This page of our PERT study guide covers operations and fractions. You will learn the step-by-step method to apply the order of operations when solving equations. We also demonstrate how to perform arithmetic with fractions.
Order of Operations (PEMDAS)
This principle requires operations to be completed in a specific sequence:
- Parentheses
- Exponents
- Multiplication and Division (done at the same time from left to right)
- Addition and Subtraction (done at the same time from left to right)
For example, in the expression $3 + 5 * (2^2) − 1$, solve the expression inside parentheses first, then evaluate the exponent, and proceed to multiplication, subtraction, and addition.
Properties of Operations
Commutative Property (addition/multiplication): For any two numbers $a$ and $b$, the operation’s order does not change the result. For example:
$a + b = b + a \, $ and $ \, ab = ba$
Associative Property: Grouping of numbers does not affect the sum or product. For example:
$(a + b) + c = a + (b + c)$
Distributive Property: Multiplying a number by a group of numbers added together is the same as doing each multiplication separately. For example:
$a(b + c) = ab + ac$
Arithmetic with Fractions
Adding/Subtracting Fractions: To add or subtract fractions, find a common denominator. Example:
$\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$
Multiplying Fractions: Multiply the numerators together and the denominators together. Example:
$\dfrac{1}{2} \ast \dfrac{2}{3} = \dfrac{1 \ast 2}{2 \ast 3} = \dfrac{1}{3}$
Dividing Fractions: Multiply by the reciprocal of the second fraction. Example:
$\dfrac{1}{2} \div \dfrac{2}{3} = \dfrac{1}{2} \ast \dfrac{3}{2} = \dfrac{3}{4}$
Key Tip: When working with fractions, always find the least common denominator for addition and subtraction, and remember to simplify your final answer.