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

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

Core Principles

Equations: Maintain balance by performing the same operation on both sides to isolate the unknown.
Pythagoras' Theorem: In a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a² + b² = c²).
Rates and Ratios: Compare quantities of the same type using ratios, and compare quantities with different units using rates.
Financial Mathematics: Understand concepts like percentages, profit, loss, and applying unitary methods to solve financial problems.

Action Steps

To solve equations, perform inverse operations to isolate the variable, ensuring balance by applying the same operation to both sides.
To find the hypotenuse using Pythagoras' Theorem, square the lengths of the two shorter sides, add them, and then take the square root of the sum.
To find a shorter side using Pythagoras' Theorem, square the hypotenuse, square the known shorter side, subtract the square of the shorter side from the square of the hypotenuse, and take the square root of the result.
To simplify ratios, divide all parts of the ratio by their highest common factor or convert to a common unit first.
To solve problems involving rates, use the unitary method to find the value of one unit (e.g., cost per kg, speed per hour) and then scale up or down as needed.
To calculate percentage increase, find the percentage of the original value and add it to the original value.
To calculate percentage decrease, find the percentage of the original value and subtract it 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, typically containing one or more variables.
Inequality: A mathematical statement that compares two expressions using symbols such as <, >, ≤, or ≥.
Pythagorean Triple: A set of three positive integers (a, b, c) that satisfy the equation a² + b² = c².
Ratio: A comparison of two or more quantities of the same kind.
Rate: A ratio that compares two quantities measured in different units.
Unitary Method: A method used to solve problems by first finding the value of a single unit.
Percentage: A fraction out of 100, represented by the symbol '%'.

Real World Examples

An orange costs 5c more than an apple, and together they cost $1.15.: This problem is solved by setting up an equation: let x be the cost of an apple, then x + 5 is the cost of an orange. The equation becomes x + (x + 5) = 115 (in cents), leading to x = 55 cents for an apple and 60 cents for an orange.
A map has a scale of 1:20000, and a distance on the map is 5 mm.: To find the actual distance, multiply the map distance by the scale factor: 5 mm × 20000 = 100000 mm = 100 m.
A truck uses 4 L of petrol to travel 36 km.: Using the unitary method, find the distance per litre: 36 km / 4 L = 9 km/L. Then, to find the distance for 70 L, multiply: 9 km/L × 70 L = 630 km.
A pair of shoes is discounted by 20%, and the sale price is $120.: If the discount is 20%, the sale price represents 80% of the original price. To find the original price, calculate $120 / 0.80 = $150.

Quiz

What is the formula for Pythagoras' Theorem?: a² + b² = c²
If a ratio is 2:3, and the first part is 10, what is the second part?: 15
If a price is increased by 10% and the original price was $50, what is the new price?: $55