Ask ten people to solve 8 - 2 x 3 + 4 and you will get more than one answer. Some will subtract first and get 22 backwards through the expression, landing on 6. Others will multiply first, as the rules require, and get 6 the correct way. A few will work strictly left to right and land somewhere else entirely. The gap between those answers is not a matter of opinion. It comes down to a small set of rules, taught in school as PEMDAS or BODMAS, that decide which operation in an expression gets evaluated first. Most people learned these rules once, used them just long enough to pass a test, and have not thought about them since. But the rules quietly govern everything from spreadsheet formulas to calculator results to the totals on a receipt, and getting them wrong produces a confidently incorrect answer every time.
What Order of Operations Actually Means
An expression like 3 + 4 x 2 contains two operations: addition and multiplication. If you read it left to right and do each step as you reach it, you get (3 + 4) x 2, which equals 14. But the agreed-upon mathematical convention says multiplication happens before addition, so the correct reading is 3 + (4 x 2), which equals 11. Both readings are internally consistent. Only one of them is the standard that every textbook, calculator, and programming language is built to follow.
The order of operations exists because mathematical notation needs to mean the same thing to everyone who reads it. Without a shared convention, the same string of numbers and symbols could be evaluated in dozens of different orders, each producing a different result. The rules are not arbitrary busywork from a math class. They are the agreement that makes written math a reliable form of communication, the same way punctuation rules make written language unambiguous.
PEMDAS, BODMAS, and Why the Names Differ Around the World
In the United States and Canada, the order of operations is usually taught as PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In the United Kingdom, Australia, and India, the same concept is taught as BODMAS or BIDMAS: Brackets, Order (or Indices), Division and Multiplication, Addition and Subtraction. Despite the different letters, both acronyms describe the exact same hierarchy. Brackets and parentheses mean the same thing. Order and indices both refer to exponents and roots. Multiplication and division share the same priority level regardless of which letter comes first in the acronym, and the same is true for addition and subtraction.
This is where a common misreading causes real errors. Multiplication does not always come before division, and addition does not always come before subtraction, even though the acronyms appear to list them in that order. Multiplication and division are tied for priority, as are addition and subtraction. When operations at the same priority level appear together, you work through them left to right in the order they appear. In 12 / 3 x 2, division is not finished before multiplication starts. You work left to right: 12 / 3 = 4, then 4 x 2 = 8. Treating multiplication as always coming first would give you 12 / 6 = 2, which is wrong.
Why Your Calculator Might Disagree With You
Type 6 / 2 x (1 + 2) into different calculator apps and you can genuinely get two different answers: 9 or 1. This is not a bug. It reflects a real disagreement about how implicit multiplication next to parentheses should be grouped, and different calculator designs resolve it differently. A basic calculator that processes operations strictly as you press buttons, in the order you press them, will produce a different result than a calculator built to parse and evaluate a full algebraic expression according to standard precedence rules.
This is one of the main reasons a dedicated scientific calculator is worth using for anything beyond simple arithmetic. A scientific calculator evaluates a full expression at once, applying exponents, multiplication, division, addition, and subtraction in the correct standard order regardless of the sequence you typed them in, and it lets you nest parentheses to force a specific grouping when the default order is not what you want. If you regularly work with multi-step expressions involving exponents, roots, or several layers of parentheses, typing the full expression into a Scientific Calculator and letting it apply the rules consistently removes the guesswork entirely, and lets you double-check your own manual work against a result you can trust.
Fractions and the Order of Operations
Fractions add a layer that often gets missed: the fraction bar itself acts as a grouping symbol, equivalent to invisible parentheses around the entire numerator and the entire denominator. The expression (4 + 6) / 2 means add 4 and 6 first, then divide by 2, giving 5. But if that fraction is written on paper as a stacked fraction with 4 + 6 on top and 2 on the bottom, it is easy to mentally simplify it as 4 + (6 / 2), which gives 7, an entirely different and incorrect answer. The visual layout of a fraction does the grouping work that parentheses do in a single line of text, and losing that grouping when you rewrite the expression is one of the most common silent errors in algebra.
