Multiplication or Division First: A Thorough Guide to the Order of Operations

For many students and even adults returning to maths after a break, the question “Should I do multiplication or division first?” is a common stumbling block. The short answer is that multiplication and division are treated as being at the same level in the order of operations, and when both appear in a calculation, you proceed from left to right. Yet there is much more to understand. This guide unpacks the nuances, offers clear rules, and provides practical examples that make the principle easy to apply in exams, real life, and daily problem solving. We’ll explore what “multiplication or division first” means in practice, how it differs across curricula, and the best ways to teach and learn this essential concept.

Understanding the basic idea: what does “multiplication or division first” really mean?

The phrase “multiplication or division first” invites a moment of interpretation. In standard mathematical conventions, the operations of multiplication and division are considered to be on the same hierarchical level. There is no universal rule to always perform multiplication before division or vice versa. Instead, you apply both by the same rule: when you have multiplication and division in the same expression, carry out operations from left to right as you encounter them. This is the heart of how the rule is usually taught in schools across the UK and many other countries.

To grasp this, consider a simple expression: 8 ÷ 4 × 2. Reading from left to right, you first do 8 ÷ 4, which equals 2, then multiply by 2 to get 4. If you insisted on “multiplication first,” you would perform 4 × 2 first, arriving at 8, and then 8 ÷ 4 equals 2. The two results differ, which demonstrates why the left-to-right convention matters. In short, the rule is not about prioritising multiplication over division in all cases; it is about treating them with equal importance and applying the operation in the order they appear from left to right.

Historical and curricular context: PEMDAS, BODMAS, and BIDMAS

Different countries and curricula use slightly different acronyms to describe the order of operations. In the United States, you might hear PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In the United Kingdom, the equivalent is BODMAS or BIDMAS: Brackets, Orders (or Indices), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). The central idea remains the same: within the group of multiplication and division, and within the group of addition and subtraction, processes are performed from left to right. The phrase “multiplication or division first” is best understood as applying the left-to-right rule rather than a fixed prioritisation of one operation over the other.

Over time, the distinction between “multiplication first” and “multiplication and division together” has become clearer in classrooms. Teachers emphasise that the left-to-right approach is consistent and logical, reducing confusion for learners who worry about special cases where a single operation seems to dominate. By keeping a universal rule for left-to-right evaluation, students build a robust framework that works across all sorts of expressions, including those with brackets and exponents.

When you have brackets, exponents, and other operations: where does multiplication or division fit in?

Brackets (parentheses) take precedence in the order of operations. Anything inside brackets must be evaluated first. After resolving the content inside brackets, you move on to exponents, then to multiplication and division, and finally to addition and subtraction. The “multiplication or division first” question becomes more straightforward once brackets are considered. For example, in the expression (3 + 5) × 2 ÷ 4, you first evaluate inside the brackets: 3 + 5 equals 8. Then you have 8 × 2 ÷ 4. Now you apply the left-to-right rule: 8 × 2 equals 16, then 16 ÷ 4 equals 4.

Another common scenario is when you have a fraction, such as 12 ÷ 3 × 6. Reading left to right, you perform 12 ÷ 3 first to get 4, then 4 × 6 to obtain 24. If you misinterpret the instruction as “always multiply before divide,” you might arrive at the incorrect result. This illustrates why the left-to-right convention is essential for consistency and accuracy.

Illustrative examples: practising the left-to-right rule with multiplication and division

Below are a few carefully chosen examples to demonstrate how “multiplication or division first” operates in practice. Each example shows the correct approach and helps cement the habit of evaluating from left to right when both operations occur.

Example 1: A straightforward left-to-right case

Compute 18 ÷ 3 × 6.

First, perform 18 ÷ 3 which equals 6. Then multiply by 6 to obtain 36. The final answer is 36.

Example 2: Mixed numbers with brackets

Calculate (4 × 5) ÷ 2.

Inside the brackets, 4 × 5 equals 20. Then you have 20 ÷ 2 which equals 10. The result is 10, illustrating how brackets can remove ambiguity and ensure multiplication has a clear starting point inside a group.

