Numbers, Operations, Fractions & Geometry

1. Large Numbers (Indian System)

Place Values

Ones. Tens. Hundreds. Thousands. Ten Thousands. LAKHS (1,00,000). Ten Lakhs. CRORES (1,00,00,000). Ten Crores.

Example: 4,56,78,900 = Four crore fifty-six lakh seventy-eight thousand nine hundred.

Roman Numerals

I(1), V(5), X(10), L(50), C(100), D(500), M(1000). Rules: If smaller BEFORE larger → SUBTRACT. If smaller AFTER larger → ADD. IX=9. XI=11.

2. The Four Operations (+, —, ×, ÷)

Addition and Subtraction (Large Numbers)

Align by place value. Add/subtract column by column. Carry/borrow as needed.

Multiplication

3-digit × 2-digit. 3-digit × 3-digit. 'Multiply each digit. Shift for tens/hundreds. Add the partial products.'

Division

Dividend ÷ Divisor = Quotient + Remainder. Dividend = Divisor × Quotient + Remainder. 'Division is the reverse of multiplication.'

3. Factors and Multiples

Divisibility Tests

HCF (Highest Common Factor) — Prime Factorisation or Long Division

LCM (Lowest Common Multiple) — Prime Factorisation. LCM × HCF = Product of the two numbers.

4. Fractions

Types: Proper (<1). Improper (≥1). Mixed (whole + proper).

Operations

• Add/Subtract: Find LCM of denominators. Convert. Add/subtract numerators.
• Multiply: (a/b) × (c/d) = (a×c)/(b×d). 'Multiply tops. Multiply bottoms.'
• Divide: (a/b) ÷ (c/d) = (a/b) × (d/c). 'Flip the second. Then multiply.'

Simplifying: Divide numerator and denominator by their HCF.

5. Decimals

Place Value: Tenths (0.1). Hundredths (0.01). Thousandths (0.001).

Operations

• Add/Subtract: Align decimal points.
• Multiply: Multiply normally. Count decimal places. Place the point.
• Divide: Move decimal to make divisor a whole number.

Conversion: Fraction ↔ Decimal. Divide numerator by denominator.

6. Geometry

Angles

Triangles

Circle

• Radius (from centre to edge). Diameter (2 × radius — through centre). Chord (joins two points). Circumference = 2πr.

Perimeter and Area

• Square: P=4s. A=s². Rectangle: P=2(l+w). A=l×w.