Order of Operations

Study Sheet

Order of Operations

Which operation goes first

The Agreed Order

Concept
The Big Idea

Without a rule, 2+3×42 + 3 \times 4 could be 2020 or 1414. Everyone agrees to do multiplication and division before addition and subtraction, with brackets first of all.

An expression like 2+3×42 + 3\times4 needs a rule for which operation goes first, or two people would get different answers. The agreement: grouping symbols first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. It is a convention — but one the whole world shares.

Example
Evaluating 2+3×42 + 3 \times 4
+23 rows of 4
2 + 3 × 4 is 2 dots plus 3 rows of 4: 14 dots. The rows must be counted as a group.

Multiply first: 3×4=123 \times 4 = 12. Then add: 2+12=142 + 12 = 14.

Example
Evaluating 18−6÷318 - 6 \div 3

Divide first: 6÷3=26 \div 3 = 2. Then subtract: 18−2=1618 - 2 = 16. Working left to right instead would give 12÷3=412 \div 3 = 4, a different number, which is exactly why the order had to be agreed.

Side note
Where it goes wrong

Treating multiplication as always before division, or addition as before subtraction. They are ties, done left to right: 8÷2×4=4×4=168 \div 2\times4 = 4\times4 = 16.

Try it
Try it: evaluate

Evaluate 7+8×2−57 + 8 \times 2 - 5.

Show the answer

Multiply first: 8×2=168 \times 2 = 16. Then add and subtract from left to right: 7+16=237 + 16 = 23, and 23−5=1823 - 5 = \mathbf{18}.

In Plain Terms & More Examples

Concept
In plain terms

Do brackets first, then multiplication and division, and only then addition and subtraction — working left to right within each step. This agreed rule is the order of operations (often remembered as PEMDAS).

Example
Evaluating 10−2×310 - 2 \times 3

Multiply first: 2×3=62 \times 3 = 6. Then subtract: 10−6=410 - 6 = 4.

Example
Evaluating (6+2)×3(6 + 2) \times 3
3 rows of (6 + 2)
Each row has 6 + 2 = 8 dots, and there are 3 rows: 24.

Brackets first: 6+2=86 + 2 = 8. Then multiply: 8×3=248 \times 3 = 24.

Side note
Where it goes wrong

Working strictly left to right. 10−2×310 - 2 \times 3 is not 8×3=248 \times 3 = 24. The multiplication comes first, 2×3=62 \times 3 = 6, and then 10−6=410 - 6 = 4.

Try it
Try it: brackets first

Evaluate (9−4)×2+1(9 - 4) \times 2 + 1.

Show the answer

Brackets: 9−4=59 - 4 = 5. Multiply: 5×2=105 \times 2 = 10. Add: 10+1=1110 + 1 = \mathbf{11}.

Nested Grouping

Concept
Inside out

With brackets inside parentheses, work from the innermost group outward. Each layer is finished before the next.

Each pair of brackets is a small problem that must be finished before its answer can be used outside. Working from the inside out guarantees that whenever you use a bracket's value, that value is complete.

Example
Finding 2×[3+(8−5)2]2\times[3 + (8 - 5)^2]

Inner: 8−5=38 - 5 = 3. Square: 99. Bracket: 3+9=123 + 9 = 12. Multiply: 2424.

Example
Finding {20−[2×(3+4)]}÷3\{20 - [2 \times (3 + 4)]\} \div 3

Innermost: 3+4=73 + 4 = 7. Square brackets: 2×7=142 \times 7 = 14. Curly braces: 20−14=620 - 14 = 6. Divide: 6÷3=26 \div 3 = 2.

Side note
Where it goes wrong

Starting from the outside. In 2×[3+(8−5)2]2 \times [3 + (8 - 5)^2], doing 2×3=62 \times 3 = 6 first breaks the bracket apart. The 22 multiplies the whole bracket, whose value, 1212, is not known until the inside is finished. Correct: 2×12=242 \times 12 = 24.

