Chapter 1
Number System & Estimation
Indian vs International Place Value
1 Million = 10 Lakhs (10^6), 1 Crore = 10 Million (10^7)
In Indian system, commas are placed after 3 digits, then every 2 digits (e.g., 1,00,00,000). In International, commas are placed every 3 digits (10,000,000).
Variables: Base 10 place-value expansion: N = d_k × 10^k + ... + d_0 × 10^0
Exam Tip: Remember Roman symbols: I=1, V=5, X=10, L=50, C=100, D=500, M=1000. Symbols V, L, D are never repeated.
Distributive Law of Multiplication
a × (b + c) = (a × b) + (a × c)
Multiplication distributes over addition, enabling quick mental calculations for large products like 12 × 105 = 12 × (100 + 5) = 1200 + 60 = 1260.
Variables: a, b, c are whole numbers
Exam Tip: Use this property to avoid lengthy long-multiplications in Olympiad exams.
Chapter 2
Fractions & Decimals
Fraction Multiplication & Division
(a/b) × (c/d) = (a × c) / (b × d) | (a/b) ÷ (c/d) = (a/b) × (d/c)
To divide by a fraction, multiply by its reciprocal. Reciprocal of a/b is b/a where product is 1.
Variables: b ≠ 0, c ≠ 0, d ≠ 0
Exam Tip: Cross-cancel common factors between numerators and denominators BEFORE multiplying to keep numbers small.
Cross-Multiplication Comparison
a/b > c/d <=> a × d > b × c
Allows comparing two fractions instantaneously without computing common LCM.
Variables: a, b, c, d are positive integers
Exam Tip: Always multiply numerator of first by denominator of second on the left side.
Chapter 3
Basic Algebra
Linear Equation Standard Form
ax + b = c => x = (c - b) / a
Whatever arithmetic operation is performed on LHS must be performed on RHS to maintain equality.
Variables: x = unknown variable, a ≠ 0, b and c are constants
Exam Tip: Always verify the root by back-substituting x into the original expression.
Chapter 4
Geometry & Mensuration Basics
Perimeter Formulas
SI: meters (m) or cmPerimeter of Rectangle = 2(l + w) | Perimeter of Square = 4 × s | Equilateral Triangle = 3 × s
Perimeter is the total boundary length of a 2-dimensional closed shape.
Variables: l = length, w = width, s = side
Exam Tip: Check that all dimensions are in the same units (e.g. convert meters to cm if mixed) before summing.
Area Formulas
SI: square meters (m²) or cm²Area of Rectangle = l × w | Area of Square = s²
Area measures the amount of 2D surface enclosed inside the boundary.
Variables: l = length, w = width, s = side length
Exam Tip: 1 m² = 10,000 cm² (because 100 cm × 100 cm). Do not confuse linear conversion with area conversion.