Simplification and BODMAS

Three or four marks in SSC CGL Tier 1 come straight from simplification — and every single one of them is lost or won on one rule: the correct order of operations.

Definition: BODMAS is the universally accepted precedence order for arithmetic operations: Brackets → Orders (powers and roots) → Division → Multiplication → Addition → Subtraction.

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Breaking Down BODMAS

B — Brackets

Four types of brackets, resolved from innermost to outermost:

Level	Symbol	Name	Resolve order
1 (innermost)	—	Vinculum (overbar)	First
2	( )	Round bracket	Second
3	{ }	Curly bracket	Third
4 (outermost)	[ ]	Square bracket	Last

Example: [ {(3 + 2) × 4} − 10 ]

• Step 1 — Round: (3 + 2) = 5
• Step 2 — Curly: {5 × 4} = 20
• Step 3 — Square: [20 − 10] = 10

Vinculum example: 3 + 4̄+6̄ × 2 (where the bar covers 4+6)

• Vinculum first: 4+6 = 10 (the bar acts like hidden brackets)
• Then: 3 + 10 × 2 = 3 + 20 = 23

O — Orders (Powers and Roots)

After all brackets are cleared, evaluate exponents and roots.

Example: 2 + 3² − √16 + 5
= 2 + 9 − 4 + 5 (evaluate 3² = 9 and √16 = 4 first)
= 12

D and M — Division and Multiplication (Equal Priority)

Critical rule: Division and Multiplication have the same priority — do them left to right as they appear.

The most common SSC mistake:

• 20 ÷ 4 × 5
• Wrong: 20 ÷ (4 × 5) = 20 ÷ 20 = 1 ✗
• Correct: (20 ÷ 4) × 5 = 5 × 5 = 25 ✓

Another example: 48 ÷ 6 ÷ 2

• Left to right: (48 ÷ 6) ÷ 2 = 8 ÷ 2 = 4 ✓
• NOT 48 ÷ (6 ÷ 2) = 48 ÷ 3 = 16 ✗

A and S — Addition and Subtraction (Equal Priority)

Similarly, Addition and Subtraction have equal priority — go left to right.

Example: 10 − 3 + 5
= (10 − 3) + 5 = 7 + 5 = 12 ✓
NOT 10 − (3 + 5) = 10 − 8 = 2 ✗

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Why "BODMAS" Works as a Mnemonic

In India, BODMAS is used universally. Some countries use PEMDAS or BIDMAS — same order, different mnemonics:

• P = Parentheses = B (Brackets)
• E = Exponents = O (Orders)
• MD = Multiplication/Division (left to right)
• AS = Addition/Subtraction (left to right)

Real-world example: Cooking recipe proportions. If a recipe calls for (2 + 3) × 4 cups of flour: you add first (inside brackets), then multiply. If you ignore brackets and do 2 + 3 × 4 = 2 + 12 = 14 instead of (2+3) × 4 = 20, you ruin the recipe.

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Common SSC CGL Question Types

Type 1 — Pure Simplification

Question: Simplify: 18 − 3 × 4 + 12 ÷ 3

Solution:
Step 1 (O): None.
Step 2 (D/M left-to-right): 3 × 4 = 12; 12 ÷ 3 = 4.
Expression becomes: 18 − 12 + 4
Step 3 (A/S left-to-right): (18 − 12) + 4 = 6 + 4 = 10

Type 2 — Nested Brackets

Question: Simplify: [5 + {3 × (8 − 4) ÷ 6}]

Solution:
Step 1 — Round bracket: (8 − 4) = 4
Step 2 — Curly bracket (left to right): 3 × 4 = 12; 12 ÷ 6 = 2. {3 × 4 ÷ 6} = 2
Step 3 — Square bracket: [5 + 2] = 7

Type 3 — Vinculum (bar notation)

Question: Simplify: 2 + 3 × 4̄+2̄ (bar over "4+2")

Solution:
Step 1 — Vinculum: 4 + 2 = 6
Step 2 — Multiplication: 3 × 6 = 18
Step 3 — Addition: 2 + 18 = 20

Type 4 — Mixed with Powers

Question: Simplify: 4² − 3 × (6 − 2) + 8 ÷ 2

Solution:
Step 1 — Brackets: (6 − 2) = 4
Step 2 — Orders: 4² = 16
Step 3 — D/M left-to-right: 3 × 4 = 12; 8 ÷ 2 = 4. Expression: 16 − 12 + 4
Step 4 — A/S left-to-right: (16 − 12) + 4 = 4 + 4 = 8

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The Five Most Common Mistakes

Mistake 1 — Treating D and M as having different priorities:
24 ÷ 3 × 2 = 8 × 2 = 16 (correct). NOT 24 ÷ 6 = 4.

