Mathematics

Equations, Factors, and Formulae 1. Equations Linear Equations in One Unknown Solving for a variable by isolating it using inverse operations: Addition $\leftrightarrow$ Subtraction Multiplication $\leftrightarrow$ Division Squaring $\leftrightarrow$ Square Root Simultaneous Linear Equations Finding a common solution $(x, y)$ for two linear equations. Substitution Method: Express one variable in terms of the other from one equation, then substitute into the second. Elimination Method: Multiply equations to make coefficients of one variable equal (or opposite), then add or subtract the equations to eliminate that variable. ...

Fractions, Percentages, and Standard Form 1. Fractions Definitions Proper Fraction: Numerator is smaller than denominator (e.g., $1/2, 3/4$). Improper Fraction: Numerator is greater than or equal to denominator (e.g., $5/4, 7/3$). Mixed Number: Combination of a whole number and a proper fraction (e.g., $1 \frac{1}{4}$). Conversions Mixed Number $\rightarrow$ Improper Fraction: $\text{Whole} \times \text{Denominator} + \text{Numerator} / \text{Denominator}$. Improper Fraction $\rightarrow$ Mixed Number: Divide numerator by denominator; quotient is whole number, remainder is numerator. Basic Operations Addition/Subtraction: Find a common denominator, then add/subtract numerators. Example: $\frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}$ Example: $\frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}$ Multiplication: Multiply numerators together and denominators together. Example: $\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$ Division: Multiply by the reciprocal of the divisor (Keep-Change-Flip). Example: $\frac{2}{3} \div \frac{5}{7} = \frac{2}{3} \times \frac{7}{5} = \frac{14}{15}$ 2. Percentages Percentage of a Quantity: $\frac{\text{percentage}}{100} \times \text{quantity}$. Percentage Increase/Decrease: $\text{Change} = \frac{\text{percentage}}{100} \times \text{original}$. $\text{New Value} = \text{original} \pm \text{change}$. Simple Interest: $I = P \times R \times T$ (Principal $\times$ Rate $\times$ Time). Compound Interest: $A = P(1 + \frac{r}{100})^n$ (Total amount after $n$ years). Reverse Percentages: Finding the original value after a percentage change. $\text{Original} = \frac{\text{New Value}}{1 \pm \frac{\text{percentage}}{100}}$. Repeated Percentage Change: Applying a percentage change multiple times sequentially. Exponential Growth and Decay: Growth: $y = a(1 + r)^x$ Decay: $y = a(1 - r)^x$ Where $a$ is initial value, $r$ is rate, and $x$ is time. ...

Collection, Organising, and Displaying Data 1. Collection and Classifying Data Data can be categorized into two main types: Qualitative (Categorical): Non-numerical data. Nominal: Categories with no inherent order (e.g., eye color, country). Ordinal: Categories with a logical order (e.g., satisfaction rating: low, medium, high). Quantitative (Numerical): Numerical data. Discrete: Values that can only take specific values, often counts (e.g., number of students in a class). Continuous: Values that can take any value within a range, often measurements (e.g., height, weight, time). 2. Organising Data Before displaying data, it must be organized to make analysis easier. ...

Lines, Angles, and Shapes 1. Lines and Angles Basic Definitions Parallel Lines: Lines in a plane that never meet, no matter how far they extend. Perpendicular Lines: Lines that intersect at a right angle ($90^\circ$). Types of Angles: Acute: $0^\circ < \theta < 90^\circ$ Right: $\theta = 90^\circ$ Obtuse: $90^\circ < \theta < 180^\circ$ Reflex: $180^\circ < \theta < 360^\circ$ Angle Relationships Angles on a straight line: Sum to $180^\circ$. Angles around a point: Sum to $360^\circ$. Vertically opposite angles: Are equal. Parallel Line Angles: Corresponding Angles: Equal (F-shape). Alternate Angles: Equal (Z-shape). Co-interior Angles: Sum to $180^\circ$ (C-shape). ...

Making Sense of Algebra 1. Using Letters for Unknowns In algebra, letters (variables) are used to represent numbers that are either unknown or can change. Expression: A mathematical phrase without an equals sign (e.g., $3x + 5$). Equation: A statement that two expressions are equal (e.g., $3x + 5 = 11$). Formula: A rule showing the relationship between different variables (e.g., $A = lw$). 2. Substitution Substitution is the process of replacing variables with specific numerical values to evaluate an expression. ...