🟡 Intermediate Level
Grade 6–7 · Ages 11–13 · Brackets · Negatives · Simple fractions & decimals · Powers

DMAS — Intermediate
Order of Operations

At this level you already know the basic DMAS priority. Now you master brackets with expressions inside them, negative numbers, simple fractions, and integer powers — the building blocks of all algebra.

6Topics
30+Examples
3Graded exams
60Questions
( )
Brackets
Always first
DMAS inside
n²
Powers
After brackets
Before ×÷
D·A
÷ and ×
Left → right
Equal priority
M·S
+ and −
Left → right
Equal priority
01

Quick Review — Core DMAS

Before diving deeper, confirm the foundation: multiplication and division always come before addition and subtraction. Equal-priority operations go left to right.

Priority order
( ) → powers → ×÷ → +−

This is the complete DMAS hierarchy. Each level must be fully resolved before moving to the next.

Left-to-right rule
6÷2×3 = 9 (not 1)

6÷2=3 first (leftmost), then 3×3=9. Never compute 2×3=6 first just because multiplication "feels" stronger.

Two separate groups
2×3 + 4×5 = 6+20 = 26

When two separate multiplication groups exist, compute both first, then combine with + or −.

The two-pass method
Pass 1: all ×÷ · Pass 2: all +−

Scan the expression twice: first resolve every × and ÷, then go back and resolve every + and −.

ReviewExample — 3 + 2 × 6 − 12 ÷ 4
1
All ×÷ left to right: 2×6=12 and 12÷4=3
2
Expression: 3 + 12 − 3
3
Left to right: 3+12=15, then 15−3=12
✓
Answer: 12
02

Brackets — Deep Dive

At intermediate level, brackets contain their own DMAS expressions. You must apply the full DMAS rule set inside each bracket before using the result.

DMAS inside brackets
(3 + 2×4) = (3+8) = 11

Inside (3+2×4): multiplication first → 2×4=8, then 3+8=11. The bracket result is 11.

Coefficient × bracket
3(4+2) = 3×6 = 18

A number touching a bracket means multiply. Solve the bracket first, then multiply.

Multiple brackets
Solve each, left to right

Two or more separate brackets: solve each one independently, then combine in the outer expression.

Nested brackets
[(2+3)×2] = [5×2] = 10

Innermost bracket first: (2+3)=5. Then outer: 5×2=10. Always work from inside out.

IntermediateExample 1 — DMAS inside bracket: 2×(10 − 3×2) + 1
1
Inside bracket, multiplication first: 3×2=6
2
10−6=4. Bracket = 4.
3
Multiply: 2×4=8
4
8+1=9
✓
Answer: 9
IntermediateExample 2 — Nested: 3×[(4+2)÷2 + 1]
1
Innermost: (4+2)=6
2
Inside outer bracket, DMAS: 6÷2=3, then 3+1=4
3
3×4=12
✓
Answer: 12
ChallengeExample 3 — Fraction bar as brackets: (3×4 − 6)÷(1 + 2)
1
Numerator: 3×4=12, then 12−6=6
2
Denominator: 1+2=3
3
6÷3=2
✓
Answer: 2
03

Negative Numbers in DMAS

DMAS priority rules apply exactly the same way with negative numbers. The only extra skill needed is knowing the sign rules for multiplication and division.

Sign rules for × and ÷
(+)×(+)=+ · (−)×(−)=+ · (+)×(−)=−

Same signs → positive result. Different signs → negative result. Apply after evaluating the numbers.

Subtracting a negative
a − (−b) = a + b

Subtracting a negative equals adding. 5−(−3)=5+3=8. Two negatives in subtraction make a positive.

Negative in brackets
(−2+5)=3

Treat negative numbers inside brackets normally. Apply the same DMAS rules inside.

Multiplying by negative
−3×(2+4) = −3×6 = −18

Solve the bracket first, then apply the negative sign in the multiplication.

IntermediateExample 1 — Negative × negative: (−2)×(−3) + 1
1
Multiplication first (DMAS): (−2)×(−3)=6 (negative × negative = positive)
2
6+1=7
✓
Answer: 7
IntermediateExample 2 — Negative bracket: −2×(3−7)+5
1
Brackets: 3−7=−4
2
Multiplication: −2×(−4)=8
3
8+5=13
✓
Answer: 13
🧠 Sign rule memory trick: Think of negative signs as "direction changes." Each − sign flips the direction once. Two flips (−×−) bring you back to positive, just like turning around twice.
04

Fractions & Decimals in DMAS

DMAS priority rules apply identically to fractions and decimals. The only additions: simplify fractions when possible, and remember dividing by a fraction = multiplying by its reciprocal.