Mistake 2 — Same for A and S:
10 − 4 + 3 = 6 + 3 = 9 (correct). NOT 10 − 7 = 3.

Mistake 3 — Ignoring vinculum:
Students treat bar-notation the same as multiplication without parentheses.

Mistake 4 — Applying operations right-to-left:
Division especially feels "natural" right-to-left, but rules require left-to-right.

Mistake 5 — Skipping the bracket hierarchy:
Solving square brackets before inner round brackets.

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Full Worked Example (SSC CGL Style)

Question: Simplify: [3 + {5 × (12 − 8)² ÷ 20} − 2]

Solution:

Step 1 — Innermost bracket (round): (12 − 8) = 4

Step 2 — Orders: 4² = 16

Step 3 — Curly bracket (D/M left-to-right): 5 × 16 = 80; 80 ÷ 20 = 4. So {5 × (12−8)² ÷ 20} = 4

Step 4 — Square bracket: [3 + 4 − 2] = [7 − 2] = 5

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SSC CGL Examination Insight

SSC CGL Tier I typically includes 3–5 simplification questions. They test:

1. Pure BODMAS order (no tricks)
2. Bracket hierarchy
3. Fraction simplification with BODMAS
4. Mixed operations with integers and decimals

Time target: 30–45 seconds per simplification question. If you find yourself spending more time, practice until the order of operations is automatic.

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Exam halls have claimed countless marks from students who knew the right method but couldn't execute fast enough — mastering these simplification formulas turns a 3-minute slog into a 30-second win.

Algebraic Identities: Your Speed-Calculation Arsenal

Algebraic identities are equations that hold true for all values of the variables. Unlike equations that you solve, identities are tools you apply — they let you expand, factor, and rearrange expressions without grinding through long multiplication.

The Standard Expansion Identities

Definition: An algebraic identity is an equality that is satisfied by every value of the variable(s) involved.

The six identities you must memorise cold:

Identity	Form	When to use
(a + b)² = a² + 2ab + b²	Perfect square sum	Expand squares of binomials
(a − b)² = a² − 2ab + b²	Perfect square difference	Same, with subtraction
a² − b² = (a + b)(a − b)	Difference of squares	Factor or evaluate quickly
(a + b)³ = a³ + 3a²b + 3ab² + b³	Cube expansion	Expand cubed binomials
a³ + b³ = (a + b)(a² − ab + b²)	Sum of cubes	Factor sum-of-cubes expressions
a³ − b³ = (a − b)(a² + ab + b²)	Difference of cubes	Factor difference-of-cubes

Why it matters: SSC CGL, RRB NTPC, and banking exams frequently ask you to find the value of expressions like a² + b² given a + b and ab. Without these identities you'd need to find a and b separately — with them you can compute the answer in one line.

Using Identities Backwards (Factoring Direction)

The real power comes from using identities in reverse. Consider 9x² − 25. Recognise this as (3x)² − (5)², which immediately factors as (3x + 5)(3x − 5). This "reverse application" is the key to simplifying complex fractions and expressions.

Real-world example (exam-style): If a + b = 7 and ab = 12, find a² + b².

Using (a + b)² = a² + 2ab + b²:
49 = a² + 24 + b²
a² + b² = 25

No need to solve for a and b individually.

Fraction Shortcuts

Division of Fractions

Definition: To divide one fraction by another, multiply the first fraction by the reciprocal of the second.

a/b ÷ c/d = a/b × d/c = ad/bc

Common misconception: Many students try to divide numerators and denominators separately — (a ÷ c)/(b ÷ d) — which gives the wrong answer unless a/c and b/d simplify cleanly. Always flip-and-multiply.

