Fundamental Arithmetical Operations — Study Notes

Overview

Fundamental Arithmetical Operations form the backbone of the Elementary Mathematics section in SSC GD. Every question—whether it's percentage, profit-loss, or time-distance—ultimately relies on your speed and accuracy in basic arithmetic and simplification. This topic tests your ability to apply the correct order of operations (BODMAS/BODMAS rule), handle fractions and decimals, and simplify complex numerical expressions within strict time constraints.

In the SSC GD exam, you'll face 3–5 direct simplification questions worth 1 mark each. More importantly, mastery here accelerates every other math topic because you'll spend less time on calculation and more on problem-solving. Students often lose marks not because they don't know the method, but because they make arithmetic slips or apply operations in the wrong order. Treating this as a "basic" topic and skipping practice is the single biggest mistake—the exam rewards speed and accuracy over theoretical knowledge.

Your goal is threefold: internalize the BODMAS sequence so it becomes automatic, develop mental math shortcuts for common operations, and practice mixed simplification problems daily until your error rate drops below 5%. This topic is not about difficulty—it's about discipline and drill.

Key Concepts

• **BODMAS Rule**: The universal order of operations is Brackets first, then Orders (powers/roots), followed by Division and Multiplication (left to right), and finally Addition and Subtraction (left to right). Many students misremember this as "do multiplication before division"—both have equal priority and are solved left to right.

• **Brackets Hierarchy**: When multiple brackets appear, solve innermost first: ( ) are solved first, then { }, then [ ]. Always work inside-out, simplifying the deepest nested operation before moving outward.

• **Left-to-Right Rule**: For operations of equal priority (division and multiplication, or addition and subtraction), always proceed from left to right. The expression 20 ÷ 4 × 5 equals (20 ÷ 4) × 5 = 25, not 20 ÷ (4 × 5) = 1.

• **Fraction Operations**: To add/subtract fractions, find the LCM of denominators. To multiply, multiply numerators and denominators directly. To divide, multiply by the reciprocal of the divisor. Mixed fractions must be converted to improper fractions before any operation.

• **Decimal Alignment**: When adding or subtracting decimals, align decimal points vertically. When multiplying, count total decimal places in both numbers and place the point accordingly in the answer. For division, shift the decimal in divisor to make it whole, then shift equally in dividend.

• **Order of Powers**: When simplifying expressions with exponents, remember powers are "orders" in BODMAS—they come after brackets but before multiplication/division. Also recall: x^a × x^b = x^(a+b), x^a ÷ x^b = x^(a-b), and (x^a)^b = x^(a×b).

• **Negative Number Rules**: Two negatives make a positive when multiplying or dividing: (–3) × (–4) = +12. When adding/subtracting, treat the negative as "opposite direction": 5 – (–3) becomes 5 + 3 = 8.

• **Zero Properties**: Any number multiplied by zero equals zero. Division by zero is undefined (never write an answer for it). Zero divided by any non-zero number equals zero: 0 ÷ 5 = 0.

Formulas / Key Facts

**BODMAS Sequence**: B (Brackets) → O (Orders/Powers) → DM (Division and Multiplication, left to right) → AS (Addition and Subtraction, left to right)

**Fraction Addition**: a/b + c/d = (ad + bc)/bd or find LCM of b and d

**Fraction Multiplication**: a/b × c/d = ac/bd

**Fraction Division**: a/b ÷ c/d = a/b × d/c = ad/bc

**Decimal Multiplication**: Count total decimal places in factors; answer has same total decimal places

**Percentage to Fraction**: x% = x/100

**Power of 10**: 10^n means 1 followed by n zeros; 10^–n means decimal with point n places left

**Reciprocal**: Reciprocal of a/b is b/a; reciprocal of x is 1/x

**Absolute Value**: |x| means distance from zero, always non-negative

Worked Examples

**Example 1**: Simplify 48 ÷ 8 + 5 × 3 – 4

*Step 1*: Apply BODMAS. No brackets or orders, so start with division and multiplication from left to right. *Step 2*: 48 ÷ 8 = 6, so expression becomes 6 + 5 × 3 – 4 *Step 3*: 5 × 3 = 15, so expression becomes 6 + 15 – 4 *Step 4*: Now only addition and subtraction remain. Work left to right: 6 + 15 = 21 *Step 5*: 21 – 4 = 17

**Answer**: 17

**Example 2**: Simplify 3/4 + 5/6 – 1/2

*Step 1*: Find LCM of denominators 4, 6, and 2. LCM = 12 *Step 2*: Convert each fraction: 3/4 = 9/12, 5/6 = 10/12, 1/2 = 6/12 *Step 3*: Now compute: 9/12 + 10/12 – 6/12 = (9 + 10 – 6)/12 = 13/12 *Step 4*: Convert to mixed fraction: 13/12 = 1 1/12

**Answer**: 1 1/12

**Example 3**: Simplify 12 + {15 – (6 + 3 × 2)}

*Step 1*: Start with innermost bracket (6 + 3 × 2). Apply BODMAS: 3 × 2 = 6 first *Step 2*: (6 + 6) = 12 *Step 3*: Now solve curly bracket: 15 – 12 = 3 *Step 4*: Finally: 12 + 3 = 15

**Answer**: 15

**Example 4**: Simplify 0.5 + 0.25 × 4

*Step 1*: Apply BODMAS. Multiplication before addition. *Step 2*: 0.25 × 4 = 1.00 *Step 3*: 0.5 + 1.00 = 1.5

**Answer**: 1.5 (Note: Many students wrongly do 0.75 × 4 = 3 by adding first)

Common Mistakes

**Mistake**: Doing addition before multiplication when no brackets are present → **Fix**: Always apply BODMAS strictly. In 5 + 3 × 2, multiply first to get 5 + 6 = 11, not 8 × 2 = 16.

**Mistake**: Treating division and multiplication as "division first" or vice versa → **Fix**: They have equal priority. Always work left to right: 12 ÷ 3 × 2 = 4 × 2 = 8, not 12 ÷ 6 = 2.

**Mistake**: Adding fractions by adding numerators and denominators separately (1/2 + 1/3 = 2/5) → **Fix**: Find common denominator first. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

**Mistake**: Forgetting to convert mixed fractions before operations → **Fix**: Always convert mixed to improper first. For 2 1/2 × 3, convert to 5/2 × 3 = 15/2, then convert back if needed.

**Mistake**: Misplacing decimal points in multiplication (0.2 × 0.3 = 0.6) → **Fix**: Count decimal places. 0.2 (1 place) × 0.3 (1 place) = 0.06 (2 places total).

Quick Reference

• **BODMAS order**: Brackets → Orders → Div/Mult (L→R) → Add/Sub (L→R)
• **Equal priority operations**: Always solve left to right
• **Fraction addition**: Common denominator required; multiplication does not need it
• **Negative × Negative = Positive**; Negative × Positive = Negative
• **Any number × 0 = 0**; division by 0 is undefined
• **Practice daily**: 20 simplification problems will build speed and eliminate careless errors