The Order of Operations That Shows Up on Every Math Test
I spent two weeks debugging a spreadsheet where the results kept coming out wrong, only to realize someone had written =5+3*2 instead of =(5+3)*2 and expected it to equal 16. It evaluates to 11 because multiplication happens before addition. That is the entire problem with PEMDAS in one sentence. Please Excuse My Dear Aunt Sally is just a memory trick for the standard order of operations used in algebra and everything built on it. The letters stand for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It tells you which part of a math expression to calculate first so you do not get two different answers for the same string of numbers. The real issue is that most people treat it as a strict left-to-right list. It is not. Multiplication and division sit on the same tier, and addition and subtraction sit on the same tier. You resolve them in the order they appear, moving left to right. That detail alone fixes about sixty percent of the errors I see in student work.
I remember grading a midterm where a student evaluated 12 divided by 3 times 2 as 12 divided by 6. They multiplied first because M comes before D in the mnemonic. The correct answer is 8. The expression parses as 12 divided by 3, which gives 4, then multiplied by 2. That mistake shows up constantly in calculus when students simplify expressions with implied multiplication next to fractions.
How to Actually Use the Order of Operations
Start by finding anything inside parentheses or other grouping symbols like brackets or fraction bars. Work from the innermost group outward if you have nested ones. Next, handle exponents and roots. Then move to multiplication and division in the exact order they appear from left to right. Finish with addition and subtraction, also left to right. Consider this expression: 4 plus 6 divided by 2 times 3 minus 1. The division and multiplication happen before the addition and subtraction. Six divided by 2 is 3. Three times 3 is 9. Four plus 9 minus 1 equals 12. If you had added first, you would get a completely different number, and you would be wrong. Here is a harder example that trips people up: 2 to the power of 3 plus 1. The exponent goes first, giving 8 plus 1, which is 9. Some students see the addition and think they should combine 2 and 1 first to get 3 to the power of 3, which is 27. That is incorrect. The exponent applies only to the 2, not to the sum.
Get the Full Details
Another edge case involves negative signs. The expression minus 3 squared is not the same as negative 3 squared. In the first case, you square 3 to get 9, then apply the negative, giving negative 9. In the second case, written as negative 3 squared with parentheses, you square negative 3 to get positive 9. Calculator behavior differs here depending on whether you type it as -3^2 or (-3)^2. This distinction costs points on standardized tests more often than any other single mistake.
Where the Mnemonic Breaks Down
Please Excuse My Dear Aunt Sally works fine for basic arithmetic and introductory algebra. It becomes less useful in computer science and advanced mathematics because different languages and fields handle operations differently. Python, JavaScript, and Excel all follow the same general rules, but some programming contexts introduce operator precedence tables that add steps like bitwise operations, logical operators, and assignment operators. The mnemonic does not cover any of that. The mnemonic also does not address implicit multiplication clearly. When you write 2x(3+4), is the multiplication by 2 meant to happen before or after the parentheses? Standard convention says the parentheses first, then the multiplication, but students sometimes treat the implied multiplication as having higher priority than explicit operations, which can lead to confusion in more complex expressions. Another limitation is that the mnemonic suggests multiplication always comes before division, which is false. They share the same precedence level. The same problem exists for addition and subtraction. If you remember it as a rigid hierarchy, you will make mistakes on expressions where division appears after multiplication or subtraction appears after addition.
I have seen teachers accept the mnemonic as gospel and then mark students wrong for evaluating an expression like 10 minus 3 plus 2 as 10 minus 5 instead of 7+ I learned this the hard way when my younger sibling was confused about why their homework kept getting red marks. The teacher had taught it as a strict order rather than explaining the left-to-right rule for same-level operations.
Practical Tips for Getting It Right
Write out each step explicitly when you are learning. Do not try to do everything in your head. Circle the operation you are about to perform, write the new value, and move to the next one. This takes more time but reduces errors significantly, especially under test conditions where you might otherwise second-guess yourself. Use parentheses freely even when they are not required. Adding extra grouping symbols makes your intent clear and removes ambiguity. The expression 5 plus 3 times 2 is correct, but 5 plus 3 times 2 leaves no room for misinterpretation. It also helps when you are writing code or documenting calculations for someone else to review. Check your work by substituting simple values into variables if the expression contains them. If you have an algebraic expression with x equals 2, plug it in and evaluate both your original and your simplified version. If they do not match, you made a mistake in the order of operations or in the simplification itself.
When using calculators or spreadsheet software, be aware of how they interpret your input. Most graphing calculators follow standard precedence rules, but simple calculators may evaluate operations in the exact order you enter them. If your answer does not match your manual calculation, check whether your tool is using the correct order of operations or just computing left to right. This usually cuts down errors from about thirty percent of attempts to less than five percent once you consistently apply the left-to-right rule for same-level operations. The improvement is not instantaneous, but it becomes automatic within a few weeks of deliberate practice.
Alternative Approaches
Some educators prefer GEMDAS, which adds Grouping symbols before Exponents. Others use BEDMAS, which swaps the order of Division and Multiplication but means the same thing. The underlying rules are identical across all variants. Pick whichever version your class uses and stick with it to avoid confusion. For computer programming, memorizing the official operator precedence table for your language is more reliable than any mnemonic. The table is static and available in documentation. It includes operators that the PEMDAS framework does not address at all. If you are working in a field where order of operations affects safety-critical decisions, such as engineering calculations or financial modeling, verify every expression with a secondary method. Re-calculate using a different tool or by hand. This adds time but catches the rare cases where a single mistake could propagate into a significant error.
