Equations, Formulas & Financial Math Essentials

Solving algebraic equations and inequalities requires precise application of inverse operations to isolate unknown variables, while understanding various income structures and financial calculations is fundamental for personal economic literacy.

Core Principles

Equation Equivalence: Maintain balance by applying identical operations (add, subtract, multiply, divide) to both sides of an equation to derive simpler, equivalent forms.
Inequality Reversal: Reverse the inequality sign (e.g., < to >) when multiplying or dividing both sides by a negative number to preserve truth.
Formula Subject: Rearrange formulas using inverse operations to isolate the desired pronumeral (the 'subject') on one side, enabling calculation of its value.
Income Diversification: Recognize distinct employment types (Salary, Wages, Casual, Commission, Piece Work) each with unique pay structures, benefits, and risks.
Additional Payments: Factor in overtime (1.5x or 2x normal rate), bonuses, holiday loading (e.g., 17.5% of 4 weeks' pay), and superannuation (e.g., 9.25% of earnings) to accurately calculate total income.

Action Steps

1. Solve Equations: Remove fractions, expand grouping symbols, collect like terms, then make the pronumeral the subject using inverse operations.
2. Translate Problems to Equations: Carefully read the problem, identify knowns and unknowns (assign a pronumeral), connect information to form an equation, then solve.
3. Evaluate Formulas: Substitute known values for pronumerals into the formula, then solve the resulting equation for the unknown subject. Alternatively, rearrange the formula first, then substitute.
4. Calculate Superannuation: Multiply monthly earnings by the superannuation rate (e.g., 0.0925 for 9.25%), ensuring monthly earnings meet the minimum threshold ($450).

Formulas

Density: $D = \frac{M}{V}$
Kinetic Energy: $K = \frac{1}{2}mv^2$
Celsius to Fahrenheit: $C = \frac{5}{9}(F - 32)$
Compound Interest: $A = P\left(1 + \frac{r}{100}\right)^n$
Pythagorean Theorem: $c = \sqrt{a^2 + b^2}$
Velocity (constant acceleration): $v = u + at$
Area of Rectangle: $A = lb$
Perimeter of Rectangle: $P = 2(l+b)$
Surface Area of Sphere: $S = 4\pi r^2$
Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Key Terms

Equation: A number sentence with one or more unknown numbers (pronumerals) that states two expressions are equal.
Inequality: A number sentence where an inequality sign (>, <, ≥, ≤) replaces the equals sign, indicating a range of solutions.
Formula: A special type of equation representing a relationship between physical quantities, always containing more than one pronumeral.
Pronumeral: A symbol, usually a letter, representing an unknown number in an equation or formula.
Subject of a Formula: The pronumeral isolated by itself on the left-hand side of a formula, whose value is determined by the other variables.
Equivalent Equations: Equations that can be transformed into one another by performing the same operation on both sides, having the same solution.
Overtime: Time worked beyond standard hours, typically paid at 1.5 or 2 times the normal rate.
Holiday Loading: An additional payment, often a fixed percentage of normal pay over a set number of weeks, given for annual holidays.
Superannuation: Employer contributions to an investment fund for an employee's retirement, typically a percentage of earnings.

Pitfalls to Avoid

Unbalancing Equations: Failing to perform the exact same operation on both sides of an equation causes an incorrect solution.
Incorrect Inequality Reversal: Not reversing the inequality sign when multiplying or dividing by a negative number leads to an incorrect solution set.
Ignoring Grouping Symbols: Incorrectly expanding or neglecting grouping symbols (parentheses) before collecting like terms will lead to errors in solving.
Miscalculating Overtime/Loading: Applying incorrect rates (e.g., 1.5x vs 2x) or percentages (e.g., 17.5%) for additional payments results in inaccurate total earnings.

Real World Examples

A rectangle has a perimeter of 192m and its length is three times its width.: Set up the equation $2(x + 3x) = 192$ to find the width $x=24$m and length $72$m.
A father is 28 years older than his son and is currently three times as old as his son.: Solve the equation $3x - x = 28$ to find the son's age $x=14$ years, making the father $42$ years old.
Car A travels at 80 km/h starting at 6:00 am. Car B travels at 100 km/h starting at 7:30 am from the same location.: Set up $80(t+1.5) = 100t$ to find that Car B catches Car A $t=6$ hours after Car B starts, which is 1:30 pm.
Paul's monthly earnings are $5500, and the superannuation rate is 9.25%.: Calculate the superannuation contribution as $0.0925 \times 5500 = $508.75.
Peter works 35 normal hours at $17.60/hr, 4 hours at time-and-a-half, and 2 hours at double-time.: Total earnings are calculated as $(35 \times 17.60) + (4 \times 17.60 \times 1.5) + (2 \times 17.60 \times 2) = $792.