Order of Operations

What is $2 + 3 \times 4$ ? Sweeping left to right: $2+3=5$ , then $5\times4=20$ . Yet mathematicians say the answer is 14. Nobody miscalculated — the expression simply obeys a set of rules everyone must share: the order of operations. Without it, one expression would have several "correct" answers, and calculators could never talk to people.

Why multiplication goes first

Picture $2 + 3 \times 4$ as a real scene: you hold 2 candies, then buy 3 bags of 4 each. The bags hold $3 \times 4 = 12$ candies, and that pack must be tied before anything else; adding your own 2 makes 14. Multiplication describes "3 packs of 4", and the pack has to be tied up first. Adding and subtracting merely move packs around, so they naturally come last.

The four-step ladder

The rules form four levels, highest first:

Brackets: computed first — the inside of a bracket is a sealed little world;
Powers (exponents): things like $2^2$ and $5^3$ , right behind brackets;
Multiply and divide: one level, left to right;
Add and subtract: one level, again left to right.

The English mnemonics PEMDAS (Parentheses, Exponents, Multiply, Divide, Add, Subtract) and BODMAS (Brackets, Orders, Divide, Multiply, Add, Subtract) describe the same ladder. Notice that in both, M stands beside D and A beside S — multiplication and division share one level, addition and subtraction another, and within a level you sweep left to right. What powers are, and why they rank so high, is the subject of exponents.

Brackets override everything

$(2 + 3) \times 4$ : the bracket orders you to compute $2+3=5$ first, then multiply by 4, giving 20. The same three numbers, one pair of brackets — the answer jumps from 14 to 20.

A full workout

String the four levels together. At every step do only the highest active level and copy the rest down unchanged.

One step at a time

Evaluate $3 + 4 \times 2^2 - 10 \div 2$ .

Powers first: $2^2 = 4$ , so the expression becomes $3 + 4 \times 4 - 10 \div 2$ ;
Multiply and divide, left to right: $4 \times 4 = 16$ and $10 \div 2 = 5$ , leaving $3 + 16 - 5$ ;
Add and subtract, left to right: $3 + 16 = 19$ , then $19 - 5 = 14$ .

The answer is 14. Never jump the queue: no multiplying until the powers are done, no adding until multiply-and-divide is cleared.

Three common traps

$2 \times 3^2$ : the power binds only to the 3 beside it, so $3^2 = 9$ and $2 \times 9 = 18$ . To square the whole product, write $(2 \times 3)^2 = 36$ ;
$12 \div 2 \times 3$ : same level, left to right — $12 \div 2 = 6$ , then $6 \times 3 = 18$ . Doing $2\times3$ first to get 2 is cutting in line;
Invisible multiplication: $3(4 + 1)$ really means $3 \times (4 + 1)$ , and the bracket still goes first — omitting the sign does not lower its rank.

Left to right only applies within a level

Left-to-right is the rule between multiplications and divisions, or between additions and subtractions — not for the whole expression. Meet a power next to a product, do the power; meet a bracket, climb inside it first.

Check yourself

Quick quiz

1. 2+3×4 = ?
2. 2×3² = ?
3. 12÷2×3 = ?