Example 3: Left-to-right with a longer chain

Evaluate 100 ÷ 5 × 8 ÷ 2.

Proceed left to right: 100 ÷ 5 equals 20. Then 20 × 8 equals 160. Finally, 160 ÷ 2 equals 80. The result is 80, showing how a sequence of multiple steps still obeys the same left-to-right rule for multiplication and division.

Example 4: A potential trap

Find the value of 7 + 4 × 3 ÷ 6.

First, handle the multiplication/division: 4 × 3 equals 12, then 12 ÷ 6 equals 2. Now add 7 to obtain 9. The final result is 9. A common mistake is to perform the addition before the multiplication/division, which would produce a different and incorrect outcome.

Edge cases: decimals, fractions, and very large numbers

Decimals and fractions can complicate mental calculation, but the fundamental rule remains unchanged. When a problem includes decimals, you still apply the left-to-right rule for multiplication and division. For instance, 2.5 × 4 ÷ 2.5: start with 2.5 × 4 = 10. Then 10 ÷ 2.5 = 4. In fractions, consider (1/2) ÷ (3/4) × 2. First, (1/2) ÷ (3/4) is equivalent to (1/2) × (4/3) = 2/3. Then multiply by 2 to get 4/3. Here again, left-to-right evaluation yields the correct result.

Another scenario involves large numbers or expressions that span multiple operations. The key is to break the expression into smaller parts, apply the left-to-right rule consistently to the multiplication and division steps, and keep careful track of units or quantities if the problem relates to real-life measurements or money.

Common pitfalls and misconceptions to watch out for

Even with clear rules, learners can trip up on a few recurring misunderstandings. Here are some of the most frequent pitfalls and how to avoid them.

1. Treating multiplication and division as inherently sequentially ordered

It’s a mistake to assume multiplication always proceeds before division. The standard rule is left-to-right when both appear in the same expression. Always check the order in which the operations appear and apply the left-to-right rule accordingly.

2. Ignoring brackets or misplacing parentheses

Brackets are the highest-priority instruction. If you skip brackets or misplace them, your entire result is likely to be wrong. When possible, rewrite long expressions by simplifying inside brackets first, then address the outside operations.

3. Overlooking implicit grouping in long calculations

In longer expressions, it can be easy to lose track of the sequence. Break the calculation into steps, write each operation on a new line, and cross-check as you go. This reduces the risk of inadvertently applying multiplication before division in the wrong segment.

4. Relying on calculators without understanding the process

Calculators can perform the left-to-right rule automatically, but relying solely on software can hinder understanding. It’s important to be able to articulate why the result is what it is, especially in exam situations where showing your method is as important as the answer.

5. Confusing “multiplication or division first” with “always do multiplication first”

Many learners confuse the phrase “multiplication or division first” with an artificial precedence. Reiterate that both operations are on the same level and the evaluation proceeds left to right. Practice with a variety of expressions to embed this rule in memory.

Practical strategies for teaching and learning this concept

Whether you are a parent helping a child with homework or a teacher planning lessons, several effective strategies can reinforce the rule of left-to-right evaluation for multiplication and division. Here are some practical approaches that work well in UK classrooms and beyond.

Use visual models to illustrate left-to-right evaluation

Number lines, arrays, or area models can help learners visually walk through each operation in the order they occur. When students see a calculation progress step by step, the left-to-right rule becomes intuitive rather than abstract.

Incorporate real-life scenarios

Clicking into real-life contexts such as cooking, budgeting, or distributing items can make the rule tangible. For instance, “If you have 6 cups of flour and you need to divide it equally among 3 bags, how many cups per bag?” followed by adding a multiplier to this scenario can help reinforce the left-to-right approach.

Encourage verbalisation of the process

Ask learners to articulate each step aloud as they work through a problem. Phrases like “I’m doing this because it comes first in the expression; now I move to this operation” reinforce the discipline of following the established order.