Question: Simplify (3/5) ÷ (9/10).
Solution: (3/5) × (10/9) = 30/45 = 2/3.

Mixed Fractions

Definition: A mixed fraction (or mixed number) combines a whole number and a proper fraction: 3½ means 3 + ½.

Conversion rule: 3½ = (3 × 2 + 1)/2 = 7/2.

General rule: a(b/c) = (ac + b)/c.

Why it matters: Exam questions often present data in mixed fractions. Converting incorrectly is one of the most common sources of careless errors.

Surds: Rules for Working with Square Roots

Definition: A surd is an irrational root that cannot be simplified to a rational number — for example, √2, √3, √7.

The Three Surd Laws

1. Product rule: √a × √b = √(ab) (only valid when a ≥ 0, b ≥ 0)
2. Quotient rule: √a / √b = √(a/b)
3. Conjugate product: (√a + √b)(√a − √b) = a − b

The third rule is especially powerful for rationalising denominators — a common exam task.

Worked Example:

Question: Rationalise 1/(√5 + √2).

Solution:
Step 1: Multiply numerator and denominator by the conjugate (√5 − √2).
Step 2: Denominator = (√5)² − (√2)² = 5 − 2 = 3.
Step 3: Numerator = (√5 − √2).
Conclusion: Result = (√5 − √2)/3.

Common misconception: √(a + b) ≠ √a + √b. This is wrong for almost all values. For example, √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. The product rule works; the addition rule does not.

Quick Decimal Equivalents to Memorise

Fraction arithmetic becomes much faster when you have common conversions memorised. These appear directly in options or are needed to compare quantities:

Fraction	Decimal	Memory hook
1/2	0.5	Base reference
1/3	0.333…	Repeating 3s
1/4	0.25	Half of half
1/5	0.2	One-fifth
1/6	0.1667	1/2 of 1/3
1/7	0.1429	7 goes ~7 times into 1
1/8	0.125	Half of 0.25
1/9	0.1111	Repeating 1s

Why it matters: Comparing options like "is 5/7 or 8/11 larger?" becomes immediate if you convert: 5/7 ≈ 0.714, 8/11 ≈ 0.727 → 8/11 is larger. No cross-multiplication needed.

Bringing It Together: A Multi-Step Simplification

Question: Evaluate [(√5 + √3)² − (√5 − √3)²] / (2√15).

Solution:
Step 1: Expand using (a+b)² and (a−b)²:
(√5 + √3)² = 5 + 2√15 + 3 = 8 + 2√15
(√5 − √3)² = 5 − 2√15 + 3 = 8 − 2√15

Step 2: Subtract: (8 + 2√15) − (8 − 2√15) = 4√15.

Step 3: Divide: 4√15 / 2√15 = 2.

Conclusion: The answer is 2.

Notice how using identities directly avoided any need to compute √5 or √3 individually.

One wrongly-placed bracket can cost you the entire mark — BODMAS is not just a rule to recall but a discipline to apply step by step, every single time.

Definition: BODMAS (also written BIDMAS or PEMDAS in other syllabi) is the standard order of operations: Brackets, Of (powers/exponents), Division, Multiplication, Addition, Subtraction. Operations of equal precedence (D & M, or A & S) are evaluated left to right.

Understanding the Bracket Hierarchy

When expressions contain nested brackets, you work from the innermost bracket outward:

Bracket Type	Symbol	Order of evaluation
Round (parentheses)	( )	First (innermost)
Curly (braces)	{ }	Second
Square	[ ]	Third (outermost)

Why it matters: Indian competitive exams — SSC CGL, SSC CHSL, RRB NTPC — consistently include 2–3 BODMAS questions per Quantitative Aptitude section. A single sign error typically shifts the answer by exactly the value that eliminates your option, making it easy to "trap" hasty solvers.

Full Worked Example 1 (Nested Brackets)

Question: Simplify 18 − [6 − {4 + (8 − 3)}].

Solution:

Step 1: Innermost bracket ( ): 8 − 3 = 5.
Expression becomes: 18 − [6 − {4 + 5}]

Step 2: Curly bracket { }: 4 + 5 = 9.
Expression becomes: 18 − [6 − 9]

Step 3: Square bracket [ ]: 6 − 9 = −3.
Expression becomes: 18 − (−3)

Step 4: Subtracting a negative equals adding: 18 + 3 = 21.

