Year 8 Mathematics: Equations, Pythagoras, Rates, Ratios & Financial Math

Mastering fundamental mathematical concepts including solving equations and inequalities, applying Pythagoras' theorem, understanding ratios and rates, and basic financial calculations is crucial for Year 8 mathematics success.

Core Principles

Equations are balanced statements; operations applied to one side must be mirrored on the other.
Pythagoras' theorem (a² + b² = c²) relates the sides of a right-angled triangle.
Ratios compare quantities of the same type, while rates compare quantities of different types.
Financial mathematics involves understanding percentages for profit, loss, discounts, and growth.

Action Steps

To solve equations, isolate the unknown variable by performing inverse operations on both sides.
To find the hypotenuse (c) using Pythagoras' theorem, calculate $c = \sqrt{a^2 + b^2}$
To find a shorter side (a or b) using Pythagoras' theorem, calculate $a = \sqrt{c^2 - b^2}$ or $b = \sqrt{c^2 - a^2}$
To simplify ratios, divide all parts by their highest common factor.
To convert between units for rates, use the unitary method to find the value of one unit.
To calculate percentage increase, add the percentage of the original value to the original value.
To calculate percentage decrease, subtract the percentage of the original value from the original value.

Formulas

Pythagoras' Theorem: $a^2 + b^2 = c^2$
Perimeter of a rectangle: $P = 2l + 2b$
Speed, distance, time: $d = s \times t$

Key Terms

Equation: A mathematical statement that asserts the equality of two expressions, containing an equals sign (=).
Inequality: A mathematical statement that compares two expressions using symbols like >, <, ≥, or ≤.
Pythagorean Triple: A set of three positive integers (a, b, c) that satisfy the equation $a^2 + b^2 = c^2$.
Ratio: A comparison of two or more quantities of the same kind, often expressed in simplest form.
Rate: A comparison of two quantities of different kinds, often expressed per unit of another quantity (e.g., km/h, $/kg).
Unitary Method: A method of solving problems by first finding the value of one unit.
Percentage: A fraction out of 100, represented by the symbol '%'.

Real World Examples

An orange costs 5c more than an apple. An orange and an apple together cost $1.15.: This problem is solved using equations to find the cost of each fruit: Apples cost 55 cents, oranges cost 60 cents.
A model car is 17 cm long, with a scale of 1:25.: The actual car's length is calculated by multiplying the model length by the scale factor: 17 cm × 25 = 425 cm or 4.25 m.
A truck uses 4 L of petrol to travel 36 km.: Using the unitary method, you can find how far it will travel using 70 L: 4 L for 36 km means 1 L for 9 km, so 70 L will travel 70 × 9 = 630 km.
A pair of shoes is discounted by 20%, and the sale price is $120.: To find the original price, recognize that $120 represents 80% of the original price. The original price is $120 / 0.80 = $150.