# Page 1 - Math Study Guide for the PERT

## General Information about the PERT Math Test

The problem with any skill, including math, is that you may learn things easily, but have trouble remembering them later. The PERT Math section is designed to test many math skills that you may not have used for a while. Be sure to refresh your memory before the test so that your score truly reflects your skills.

## Terms to Know

You will encounter certain math terms while taking the PERT. Knowing what they are and how to work with them will help you answer more questions correctly.

### Order of Operations

Simple math operations if presented in a series can be mind-boggling, such as this one:

\[4 \times 3 - 15 \div 3 \times 2 +1\]Follow the rule abbreviated as PEMDAS. Perform operations inside Parentheses first, evaluate Exponents next, perform Multiplication and Division from left to right, then Addition and Subtraction, also from left to right. Follow that order to arrive at the answer of 3.

One easy way to remember PEMDAS is to learn the phrase: “Please Excuse My Dear Aunt Sally.”

### Exponents and Square Roots

Exponents, powers, and indices — all 3 terms mean the same thing. “Squared” and “cubed” mean raised to the power of 2 and 3, respectively.

A positive exponent (for instance 5 in the term \(x^5\)) means x multiplied by itself five times.

A negative exponent (for instance -5 in the term \(x^{-5}\)) means 1 divided by x five times, or simply the reciprocal of the term but with a positive exponent (\(\frac {1}{x^5}\)).

A fractional exponent refers to radical expressions. Here are important rules to remember when dealing with roots or radical expressions:

\[\sqrt {xy} = (xy)^{½}\]Example:

\[\sqrt {9a} = (9a)^{½}\] \[\sqrt {xy} = \sqrt {x} \cdot \sqrt {y}\]Example:

\[\sqrt {9a} = \sqrt {9} \cdot \sqrt {a} = 3 \sqrt{a}\] \[\frac {\sqrt x}{\sqrt y} = \sqrt {\frac {x}{y}}\]Example:

\[\frac {\sqrt 100}{\sqrt 25} = \sqrt {\frac {100}{25}} = \sqrt {4} = 2\]is the same as:

\[\frac {\sqrt 100}{\sqrt 25} = \frac {10}{5} = 2\]### Prime Numbers

A prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself. Prime numbers are often encountered in math questions involving prime factorization, or finding the prime factors of another number. The prime factorization of 90, for example, is: 2, 3, 3, 5 (notice that \(2*3^2*5 = 90\)).

### Percent

A percent indicates a number out of 100; so 5% means 5 out of 100, or \(\frac{5}{100}\). You can think of percent as “per 100.” It can also be understood as a part of a whole. So to find out what percent of 60 is 20, write it first as \(\frac{20}{60}\). Divide to find the ratio and then multiply by 100 to calculate the percentage. The correct answer is 33.3%

### Equivalence

Math questions involving fractions often entail reducing answers to their simplest form. The fraction \(\frac {15}{45}\) can be reduced to \(\frac {1}{3}\).

This can also be seen with the fractions \(\frac{4}{12}\), \(\frac{6}{18}\) and \(\frac{2}{6}\). They are equivalent fractions because they all reduce to the same fraction. Equivalent fractions can be found by dividing the numerator and denominator by a common factor, or by multiplying the numerator and denominator by a fraction equal to 1, \(\frac{3}{3}\), for example.

### Slope of a Line

The slope of a line is defined as the change in y over the change in x. The formula for slope (m) is:

\(m = \frac {(y - y_1)}{(x - x_1)}\), where \((x, y)\) and \((x_1, y_1)\) are 2 points on the line.

### Functions

In math, a function describes the relationship between inputs and outputs. A function has an input, a relationship and an output. In the function:

\[f(x) = x^3\]we see that for every input of \(x\), there is an output of \(x^3\).

### Absolute Value

The absolute value of a number is the number’s distance from zero. Consequently, absolute values are always positive. Here are important properties to remember when dealing with absolute values:

\[\mid x \cdot y \mid = \mid x \mid \cdot \mid y \mid\]and

\[\mid u \mid = x\]is the same as:

\[u = \pm x\]Solve \(\mid y + 3 \mid = 8\)

\[y + 3 = \pm 8\] \[y = -11\] \[y = 5\]### Standard Form

Scientific notation, or “standard form” in Britain, is a method for writing very large or very small numbers. Scientific notation contains two parts: the first digit followed by a decimal point and the rest of the digits x 10 raised to an exponent which indicates the number of places the decimal is moved

The number 123,000,000 is written as \(1.23 \times 10^8\).

