My Dear Aunt Sally Math

Order of Operations and Why You Keep Getting Wrong Answers

You learned this in seventh grade and forgot it by eighth grade, which is normal. The expression 3 + 4 × 2 will always come back to haunt you if you add first, giving you 14 instead of the correct 11. Everyone hits this wall at some point. The mnemonic that stuck for most people is My Dear Aunt Sally Math, which maps directly to Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It is simple enough to remember but easy to mess up in practice because people treat it like a strict left-to-right checklist rather than a hierarchy of operations. Here is the method in plain terms. When you encounter an expression, you do not just work left to right. You scan the whole thing first and identify what needs to be resolved at each level. Start with anything inside grouping symbols — parentheses, brackets, or fraction bars count as grouping. Then handle exponents and roots. After that, multiplication and division, which sit at the same tier. Finally, addition and subtraction at the lowest tier. The part people consistently get wrong is the multiplication-division and addition-subtraction pairing. They are not separate steps. You handle them together, moving left to right through the expression. So in something like 12 ÷ 3 × 2, you divide first because it comes first on the left, giving you 4 × 2 = 8. If you multiplied first just because multiplication appears before division in the acronym, you would get 12 ÷ 6 = 2, which is wrong. Same logic applies to addition and subtraction.

I ran into this repeatedly when reviewing homework from students taking pre-algebra. One kid would write 8 3 + 2 and confidently arrive at 3 because they added first. You see this pattern constantly. The acronym reinforces the mistake because it separates M from D and A from S, making them look like independent rules rather than paired operations that share equal priority. Another layer people miss is the implicit multiplication that shows up everywhere. In algebra, 2(x + 3) means 2 × (x + 3), and the parentheses still get evaluated first before the multiplication happens. When expressions get written in inline format like 1/2x, it becomes genuinely ambiguous whether that means (1/2)x or 1/(2x). In professional mathematics, that notation is avoided precisely because it causes arguments. If you are working in a programming language, the same ambiguity exists and the compiler will resolve it one way or the other, so you need to know which way your tool resolves it.

The Practical Walkthrough

Take an expression like 5 + 2² × (8 3) ÷ 5. You work through it in stages. First, the parentheses: 8 3 = 5. The expression is now 5 + 2² × 5 ÷ 5. Next, the exponent: 2² = 4. Now you have 5 + 4 × 5 ÷ 5. Multiplication and division go left to right, so 4 × 5 = 20, then 20 ÷ 5 = 4. Finally, 5 + 4 = 9. That is the answer. If you had done anything out of order, even slightly, you would have landed somewhere else. Expressions with nested parentheses require another pass through the same logic. Something like 3 × [2 + (4 1)²] ÷ 5 starts inside the innermost parentheses: 4 1 = 3. Then the exponent: 3² = 9. Then the bracket content: 2 + 9 = 11. Then multiplication and division: 3 × 11 = 33, then 33 ÷ 5 = 6.6. The nesting just repeats the same steps at a smaller scale. I deal with this stuff professionally in technical documentation work, and one edge case still catches me occasionally. You encounter expressions where a fraction bar acts as an implicit grouping symbol, like (a + b) / (c d), and people will drop the parentheses around the denominator when they type it into a calculator and get a wrong result. This happens in real-world data work where someone pastes a formula into Excel or a Python script without braces. The workaround is to always write denominators as (denominator) even when they are single terms, and to mentally treat every fraction bar as if it has parentheses around both the numerator and the denominator. This habit alone prevented a batch of incorrect reports I was reviewing last year where the source calculations had been entered into a spreadsheet without that protection.

When the Mnemonic Breaks Down

The main limitation of My Dear Aunt Sally Math is that it does not scale well beyond basic arithmetic. Once you introduce functions like sine, logarithms, or absolute value bars, the rules shift. The argument of a function gets evaluated first, which behaves like parentheses but is not actually the same thing. Absolute value bars |5 3| also act as grouping symbols. Square root symbols carry their own implicit grouping over everything under the radical. The mnemonic does not cover any of this. It also does not help with order of operations in programming, where operator precedence follows its own rules that differ slightly from PEMDAS. In C and JavaScript, for example, multiplication and division share precedence with addition and subtraction in ways that can produce unexpected results if you assume pure mathematical ordering. If you write code, you need to consult the language reference for its specific precedence table rather than relying on the mnemonic. For advanced mathematics, especially when dealing with limits, series, or piecewise functions, the concept of order of operations expands into domain considerations and convergence rules that have nothing to do with PEMDAS. The mnemonic is a tool for arithmetic and introductory algebra, not a universal law of computation.

What Actually Helps People Remember This

Practice with deliberately tricky expressions is more useful than memorizing the acronym. Work through problems that mix all four operations with exponents and parentheses. The ones that trip people up most are expressions where a subtraction sits immediately after a division or where an exponent applies to only part of a grouped term, like 5 (2 1)² versus (5 2 1)². Those two look similar but produce very different results because of where the grouping lands. Writing expressions in a fully parenthesized form before evaluating them is a habit that eliminates most errors. You force yourself to make the grouping explicit, which removes the ambiguity that causes mistakes. It takes a few extra seconds but it prevents the kind of error that forces you to redo an entire problem set. The method is straightforward once you stop treating multiplication as strictly before division and addition as strictly before subtraction. They are paired operations. Work through them left to right. Handle grouping first, then exponents, then the paired tiers. Anything beyond that level of arithmetic requires tools the mnemonic was never designed to support.