Fraction × integer
(3/4) × 8 = 6

Multiply numerator by integer: 3×8=24, divide by denominator: 24÷4=6. Or: 8÷4=2, then 3×2=6.

Dividing by fraction
8 ÷ (1/2) = 8 × 2 = 16

Dividing by a fraction = multiplying by its reciprocal. Flip and multiply. 8÷(1/2)=8×(2/1)=16.

Decimal priority
DMAS applies identically

2.5×4−3: same rules. Multiply first: 2.5×4=10. Then 10−3=7. Decimals don't change the priority.

Simplify first (hack)
6/12 × 4 → 1/2 × 4 = 2

Before multiplying a fraction, simplify it. 6/12=1/2 first makes the calculation much easier.

IntermediateExample 1 — Fraction multiplication: 3/4 × 8 − 2
1
Multiplication first: 3/4 × 8 = 6
2
6−2=4
✓
Answer: 4
IntermediateExample 2 — Dividing by fraction: 12÷(1/3)+1
1
Brackets: 1/3. Division by fraction = multiply by reciprocal: 12×3=36
2
36+1=37
✓
Answer: 37
ChallengeExample 3 — Decimal: (1.5+0.5)×(4−1)÷3
1
Both brackets: (1.5+0.5)=2 and (4−1)=3
2
Left to right: 2×3=6, then 6÷3=2
✓
Answer: 2
05

Nested Brackets

Nested brackets are brackets inside other brackets. The rule is simple: always solve the innermost bracket first, then work outward one layer at a time.

IntermediateExample 1 — 2×[(3+1)×2+4]
1
Innermost: (3+1)=4
2
Inside outer bracket: 4×2=8, then 8+4=12
3
Outer multiply: 2×12=24
✓
Answer: 24
ChallengeExample 2 — [(2+3)×2]÷5 + (1+4)
1
Left group: innermost (2+3)=5, then 5×2=10
2
Right bracket: (1+4)=5
3
Left to right: 10÷5=2, then 2+5=7
✓
Answer: 7
06

Powers (Exponents) in DMAS

Powers (exponents) are evaluated after brackets but before ×÷ and +−. The full international order: Brackets → Powers → ×÷ → +−. This matches BODMAS and PEMDAS.

Power comes after brackets
(2+1)² → 3² = 9

Solve the bracket first to get the base, then apply the power.

Power before ×÷
3 × 2² = 3 × 4 = 12

The power 2² applies only to the 2, not to 3×2. Evaluate the power first, then multiply.

Negative base squared
(−3)² = 9 but −3² = −9

(−3)² means (−3)×(−3)=9. But −3² means −(3²)=−9. The brackets matter enormously.

Power of zero
n⁰ = 1 (n≠0)

Any non-zero number to the power of zero equals 1. Useful simplification in complex expressions.

IntermediateExample 1 — 2 + 3² × 2
1
Power first: 3²=9
2
Multiply: 9×2=18
3
2+18=20
✓
Answer: 20
IntermediateExample 2 — (−2)² vs −2²
A
(−2)²=(−2)×(−2)=4 — the negative is inside the base
B
−2²=−(2²)=−4 — the negative is outside the power
✓
Brackets change everything: (−2)²=4, −2²=−4
ChallengeExample 3 — (3+1)²÷(2³−4)×2
1
Both brackets: (3+1)=4 and (2³−4)
2
Power inside second bracket: 2³=8, then 8−4=4
3
Powers: 4²=16
4
Left to right: 16÷4=4, then 4×2=8
✓
Answer: 8
🧠 Complete intermediate DMAS checklist:
1️⃣ Brackets — innermost first, DMAS inside each
2️⃣ Powers — evaluate all exponents
3️⃣ × and ÷ — left to right
4️⃣ + and − — left to right
07

Practice Exams

Three exams — each 20 questions. Covers brackets, powers, negatives and fractions. Use 💡 hints whenever you need guidance.

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