## Specific Skills to Practice

The better you are at working with numbers, the more successful you will be on the PERT Math section. Some of these may have been taught in past math classes, but it’s harder to remember them as time passes. Practice these skills until they come back to you easily.

### Working with Fractions

You will need to know terms, like numerator and denominator. In a fraction, the numerator is the top number, or the number that represents the part. The bottom number is the denominator and it represents the whole.

#### Simplify/Lowest Terms

Simplifying fractions make it easier to work with them. To simplify \(\frac {54}{81}\):

\[\frac {54}{81} = \frac {2 \cdot 3 \cdot 9}{3 \cdot 3 \cdot 9} = \frac {2}{3}\]#### Adding and Subtracting

To add fractions with the same denominators, add the numerators and copy the denominator.

\[\frac {2}{5} + \frac {1}{5} = \frac {3}{5}\]To add fractions with different denominators, find the least common denominator, express the fractions in equivalent fractions having the least common denominator, then proceed to addition.

\[\frac {3}{7} + \frac {1}{3} = \frac {9}{21} + \frac {7}{21} = \frac {16}{21}\]Subtraction of fractions follows the same procedure as addition involving fractions.

#### Multiplying

The easiest operation for fractions is multiplication. Simply multiply the numerators then multiply the denominators. Reduce to lowest terms if applicable.

#### Dividing

To perform division involving two fractions, rewrite the second fraction in its reciprocal form and proceed to multiplication.

\[\frac {5}{9} \div \frac{4}{3} = \frac {5}{9} \cdot \frac{3}{4} = \frac {15}{36} = \frac {5}{12}\]### Solving Equations and Inequalities

#### Linear Equations

Linear equations are the algebraic expressions of lines. A linear equation involves one or more variables, with exponents of 0 or 1. Test questions may ask where two lines meet, or the coordinates (x, y) where the two equations are true.

#### Linear Inequalities

Linear inequalities use the symbols \(\lt\), \(\gt\), \(\le\) and \(\ge\) instead of the equal sign. Manipulating the inequality is similar to equations except for the following:

multiplication and division involving a negative sign changes the direction of the sign of inequality Swapping the values on the right and left of the inequality has the same effect

*Example:*

Solve for the value of \(a\):

\[2(3 - a) \le 8\] \[6 - 2a \le 8\] \[-2a \le (8 - 6)\] \[(-\frac{1}{2} \cdot (-2a) \ge 2 \cdot (-\frac{1}{2})\] \[a \ge -1\]#### Quadratic Equations

Quadratic equations are equations of degree 2, or equations that have their variables “squared”, hence, the term “quad”. The standard form of a quadratic is:

\[ax^2 + bx + c = f(x)\]To solve for the unknown variable x, or find the zeroes of the equation, first set the function equal to zero and then you may factor the expression or use the quadratic formula:

\[x = \frac {-b \pm \sqrt {b^2 - 4ac}}{2a}\]#### Literal Equations

Solving a formula or equation for a certain variable is called solving a literal equations. The formula for the area of a rectangle, for instance, is given by:

\[A = bh\]where A is the area, b is the base and h is the height.

To solve the given formula for the variable h, the formula becomes:

\[h = \frac {A}{b}\]That’s how you solve a literal equation for a variable. It’s like using a formula in a form other than its standard form.

### Evaluating Expressions in Algebra

Algebraic expressions consist of constants, variables and exponents, and are joined together by mathematical operations. They can be monomials, binomials or polynomials.

### Polynomials

#### Factoring

Factoring a polynomial is one way of finding its roots, or solving for the values of the variable.

Find the roots of \(x^2 - 9x + 20\)

First, we factor:

\[(x - 4)(x - 5) = 0\]The factors are: \((x - 4)\) and \((x - 5)\)

And the roots of the polynomial are: 4 and 5. This is because \(x - 4 = 0\) yields \(x = 4\) and \(x-5 = 0\) yields \(x = 5\)

#### Simplifying

Simplify polynomials by combining like terms. “Like terms” refers to terms that have variables and exponents that match.

