Mathematics is all about numbers, calculations and patterns. Calculating the answers is easy when it comes to simple equations of two numbers you add, subtract, multiply or divide. What happens if an equation has multiple operations? Mathematicians devised a set of conventions, or basic rules, to specify an order of calculating multiple operations in an equation. The order of operation rules is easily remembered by recalling an acronym. In Canada and the United States, the acronym is PEMDAS, while in other countries of the world, the acronym is BODMAS or other similar variants.
Both PEMDAS and BODMAS are essentially the same calculation order. They specify the order of calculation as:
- Parenthesis and Brackets
- Exponents and Orders (powers and roots)
- Multiplication and Division from left to right
- Addition and Subtraction from left to right
Educators in Canada and the United States went a step further and created a useful memory tool to assist in remembering the PEMDAS calculation order, which is "Please Excuse My Dear Aunt Sally."
Parenthesis and Brackets First
When an equation includes a set of parentheses or brackets, the rule is to calculate first within those parentheses or brackets. Take the example equation of 5(-2+7). Solve the -2+7 within the parentheses first to arrive at 5. Then multiply 5(5) for the final answer of 25. What if the equation is 75-(8+10x3)? Then within the parenthesis, you still calculate the multiplication section first, as the order of operations rules advises, to arrive at the parenthetical value of (8+30). Thus, the answer is 75-38, or 37.
Solve Exponents or Order Second
Following parenthetical operations, the next operation to solve for in an equation involves exponents or orders. Consider the equation of [(25 ÷ (10+6)]. The parentheses is solved first, so that (10+6) becomes (16), and then the exponential operation is solved second to arrive at 2×2×2×2×2, which equals 32. Hence, the final stage is to solve 32 ÷ 16, which equals 2.
Solving for exponents or orders also refers to solving for roots. Another illustration can be found in the equation [√25 +((72+1)]. Again the parenthetical operation is solved first so that ((72 +1) becomes (49+1), or 50. Then the square root of 25 is solved to arrive at 5. Consequently, the final stage of solving becomes 5+50, which equals 55.
Next Come Multiplication and Division
When an equation has several operations involving multiplication and division, they must all be solved from left to right. For example, when evaluating [6(3) ÷ 9]+[(81 ÷ 9)-(22)], you first solve within the parentheses, then solve the exponent, followed by multiplication and division operations from left to right. So calculations are [18 ÷ 9]+[(9)-(4)], which is simplified to +, which equals 7.
Addition and Subtraction Are Solved Last
Addition and Subtraction are solved last from left to right. For example, in the equation [144 ÷ (2×6)]+[(2×5)-2], all the inner parenthetical operations are calculated first. As a result, the equation becomes [144 ÷ (12)]+[(10)-2], or +, which is 20. In another example, with the equation 100+(86+(74+2((52)+7))), the innermost parenthetical operations solved first, so that ((52) +7) becomes (25+7), which transforms the equation to 100+(86+(74+2(32))), or 100+(86+(74+64)). Solving within the parenthesis once more, you arrive at 100+(86+(138)), or 100+(224), for a final answer of 324.