Provide a consistent framework for different curricula

Across the UK, students may encounter variations like PEMDAS or BODMAS. Teach the universal core: brackets, exponents, then multiplication and division from left to right, followed by addition and subtraction from left to right. This consistency helps students transfer knowledge across different courses and exams.

Practice with progressively challenging problems

Start with clear, simple expressions and gradually introduce longer chains with several steps, fractions, decimals, and negative numbers. Regular practice builds fluency and confidence in applying the left-to-right rule in diverse contexts.

Common tools and resources to support learning

There are many resources available to reinforce the concept of multiplication or division first. In addition to traditional worksheets, learners can benefit from interactive platforms, video tutorials, and problem sets that emphasise the left-to-right rule. When choosing tools, look for features such as step-by-step solutions, immediate feedback, and the ability to show the method used, not just the final answer.

Digital resources

Online practice sites often offer timed drills focusing on the left-to-right rule for multiplication and division. Look for interactive tasks that require learners to explain their reasoning as they proceed through a sequence of operations. This reinforces both procedural fluency and conceptual understanding.

Printed practice

Printing worksheets with a mix of single-step and multi-step problems helps students become familiar with the left-to-right progression. Include problems with and without brackets to ensure learners apply the rule consistently across different formats.

Low-stress assessment ideas

Short, low-stress quizzes can check understanding without creating anxiety. Ask learners to show the steps for 4–6 problems, then review any mistakes together, focusing on how they navigated from left to right through multiplication and division.

The real value: why getting this right matters

Mastery of the order of operations — particularly the rule governing multiplication and division — is more than a school exercise. It underpins higher-level mathematics such as algebra, calculus, and statistics. In everyday life, the ability to evaluate expressions correctly influences budgeting, cooking, engineering, and data interpretation. When students understand that multiplication and division are on the same level and are processed from left to right, they build a foundation for tackling complex problems with clarity and confidence.

Moreover, the skill supports logical thinking and reasoning. It trains learners to stop and identify the structure of a problem, which is the first step toward developing robust problem-solving strategies. In an era where mathematics underpins technology, science, and finance, a strong grasp of the order of operations is a practical literacy skill with long-term benefits.

Common misinterpretations in different contexts

In certain computational settings, people encounter expressions that might appear ambiguous or misleading if read too quickly. For example, in some programming languages, operators can be overloaded, or evaluation rules may differ when working with matrices or functions. While those topics go beyond basic arithmetic, the core principle still holds: identify the exact operations involved, apply any grouping (brackets) first, and then resolve multiplication and division from left to right. For learners encountering this in later study, it helps to be aware that computer languages can have their own conventions, which sometimes deviate from standard math notation, but in straightforward arithmetic, the left-to-right rule remains the anchor.

How to frame this rule in exams and assessments

In tests, it is crucial to demonstrate both the final answer and the method. When marking, examiners look for a clear sequence of operations and accurate application of the left-to-right rule for multiplication and division. Encouraging students to write out each step, even when using a calculator, improves accuracy and provides evidence of understanding. A typical mark scheme rewards the correct final result along with correct procedural steps showing the left-to-right approach in expressions containing both multiplication and division.

Summary: returning to the core idea

So, is it better to do multiplication before division? The short answer is no, not universally. The rule is: treat multiplication and division as on the same level, and evaluate expressions from left to right. The same logic applies to addition and subtraction: perform from left to right within that group. When brackets or exponents enter the picture, resolve those first, then address multiplication and division as they appear from left to right. With this understanding, the question “multiplication or division first” becomes a straightforward decision in almost all calculations, from the simplest to the most complex.

Final thoughts: building confidence with the order of operations

Developing fluency with the order of operations, including the rule for multiplication and division, takes time and deliberate practice. The most effective approach combines clear explanations, consistent practice, and opportunities to apply the rule in real-world contexts. By using brackets strategically, emphasising left-to-right evaluation for the multiplication/division group, and reinforcing through varied problems, learners grow increasingly confident in solving ever more challenging mathematical expressions. The result is not merely exam readiness but a transferable mathematical mindset that supports logical thinking across disciplines.