Why an Order Exists at All
Without an agreed sequence, 2 + 3 x 4 has two defensible answers. Working left to right gives 20. Multiplying first gives 14. Mathematics settled on 14, and every calculator, spreadsheet and programming language on earth now follows that same convention, because a shared wrong answer is worse than useless.
The order is not a law of nature. It is an agreement, chosen so that expressions can be written without brackets around every single operation.
PEMDAS and BODMAS Are the Same Rule
PEMDAS is parentheses, exponents, multiplication and division, addition and subtraction. BODMAS is brackets, orders, division and multiplication, addition and subtraction. The wording differs by country; the behaviour does not. Orders and exponents mean the same thing, as do brackets and parentheses.
Notice that PEMDAS lists multiplication before division and BODMAS lists division first. Neither is claiming priority, which brings us to the part the acronym hides.
The Part the Acronym Gets Wrong
Multiplication and division share a rank. So do addition and subtraction. When operations of equal rank sit next to each other, you work from left to right, and nothing else.
That is why 8 / 2 x 4 is 16, not 1. The division is on the left, so it happens first: 8 divided by 2 is 4, then times 4 is 16. Reading the M in PEMDAS as "multiply before divide" is what produces the wrong answer, and it is the single most common mistake on this topic.
Subtraction Behaves the Same Way
10 - 4 + 3 is 9. Working left to right, 10 minus 4 is 6, then plus 3 is 9. Treating the addition as higher priority gives 3, which is wrong for exactly the same reason.
A reliable trick is to read subtraction as adding a negative. 10 + (-4) + 3 can then be added in any order and still gives 9.
Worked Through
Take 20 - 3 x 2 squared + (6 - 2).
Brackets first: 6 minus 2 is 4, so the expression is 20 - 3 x 2 squared + 4. Exponents next: 2 squared is 4, giving 20 - 3 x 4 + 4. Multiplication: 3 times 4 is 12, giving 20 - 12 + 4. Then left to right: 20 minus 12 is 8, plus 4 is 12.
When to Just Use Brackets
If an expression needs a paragraph of explanation to be read correctly, add brackets. Being unambiguous is worth more than being clever, and no examiner has ever deducted a mark for a bracket that was not strictly required.
