygon.
The inscribed angle of two chords meeting on a circle is half the central angle that joins those two points.
There are 12 edges and 8 vertices to a rectangular prism.
A rectangular solid with six square faces is a cube.
The volume of a rectangular solid is \(length * width * height\)
The volume of a right angled cylinder is: \[ V = \pi r^2 h \] Its surface area is: \[ A = 2(\pi r^2) + 2 \pi r h \]
Data Analysis
To find the mean:
- order the numbers from least to greatest
- The median is the \(n + 1 \over 2\)th value if the number of values is odd or the average of the bounding values if the number of values is even.
The range is the difference between the greatest number and the least number.
The interquartile range is the difference between the third and first quartiles.
Standardization is the process of taking the difference between an observation and the mean and dividing it by the standard deviation.
In any group of data most of the data are within 3 standard deviations of the mean.
A list is like a set but its elements are ordered and elements can be repeated.
The number of elements in a set is denoted as \(|s|\)
Two sets are disjoint if they have no elements in common.
The multiplication principle states that if there are multiple independent and sequential choices, the number of possible combinations is the product of the number of choices for each choice.
Permutations: \(n! \over (n - k)!\)
Combinations: \(n! \over k! * (n - k)!\)
\(P(E \vee F) = P(E) + P(F) - P(E \wedge F)\)
If E and F are independent then: \(P(E \wedge F) = P(E)*P(F)\)
68 - 95 - 99.7 rule for one two and three standard deviations from the mean