Grade 9 Mathematics NCAPS Core Concepts and Reference Card

Rational Numbers ($\mathbb{Q}$)

Numbers expressible in the form $\frac{a}{b}$ where $a, b \in \mathbb{Z}$ and $b \neq 0$. They include integers, terminating decimals, and recurring decimals.

Irrational Numbers ($\mathbb{Q}'$)

Non-terminating, non-recurring decimals that cannot be expressed as a ratio of two integers, such as $\pi$ and roots of non-perfect powers.

Highest Common Factor (HCF)

The product of the lowest powers of all common prime factors among two or more numbers.

Lowest Common Multiple (LCM)

The product of the highest powers of all prime factors appearing across the given numbers.

Direct Proportion

A relationship between two variables where their quotient remains constant ($\frac{y}{x} = k$ or $y = kx$), producing a straight-line graph passing through the origin.

Inverse Proportion

A relationship between two variables where their product remains constant ($x \times y = k$ or $y = \frac{k}{x}$), producing a hyperbolic curve.

Simple Interest Formula

$A = P(1 + \frac{r \times n}{100})$, where $P$ is the principal, $r$ is the annual interest rate percentage, and $n$ is the time in years.

Compound Interest Formula

$A = P(1 + \frac{r}{100})^n$, where interest earned each period is capitalized onto the principal.

Zero Exponent Law

Any non-zero base raised to the power of zero equals one: $a^0 = 1$ where $a \neq 0$.

Linear Sequence ($n^{\text{th}}$ Term)

$T_n = dn + c$, where $d$ is the constant first difference ($d = T_2 - T_1$) and $c = T_1 - d$.

Difference of Two Squares (DOTS)

The algebraic factorisation identity given by $a^2 - b^2 = (a - b)(a + b)$.

Zero-Factor Law

If the product of two factors is zero ($a \times b = 0$), then at least one factor must be zero ($a = 0$ or $b = 0$).

Conditions for Triangle Congruency ($\equiv$)

The four minimal criteria: $\text{SSS}$ (three sides), $\text{SAS}$ (two sides and included angle), $\text{SAA}$ / $\text{AAS}$ (two angles and one side), and $\text{RHS}$ (right angle, hypotenuse, side).

Conditions for Triangle Similarity ($\parallel\vert$)

Two triangles are similar if corresponding angles are equal (equiangular) or corresponding sides are in the same proportion ($\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$).

Theorem of Pythagoras

In any right-angled triangle with legs $a$ and $b$ and hypotenuse $c$, the square of the hypotenuse equals the sum of the squares of the legs: $c^2 = a^2 + b^2$.

Dimension Scaling Rule

Multiplying linear dimensions of a solid by factor $k$ multiplies perimeter by $k$, surface area by $k^2$, and volume by $k^3$.

Gradient ($m$)

The measure of slope calculated as vertical change divided by horizontal change: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Euler's Law for Polyhedra

A formula connecting faces ($F$), vertices ($V$), and edges ($E$) of any closed polyhedron: $F + V - E = 2$.

Theoretical Probability

The ratio of favourable outcomes to total possible outcomes in a uniform sample space: $P(E) = \frac{n(E)}{n(S)}$.