Math Study Guide for the PERT
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General Information
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.
You will encounter various math terms while taking the PERT. Knowing what they are and how to work with them will help you answer more questions correctly. So, pay attention to the math terms discussion in this study guide, as they will be used in questions.
Numeration Concepts
Numeration includes the contents of our number system and all of the various operations that can be performed on them. There are many specific procedures and guidelines for working with numbers.
The Order of Operations
When presented in a series, simple math operations can be mind-boggling. Consider this one:
\[4 \times 3 - 15 \div 3 \times 2 +1\]While this might not look particularly complex, to get the correct answer, you must be sure to follow the order of operations: parentheses, exponents, multiplication, division, addition, and subtraction. This order is commonly abbreviated as PEMDAS. Perform operations inside parentheses first, evaluate exponents next, perform multiplication and division from left to right, and finally, do addition and subtraction, also from left to right.
If you follow that order with the problem above, you will arrive at the answer, \(3\). This process is shown below:
\[4 \times 3 - 15 \div 3 \times 2 +1\] \[12 - 5 \times 2 +1\] \[12 - 10 +1\] \[3\]One easy way to remember PEMDAS is to learn the phrase, “Please Excuse My Dear Aunt Sally.”
Prime and Composite 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, which means finding the prime factors of a number. The prime factorization of \(90\), for example, is \(2, \,3, \,3, \,5\) (\(2 \cdot 3^2 \cdot 5 = 90\)).
A composite number is simply a number that is not prime. It has divisors besides \(1\) and itself.
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:
\[\vert x \cdot y \vert = \vert x \vert \cdot \vert y \vert\]and
\[\vert u \vert = x\]is the same as:
\[u = \pm x\]\[y + 3 = \pm 8\] \[\begin{array}{ll} & y= +8 - 3 \quad \quad & y=-8-3\\ &y = 5 \quad \quad & y = -11\\ \end{array}\] \[y = -11\] \[y = 5\]Solve \(\vert y + 3 \vert = 8\)
Scientific Notation
Scientific notation is a method for writing very large or very small numbers. It contains two parts:
- the first non-zero digit followed by a decimal point
- the rest of the digits \(\times 10\) raised to an exponent that indicates the number of places the decimal is moved
In scientific notation, the number \(123 \text{,}000 \text{,}000\) is written as \(1.23 \times 10^8\).
Numbers less than \(1\) (for example, \(0.00037\)), will have negative powers of \(10\). . For \(0.00037\), to get the first non-zero digit to the left of the decimal point, the point will have to be moved four places to the right. We write the result as \(3.7 \times 10^{-4}\).
Basic rule: If the decimal is moved to the left, the \(10\) will have a positive exponent, and if the decimal is moved to the right, the \(10\) will have a negative exponent.
Exponents and Square Roots
Exponents, powers, and indices are three terms that mean the same thing. They are all names for the small, raised number after a letter or a number, such as the \(2\) in the number \(4^2\). Squared and cubed mean raised to the power of \(2\) and \(3\), respectively.
A positive exponent (for instance, the \(5\) in the term \(3^5\)) means the product of five \(3\)s.
A negative exponent (for instance, \(-5\) in the term \(3^{-5}\)) means \(1\) divided by the product of five \(3\)s (\(3^{-5}\) is simply the reciprocal of \(3x^5\)).
A fractional exponent refers to radical expressions. Here are important rules to remember when dealing with roots or radical expressions:
\[\sqrt {xy} = (xy)^{\frac{1}{2}}\] \[\sqrt[3] {xy} = (xy)^{\frac{1}{3}}\]Example:
\[\sqrt {9a} = (9a)^{\frac{1}{2}}\] \[\sqrt {xy} = \sqrt {x} \cdot \sqrt {y}\]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:
\[\frac {\sqrt 100}{\sqrt 25} = \frac {10}{5} = 2\]All Study Guides for the PERT are now available as downloadable PDFs