Conclusion: The answer is 21.

Common misconception corrected: "Subtracting a negative" trips up the majority of students. When you see − (−3), think of it as removing a debt of 3 — you gain 3. Algebraically: −(−3) = +3. This single rule accounts for a large fraction of wrong answers in BODMAS questions.

Full Worked Example 2 (Order of Operations, No Extra Brackets)

Question: Evaluate 5 + 3 × 4 − 2.

Solution:

Step 1: Multiplication before Addition/Subtraction: 3 × 4 = 12.
Expression becomes: 5 + 12 − 2.

Step 2: Left to right: 5 + 12 = 17, then 17 − 2 = 15.

Conclusion: The answer is 15 — NOT 30.

Common misconception corrected: A very common error is to read left-to-right naively: 5 + 3 = 8, then 8 × 4 = 32, then 32 − 2 = 30. This is wrong because multiplication must be performed before addition regardless of position. BODMAS applies universally.

Worked Example 3 (Mixing All Operations)

Question: Simplify 36 ÷ 6 + 2² × 3 − (8 − 5).

Solution:

Step 1: Brackets first: (8 − 5) = 3.
Expression: 36 ÷ 6 + 2² × 3 − 3.

Step 2: "O" — orders/exponents: 2² = 4.
Expression: 36 ÷ 6 + 4 × 3 − 3.

Step 3: Division and Multiplication (left to right):
36 ÷ 6 = 6; 4 × 3 = 12.
Expression: 6 + 12 − 3.

Step 4: Addition and Subtraction (left to right):
6 + 12 = 18; 18 − 3 = 15.

Conclusion: The answer is 15.

Worked Example 4 (The "Of" Trap)

Question: Find: 1/2 of 48 + 6 × 2 − 10.

Solution:

Step 1: "Of" acts like multiplication and comes after brackets, same rank as ×:
1/2 of 48 = 24; separately 6 × 2 = 12.
Expression: 24 + 12 − 10.

Step 2: Left to right: 24 + 12 = 36; 36 − 10 = 26.

Conclusion: The answer is 26.

Why it matters: "Of" in problems like "one-third of 90" is multiplication in disguise. Students who treat it as addition last are consistently wrong.

A Framework for Error-Free BODMAS

Use this checklist mentally before writing each step:

1. Scan the entire expression — identify all bracket layers.
2. Peel from inside out: ( ) first, then { }, then [ ].
3. Check for exponents immediately after all brackets are resolved.
4. Do all × and ÷ left-to-right in one pass.
5. Do all + and − left-to-right in one pass.
6. Verify sign whenever subtracting — a −(−n) is always +n.

Quick-revision notes for Simplification and BODMAS — the must-know points for SSC CGL Tier-I/II.

• BODMAS gives the order of operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.
• Division and multiplication have equal priority — do them left to right; same for addition and subtraction.
• "of" means multiplication and is done with orders/brackets before division.
• Simplify brackets in order: ( ), { }, [ ] from innermost outward.
• Convert mixed fractions and decimals carefully before combining.
• Use approximation for "approximate value" questions to save time.

Cover the answer, recall, then check. 7 cards on the must-know Simplification and BODMAS facts for SSC CGL.

Q1. What does BODMAS stand for?
A1. Brackets, Orders, Division, Multiplication, Addition, Subtraction.

Q2. Between division and multiplication, which is done first?
A2. Neither has priority — perform them left to right in order of appearance.

Q3. Evaluate 8 + 2 × 5.
A3. 18 — multiplication first (2×5 = 10), then 8 + 10.

Q4. What operation does "of" represent in simplification?
A4. Multiplication (evaluated with orders/brackets, before division).

Q5. In what order do you open brackets?
A5. Innermost first: ( ), then { }, then [ ].

Q6. Evaluate (6 + 4) ÷ 2 × 3.
A6. 15 — brackets: 10; then 10÷2 = 5; then 5×3 = 15.

Q7. Evaluate 12 ÷ 4 + 3 × 2.
A7. 9 — 3 + 6 = 9.