Order of
Operations
Master PEMDAS: the rules that tell you which calculations to do first so every expression gives one correct answer.
SSAT Middle Level: Quantitative
Why Does Order Matter?
Consider the expression \( 3 + 4 \times 2 \). If you add first, you get \( 7 \times 2 = 14 \). If you multiply first, you get \( 3 + 8 = 11 \). Two different answers from the same expression!
Meet PEMDAS
PEMDAS is a mnemonic that tells you the priority of each operation. Work through the expression from the highest priority to the lowest.
Parentheses
Grouping symbols first: \( ( ),\ [ ],\ {\} \)
Exponents
Powers and roots: \( 2^3,\ \sqrt{9} \)
M / D
Multiply and Divide, left to right
A / S
Add and Subtract, left to right
The Priority Ladder
The Core Rule
Order of Operations
Parentheses → Exponents → Multiply / Divide → Add / Subtract
1
Parentheses first. Simplify everything inside grouping symbols.
2
Exponents next. Evaluate all powers and roots.
3
Multiply and Divide. Work from left to right: whichever comes first.
4
Add and Subtract. Work from left to right: whichever comes first.
Same Level Means Left to Right
When two operations share the same priority level, do not default to "multiply before divide." Instead, read the expression left to right and perform whichever operation you encounter first.
MD vs. AS: Equal, Not Sequential
| Multiply / Divide | Add / Subtract | |
|---|---|---|
| Priority | Same as each other | Same as each other |
| Tie-breaker | Left to right | Left to right |
| Common Myth | "Always multiply before divide" | "Always add before subtract" |
| Truth | Whichever appears first wins | Whichever appears first wins |
Step-by-Step Checklist
1
Scan for parentheses. Simplify the innermost group first. If groups are nested, work inside-out.
2
Handle exponents. Evaluate any powers or roots remaining.
3
Sweep left to right for × and ÷. Perform each one as you meet it.
4
Sweep left to right for + and −. Finish the remaining operations.
5
Double-check. Re-read the original expression to make sure you haven't skipped a step.
Example 1: Basic Expression
Evaluate
\( 5 + 3 \times 4 - 1 \)
1
No parentheses or exponents to handle. Move to M/D.
2
Multiply: \( 3 \times 4 = \) 12. The expression becomes \( 5 + 12 - 1 \).
3
Add/Subtract left to right: \( 5 + 12 = 17 \), then \( 17 - 1 = \) 16.
Example 2: Parentheses and Exponents
Evaluate
\( (6 + 2)^{2} \div 4 - 3 \)
P
Parentheses: \( 6 + 2 = \) 8. Expression becomes \( 8^{2} \div 4 - 3 \).
E
Exponent: \( 8^{2} = \) 64. Now: \( 64 \div 4 - 3 \).
D
Divide: \( 64 \div 4 = \) 16. Now: \( 16 - 3 \).
S
Subtract: \( 16 - 3 = \) 13.
Example 3: Nested Parentheses
Evaluate
\( 2 \times [3 + (10 - 6)^{2}] - 5 \)
1
Innermost parentheses: \( 10 - 6 = \) 4. Now: \( 2 \times [3 + 4^{2}] - 5 \).
2
Exponent inside brackets: \( 4^{2} = \) 16. Now: \( 2 \times [3 + 16] - 5 \).
3
Remaining brackets: \( 3 + 16 = \) 19. Now: \( 2 \times 19 - 5 \).
4
Multiply: \( 2 \times 19 = \) 38. Now: \( 38 - 5 \).
5
Subtract: \( 38 - 5 = \) 33.
Example 4: Left-to-Right Trap
Evaluate
\( 18 \div 3 \times 2 + 4 \)
1
No P or E. Move to M/D, reading left to right.
2
Divide first (it appears before ×): \( 18 \div 3 = \) 6. Now: \( 6 \times 2 + 4 \).
3
Multiply: \( 6 \times 2 = \) 12. Now: \( 12 + 4 \).
4
Add: \( 12 + 4 = \) 16.
Common Mistakes
✕ Always Multiplying Before Dividing
M and D share the same level. Evaluate whichever appears first when reading left to right.
✕ Ignoring Parentheses
Parentheses override everything. \( 2 \times (3 + 4) = 14 \), not \( 10 \).
✕ Applying Exponent to the Wrong Base
In \( 2 \times 3^{2} \), the exponent applies only to 3, not to \( 2 \times 3 \). The answer is 18, not 36.
PEMDAS Quick-Reference Flowchart
Summary Table
| Step | Operation | Tie-breaker | Example |
|---|---|---|---|
| 1 | Parentheses | Inside-out | \( (2+3) = 5 \) |
| 2 | Exponents | Left to right | \( 5^2 = 25 \) |
| 3 | × and ÷ | Left to right | \( 12 \div 3 = 4 \) |
| 4 | + and − | Left to right | \( 4 + 7 = 11 \) |
Question 1 of 5
What is the value of \( 4 + 6 \times 3 \)?
Multiply first: \( 6 \times 3 = 18 \). Then add: \( 4 + 18 = 22 \). Choosing 30 means you added before multiplying.
Question 2 of 5
Evaluate: \( (8 - 3)^{2} + 7 \)
Parentheses: \( 8 - 3 = 5 \). Exponent: \( 5^{2} = 25 \). Add: \( 25 + 7 = 32 \).
Question 3 of 5
What is \( 20 \div 4 \times 5 \)?
Division and multiplication share the same level, so go left to right: \( 20 \div 4 = 5 \), then \( 5 \times 5 = 25 \). Choosing 1 means you multiplied first: \( 4 \times 5 = 20 \), then \( 20 \div 20 = 1 \).
Question 4 of 5
Evaluate: \( 3 + 2 \times (7 - 4)^{2} - 1 \)
Parentheses: \( 7 - 4 = 3 \). Exponent: \( 3^{2} = 9 \). Multiply: \( 2 \times 9 = 18 \). Add/Subtract left to right: \( 3 + 18 = 21 \), then \( 21 - 1 = 20 \).
Question 5 of 5
Which expression equals \( 50 \)?
Check each: A: \( 10 + 30 = 40 \). B: \( 15 \times 6 = 90 \). C: \( 2 \times 25 = 50 \) ✓. D: \( 25 + 5 = 30 \).