Mathematics: Key Facts, Formulas and Solved Examples (Grades 1–10)

• Natural numbers 1, 2, 3…; whole numbers 0, 1, 2…; integers …–2, –1, 0, 1, 2…
• Prime number – exactly two factors (2, 3, 5, 7, 11…). 2 is the only even prime. 1 is neither prime nor composite.
• HCF × LCM = product of two numbers.

Solved example: LCM of 12 and 18. 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 36. HCF = 2 × 3 = 6. Check: 6 × 36 = 216 = 12 × 18. ✓

• To add fractions, make denominators equal: 3/4 + 1/8 = 6/8 + 1/8 = 7/8.
• Percentage = (part ÷ whole) × 100. 25% of 360 = 360 × 25/100 = 90.
• Discount: sale price = marked price × (100 – discount%) ÷ 100.
• Profit % = profit ÷ cost price × 100.
• Simple interest = P × R × T ÷ 100.

Solved example: after a 20% discount a shirt costs Rs 800. Marked price = 800 × 100 ÷ 80 = Rs 1,000.

• Rectangle: area = l × w; perimeter = 2(l + w)
• Square: area = s²; perimeter = 4s
• Triangle: area = ½ × base × height
• Circle: circumference = 2πr; area = πr² (π ≈ 22/7 or 3.14)
• Cuboid: volume = l × w × h; Cube: volume = a³
• Sum of interior angles of a polygon = (n – 2) × 180°; triangle 180°, quadrilateral 360°.

Solved example: circumference of a circle with r = 7 cm = 2 × 22/7 × 7 = 44 cm.

• Solve linear equations by doing the same operation on both sides: 3x – 7 = 11 → 3x = 18 → x = 6.
• (a + b)² = a² + 2ab + b²; (a – b)² = a² – 2ab + b²; a² – b² = (a + b)(a – b)
• Quadratic x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0 → x = 2 or 3.
• Slope of a line through (x₁, y₁), (x₂, y₂) = (y₂ – y₁) ÷ (x₂ – x₁).
• Mean = sum ÷ number of values; median = middle value of ordered data; mode = most frequent value.