Simplify the polynomial \(3x^2 + 4x - 5x^2 + 16 - 7x\)

By combining like terms, we find: \(-2x^2 - 3x + 16\)

#### Adding and Subtracting

Add polynomials by placing like terms together then adding them up. This can be done with two or more polynomials. Addition is quite simple.

With subtraction, you have to be more careful because the signs of the terms in the subtrahend will all change.

\[(3a^2 + 4a + 25) - (2a^2 + 4a - 5)\]The answer is \(a^2 + 30\)

#### Multiplying

Multiply each term of the first polynomial with each term of the second polynomial. Add all the resulting terms and simplify.

#### Dividing

Dividing a polynomial with another polynomial can be simplified by first factoring each polynomial, cancelling out similar terms or factors, and then simplifying.

### Dividing by Monomials and Binomials

It is easiest to divide polynomials by a monomial. Do so by dividing each term of the polynomial with the divisor.

\[\frac {(12x^3 + 8x^2 - 2x + 6)}{4x} = 3x^2 + 2x - \frac{1}{2} + \frac{3}{2x}\]Dividing by a binomial or another polynomial may entail long division. This can sometimes be avoided by first factoring the polynomial (if possible).

\[\frac {x^2 - 4x - 45}{x^2 + 6x +5} = \frac {(x-9)(x+5)}{(x+1)(x+5)} = \frac {x-9}{x+1}\]### Applying Algorithms and Concepts

All the concepts and sets of rules learned come into play as they are applied to various mathematical problems. And, the best way to become a master of these concepts is to practice.

### Working in the Coordinate Plane

#### Translating Lines

Linear equations are equations that have constants and variables of the first degree. The standard form for writing linear equations is:

\[Ax + By + C = 0\]where A and B are simple coefficients.

Linear equations can also be written in slope-intercept form:

\[y = mx + b\]where m is the slope and b is the point where the line crosses the y-axis (y-intercept).

Another form for a linear equation is point-slope form:

\[y - y_1 = m(x - x_1)\]where m is slope and \((y - y_1)\) shows the change in rise or y, and \((x - x_1)\) shows the change in run or x.

#### Inspecting Equations

Just by looking at a linear equation, you can deduce several facts about it. A positive slope indicates a line that increases from left to right on the cartesian plane. A negative slope decreases from left to right on the cartesian plane.

Parallel lines have slopes that are equal. A line perpendicular to another line has a slope that is negative of the reciprocal of the other line’s slope, \(-2\) and \(½\), for example.

To find where a line crosses the x-axis, solve the equation when \(y = 0\). To solve for the y-intercept, solve the equation when \(x = 0\).

The linear equation \(2y = -6x + 3\) can be written as \(y = -3x + \frac{3}{2}\)

By inspection, we know that this line slopes downward from left to right. It crosses the y-axis at 1.5 and crosses the x-axis at 0.5.

### Linear Equations

#### Simultaneous

A system of linear equations can be solved to determine the number of times 2 lines intersect, and the point where this intersection occurs. They can be solved using substitution, elimination, or by graphing.

#### Two Variables

The two variables in a linear equation can be solved as long as there are two equations given.

Find the coordinates of the point of intersection of lines \(2y - 5x = 12\) and \(y = 3x + 2\).

The coordinates where the two lines meet: (8, 26). This can be verified by substituting the x and y values into either equation and confirming the equality.

### Creating/Identifying the Equation to Solve a Word Problem

Reading a mathematical problem and being able to translate it into English takes a lot of practice. These tips can help you think clearly:

- Read the entire question first, rereading if necessary.
- Note the given information.
- Identify what is being asked. Assign a variable for the unknown.
- Make a sketch to help you visualize.
- Focus on keywords that have mathematical meaning.
- Recall your formulas. You may need to solve for literal equations instead of the obvious or standard form of the equation.

It is very important to understand the concepts behind math and not merely memorize formulas.

## Tips and Tricks

Be sure to review all levels of math concepts, including those that may seem easy to you. If you are currently in a higher-level math course, you may have forgotten the procedures for solving lower-level problems and it’s a good idea to review them for this test. If you miss lower level questions, the test will automatically steer you toward even easier questions and you will not have a chance to prove your skills at the higher levels.