Try it
Try it: evaluate

Compute 36÷(2+4)×3−136 \div (2 + 4)\times3 - 1.

Show the answer

Parentheses: 36÷6×3−136 \div 6\times3 - 1. Left to right: 6×3=186\times3 = 18, then 18−1=1718 - 1 = \mathbf{17}.

Writing Expressions

Concept
Words into symbols

Sum means add, difference means subtract, product means multiply, quotient means divide. When the words say "then" or "the result", the earlier part goes in brackets.

Brackets record the order the words describe. "Add 55 and 77, then double" does the adding first, so the adding gets brackets: 2×(5+7)2 \times (5 + 7). Without them, the agreed order would double first.

Example
From words

"Add 55 and 77, then double the result" is 2×(5+7)=242\times(5 + 7) = 24. Without the parentheses, 2×5+7=172\times5 + 7 = 17 says something different.

Example
Comparing without calculating
182518251825×3
Three copies of the bar 18 + 25.

How does 3×(18+25)3 \times (18 + 25) compare with 18+2518 + 25? It is 33 copies of the same sum, so it is three times as large. There is no need to work out 18+25=4318 + 25 = 43 first.

Side note
Where it goes wrong

Reversing a subtraction. "Subtract 44 from 1010" is 10−4=610 - 4 = 6, not 4−104 - 10. The number you subtract from comes first.

Try it
Try it: place the parentheses

Insert parentheses so that 4+2×3−14 + 2\times3 - 1 equals 1717.

Show the answer

(4+2)×3−1=6×3−1=17(4 + 2)\times3 - 1 = 6\times3 - 1 = \mathbf{17}.

Ties Go Left to Right

Concept
Partners share a step

Multiplication and division are partners: do them in the order they appear, from left to right. Addition and subtraction are partners too, also done left to right. "PEMDAS" does not mean multiplication before division.

Dividing by 44 is the same as multiplying by 14\tfrac14, and subtracting is adding the opposite, so neither really comes "after" its partner. Reading left to right is simply the agreement that makes everyone get the same answer.

Example
Finding 24÷4×224 \div 4 \times 2
666624two of the parts: 12
24 split into 4 parts of 6, then 2 of those parts.

Left to right: 24÷4=624 \div 4 = 6, then 6×2=126 \times 2 = 12. Doing the multiplication first would give 24÷8=324 \div 8 = 3, which is wrong.

Example
Finding 15−6+215 - 6 + 2

Left to right: 15−6=915 - 6 = 9, then 9+2=119 + 2 = 11. Adding first would give 15−8=715 - 8 = 7, which is wrong.

Example
Finding 48÷6÷248 \div 6 \div 2

Left to right: 48÷6=848 \div 6 = 8, then 8÷2=48 \div 2 = 4. Working out 6÷26 \div 2 first would give 48÷3=1648 \div 3 = 16, which is wrong.

Side note
Where it goes wrong

Reading PEMDAS as a strict order. "M before D" would turn 24÷4×224 \div 4 \times 2 into 24÷8=324 \div 8 = 3. M and D share one step; so do A and S.

Try it
Try it: a long one

Evaluate 36÷3×2−10+436 \div 3 \times 2 - 10 + 4.

Show the answer

Left to right for ÷\div and ×\times: 36÷3=1236 \div 3 = 12, 12×2=2412 \times 2 = 24. Then left to right for −- and ++: 24−10=1424 - 10 = 14, 14+4=1814 + 4 = \mathbf{18}.

Why It Works

The rule exists because multiplication is repeated addition: 2+3×42 + 3\times4 means 2+4+4+42 + 4 + 4 + 4, and the additions hidden inside the multiplication must be finished before the 22 is added. Parentheses override this when a problem needs it, which is exactly why they exist: (2+3)×4(2 + 3)\times4 means "four copies of the sum". Powers come before multiplication for the same reason one level up — a power is repeated multiplication.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.