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PEMDAS: The Rules of the Road

Understanding the Order of Operations in Mathematics

Imagine if traffic lights had no rules. If green meant "go" for some people and "stop" for others, the roads would be chaos. Mathematics works the same way. If we didn't have a strict set of rules for the order in which we solve problems, everyone would get a different answer to the same equation.

This set of rules is known as the Order of Operations. In the United States, we remember it with the acronym PEMDAS.

[Image of PEMDAS order of operations chart]

1. What Does PEMDAS Stand For?

Each letter represents an operation. You must scan the math problem and perform them in this specific order:

  • P - Parentheses ( ) [ ] { }
  • E - Exponents (Powers and Roots)
  • M - Multiplication (x)
  • D - Division (/)
  • A - Addition (+)
  • S - Subtraction (-)

A popular mnemonic to remember this is: "Please Excuse My Dear Aunt Sally."

2. The "Left to Right" Rule

There is a massive trap in PEMDAS that tricks even adults. You do NOT always do Multiplication before Division, nor Addition before Subtraction.

The Rule: Multiplication and Division are tied in priority. Addition and Subtraction are tied in priority. When you have a tie, you solve from Left to Right.

Example of the Trap

10 - 3 + 2

Wrong Way: Do Addition first (3+2=5), then subtract (10-5=5).

Right Way: Do Subtraction first because it is on the left (10-3=7), then add (7+2=9).

3. Step-by-Step Breakdown

Step 1: Parentheses (Groupings)

Always solve what is inside the brackets first. If there are nested parentheses (parentheses inside parentheses), start from the innermost one and work your way out.

Example: 5 x (3 + 4) = 5 x 7 = 35.

Step 2: Exponents

After clearing parentheses, handle powers and roots.

Example: 5 + 32 = 5 + 9 = 14.

Step 3: Multiplication & Division

Scan the problem from left to right. Do whichever one comes first.

Example: 20 / 5 x 2
Division is first: 20 / 5 = 4.
Then multiply: 4 x 2 = 8.
(If you multiplied first, you would get 20 / 10 = 2, which is wrong!)

Step 4: Addition & Subtraction

Finally, scan from left to right and do whichever comes first.

4. A Complex Example

Let's solve this monster problem together:

4 + 3 x (6 - 2)2 / 2
  1. Parentheses: Solve (6 - 2) first.
    4 + 3 x (4)2 / 2
  2. Exponents: Solve 42.
    4 + 3 x 16 / 2
  3. Multiplication/Division: Scan left to right. 3 x 16 comes first.
    4 + 48 / 2
    Now do the division: 48 / 2 = 24.
    4 + 24
  4. Addition: Finish it off.
    4 + 24 = 28.

5. Why Does This Matter?

Computers and calculators are programmed to follow PEMDAS strictly. If you type an equation into a spreadsheet or coding language without understanding these rules, the computer will give you a result you didn't expect, leading to errors in finance, engineering, and science.

6. Different Acronyms (BODMAS, GEMDAS)

Different countries teach slight variations, but the math is identical.

  • BODMAS (UK/Australia/India): Brackets, Orders, Division/Multiplication, Addition/Subtraction.
  • GEMDAS: Grouping symbols, Exponents, etc. (More inclusive of brackets and braces).

Regardless of the name, the logic of "Groups first, Powers second, Multiply/Divide third" is universal.

7. Conclusion

PEMDAS is the grammar of mathematics. Just as punctuation tells you how to read a sentence, the order of operations tells you how to read an equation. By mastering this hierarchy, you ensure that you—and everyone else—speak the same mathematical language.