Order of Operations: PEMDAS And Common Mistakes

When a maths expression contains several operations, the order used to calculate it can change the answer completely. Order of operations gives every solver a consistent method for handling brackets, powers, multiplication, division, addition and subtraction.

PEMDAS is a familiar memory aid: Parentheses, Exponents, Multiplication, Division, Addition and Subtraction. In Australia, students may also encounter BIDMAS or BODMAS, where brackets come first and indices represent powers. The names differ slightly, but the underlying rules are the same.

These skills appear throughout Australian school maths, from Year 7 algebra to senior secondary work involving equations, graphs and statistics. They are also useful in everyday calculations, such as checking a $45 shopping bill, comparing mobile plans in Sydney or estimating travel costs between Melbourne suburbs.

The main difficulty is rarely the arithmetic itself. Most errors happen when a student treats the mnemonic as a rigid left-to-right command, overlooks a bracket, or assumes multiplication always comes before division. A clear process makes complicated expressions far less intimidating.

Why The Calculation Order Matters

Mathematical notation is designed to communicate one precise value. Consider the expression (6 + 4 \times 3). If you add first, you get 30, but if you multiply first, you get 18. The conventional order tells us to calculate (4 \times 3) before adding 6, so the correct result is 18.

Without agreed rules, the same expression could produce different answers depending on who solved it. This would cause obvious problems in algebra, spreadsheets, calculator programs and practical tasks. Order of operations prevents ambiguity by assigning priority to different parts of an expression.

Brackets can deliberately change that priority. In ((6 + 4) \times 3), the addition is completed first because it is enclosed, giving 30. The brackets do not simply make an expression look organised; they tell you which calculation must be treated as a unit.

How PEMDAS Actually Works

Start with parentheses, or brackets, and solve the innermost grouping first. Then evaluate exponents, which include squares, cubes and other powers. After that, handle multiplication and division from left to right. Finish with addition and subtraction, also moving from left to right.

The left-to-right rule is essential. Multiplication does not automatically outrank division, and addition does not automatically outrank subtraction. For example, (24 \div 6 \times 2) becomes (4 \times 2 = 8), because division appears first. Reversing the operations would incorrectly produce 2.

The same principle applies to addition and subtraction. In (18 - 7 + 3), subtract 7 first and then add 3, resulting in 14. PEMDAS is a guide to levels of priority, not a direction to perform every multiplication before every division.

Reading Brackets, Powers And Fractions

Nested brackets should be handled from the inside out. For (3[2 + (5 - 1)]), begin with (5 - 1), then calculate (2 + 4), and finally multiply by 3. Writing one intermediate line at a time reduces the risk of losing a sign or copying a number incorrectly.

Powers can create a common trap. In (2 + 3^2), square 3 before adding, so the answer is 11. A negative sign may need extra attention: (-3^2) normally means the negative of (3^2), which is (-9), while ((-3)^2) equals 9 because the brackets include the sign.

Fractions often contain an entire expression in the numerator or denominator. Treat each part as grouped. For example, in (\frac{8+4}{3}), add 8 and 4 before dividing by 3. When using a calculator, enter brackets around the numerator and denominator if the display does not clearly show the fraction structure.

Common Mistakes Students Make

A frequent error is processing everything strictly from left to right. This works only when all operations have equal priority. In (5 + 2 \times 6), calculate the multiplication first, giving 17 rather than 42. Another mistake is believing that PEMDAS means multiplication must always happen before division; these operations share the same priority.

Students also sometimes ignore an invisible multiplication sign. In (4(3+2)), the 4 multiplies the bracketed expression, so the value is 20. The same idea appears in algebraic notation such as (2x), where the number and variable are multiplied even though no symbol is shown.

Calculator entry can introduce a separate problem. Some scientific calculators follow standard precedence, while a basic calculator may calculate each button press in sequence. A student checking work on a calculator should understand the expression first, use brackets where necessary and compare the result with a handwritten calculation.

A Practical Method For Reliable Answers

Write the original expression clearly, then mark the sections that must be resolved first. Calculate one stage at a time and copy the remaining terms carefully. Keeping each line aligned makes it easier to identify where an incorrect answer began.

Estimate the result before finishing. If (7 + 9 \times 2) is entered as 32, the estimate should raise a concern because multiplying 9 by 2 gives 18, with only 7 more expected. This habit is valuable in homework, tutoring sessions and timed assessments, including senior maths tasks linked to ATAR preparation.

Use these habits when solving expressions:

Building Confidence With Mixed Expressions

Practice should progress from simple numerical examples to expressions containing brackets, powers, fractions and variables. A learner who can solve (8 + 3 \times 4) accurately is ready to try (2(5^2 - 7) \div 3), provided each stage is written down rather than performed mentally.

Australian students may see different language depending on their school, textbook or online resource. One teacher might use PEMDAS, while another uses BIDMAS, BODMAS or the phrase “brackets, orders, division and multiplication”. Focus on the priority rules rather than memorising a particular acronym.

Online maths tools can help verify a result, but a final answer alone does not reveal whether the method was sound. Step-by-step solutions are more useful because they show how brackets were interpreted and why operations were completed in a particular order. For a student working independently in Brisbane, Perth or regional Australia, this reasoning is more valuable than simply copying a number.

Order of operations becomes dependable when it is treated as a sequence of decisions: group first, evaluate powers, move left to right through multiplication and division, then move left to right through addition and subtraction. The practical takeaway is simple: slow down, write each stage, and use the rules consistently before checking the result.