This matters in everyday situations too, not just textbook problems. Splitting a bill, adjusting a recipe, or scaling a measurement often involves combining fractions with other operations, and the order in which you simplify changes the result. If you are working with fractions directly and want to add, subtract, multiply, divide, or simplify them without tracking every common denominator by hand, a Fraction Calculator handles the arithmetic correctly and shows the simplified result, which is especially useful for checking whether your manual grouping matches the correct order of operations.
Percentages Inside Larger Expressions
Percentages create their own version of this problem because a percent sign is really shorthand for "divide by 100," and that hidden division has to be evaluated in the right place relative to everything else in the expression. The phrase "20 percent of 50 plus 10" can be read two different ways depending on where the addition falls. If it means (20% x 50) + 10, the answer is 10 + 10 = 20. If it means 20% x (50 + 10), the answer is 0.2 x 60 = 12. The words alone are ambiguous, but the moment the expression is written with explicit parentheses, the order of operations makes the answer unambiguous.
Percent increases and decreases compound the issue further, because "increase by 10 percent, then decrease by 10 percent" does not return you to the original number. Each percentage operates on the result of the previous step, not on the original value, and the two changes are not symmetrical. A value of 100 increased by 10 percent becomes 110. That 110 decreased by 10 percent becomes 99, not 100. When you need to work through percentage changes step by step and confirm the order is producing the result you expect, a Percentage Calculator lets you check each stage of the calculation individually before combining them.
Real-World Math: Discounts, Tax, and Multi-Step Totals
Few places make order of operations more concrete than a shopping receipt. A discount and a tax both apply as percentages, but they apply to different amounts and in a specific sequence, and reversing that sequence changes the final price. Discounts are applied to the original price first. Sales tax is then applied to the discounted price, not the original price. Calculating tax on the original price and then subtracting the discount separately, or adding the two percentages together before applying either one, both produce the wrong total.
Take a $100 item with a 20 percent discount and 8 percent sales tax. The correct order is: apply the discount first, giving $80, then apply tax to that $80, giving a final price of $86.40. Adding the percentages together first (20 percent off, then adding 8 percent of the original $100) would produce a different, incorrect number. The order in which percentages are applied is itself an order-of-operations problem, just dressed up as a shopping question instead of an algebra problem.
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Try the Discount CalculatorCommon Mistakes That Trip People Up
Treating multiplication and division as strictly ordered
As covered earlier, multiplication does not automatically come before division just because the M appears first in PEMDAS. They share a priority level and are resolved left to right. The same is true for addition and subtraction.
Forgetting that a negative sign can change the grouping
An expression like -3 squared is often typed or written ambiguously. Depending on convention, -3 squared can mean -(3 squared), which is -9, or it can mean (-3) squared, which is 9. Most standard math notation treats it as -(3 squared) unless the negative sign is explicitly inside parentheses, but many calculators and programming languages handle this differently, which is exactly why testing an ambiguous expression on a calculator you trust matters.
Losing grouping when simplifying fractions or percentages
As shown above, both fractions and percentages carry implicit grouping that disappears the moment the expression is rewritten on a single line without parentheses. Whenever you convert a stacked fraction or a worded percentage problem into a linear expression, add parentheses explicitly around anything that was grouped in the original form.
Assuming left to right always works
Left to right is only correct within a single priority level. Across different operations, the priority hierarchy overrides reading order. An expression is not a sentence to be read start to finish; it is a structure with an internal hierarchy that has to be resolved before reading order applies.
Summary
Order of operations is not a quirky classroom rule; it is the shared agreement that lets a written expression mean exactly one thing to everyone who reads it. PEMDAS and BODMAS describe the same hierarchy under different names: grouping symbols first, then exponents and roots, then multiplication and division left to right, then addition and subtraction left to right. The places this trips people up most often are not exotic edge cases. They are everyday situations: fractions written on paper, percentages applied in sequence, and calculators that interpret an ambiguous expression differently than you expect. When in doubt, add parentheses to make your intended grouping explicit, and check ambiguous results against a calculator built to apply the standard order consistently.