This page of our PERT Study Guide covers operations and fractions. We will demonstrate the step-by-step method to apply the order of operations when solving equations. We also cover 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 from left to right)
- Addition and Subtraction (done from left to right)
For example, in the expression $3 + 5 \times (8 + 4) \div (2^2) − 1$, solve the expression inside parentheses first, evaluate the exponent next, and then complete the multiplication, division, addition, and subtraction, respectively.
Properties of Operations
The following are properties of mathematical operations, or things that are universally true for any equation.
Commutative Property (for addition and multiplication): For any two numbers $a$ and $b$, the operation’s order does not change the result.
$a + b = b + a \, $ and $ \, ab = ba$
Associative Property (for addition and multiplication): The grouping of numbers does not affect the sum or product. For example:
$(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$
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
To add or subtract fractions, find a common denominator. For example:
$\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$
To multiply fractions, multiply the numerators together and the denominators together. For example:
$\dfrac{1}{2} \times \dfrac{2}{3} = \dfrac{1 \times 2}{2 \times 3} = \dfrac{1}{3}$
To divide fractions, multiply by the reciprocal of the second fraction. For example:
$\dfrac{1}{2} \div \dfrac{2}{3} = \dfrac{1}{2} \times \dfrac{3}{2} = \dfrac{3}{4}$
Key Tip: When working with fractions, always find the least common denominator for addition and subtraction, and remember to simplify your final answer.