Is it true that All equations have only one solution.?

Simple quadratic equations can have zero, one, or two solutions (e.g., x² = 16 has solutions x = 4 and x = -4; x² = 0 has one solution x = 0; x² = -3 has no real solutions).

Is it true that A ratio like 2:3 is always smaller than a ratio like 4:5.?

Ratios must be compared after simplification or by converting them to a common format (e.g., fractions or decimals). 2:3 (or 0.66...) is greater than 4:5 (or 0.8).

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

Mastering fundamental algebraic equations, geometric theorems, proportional reasoning, and financial calculations provides a robust foundation for advanced mathematical concepts and real-world problem-solving.

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 using equivalent ratios and simplify rates to their most basic form.
Financial Mathematics: Understand percentages, mark-ups, discounts, and apply unitary methods for financial calculations.

Action Steps

Solve equations by performing inverse operations on both sides to isolate the variable.
Apply Pythagoras' Theorem to find the length of a missing side in a right-angled triangle.
Simplify ratios by dividing by the Highest Common Factor (HCF) or converting to whole numbers.
Calculate percentage increase or decrease by finding the difference and dividing by the original amount, then multiplying by 100.
Use the unitary method to find the value of one unit, then scale up or down to find the required quantity.

Formulas

Pythagoras' Theorem: $a^2 + b^2 = c^2$
Perimeter of a rectangle: $P = 2l + 2b$
Speed: $s = \frac{d}{t}$

Key Terms

Pronumeral: A letter used to represent an unknown number or quantity in an equation or expression.
Equivalent Equations: Equations that have the same solution set, achieved by performing the same operation on both sides of the original equation.
Pythagorean Triple: A set of three positive integers (a, b, c) that satisfy the equation a² + b² = c², such as (3, 4, 5).
Unitary Method: A method of solving problems by first finding the value of one unit, then using that to find the value of any number of units.
Scale Factor: The ratio of a length on a scaled drawing or model to the corresponding length on the original object.

Pro Tips

When solving equations with negative coefficients, multiply both sides by -1 to make the coefficient positive, remembering to reverse the inequality sign if applicable.
For problems involving scale, ensure units are consistent before calculating actual or scaled distances.
When dealing with financial calculations, clearly identify the cost price, selling price, profit, loss, mark-up, or discount to apply the correct formula.

Pitfalls to Avoid

Forgetting to reverse the inequality sign when multiplying or dividing by a negative number causes incorrect solutions for inequalities.
Using inconsistent units in ratio or rate problems leads to inaccurate comparisons and calculations.
Incorrectly applying percentage formulas, such as confusing mark-up with margin, results in wrong financial outcomes.

Myth vs Reality

All equations have only one solution.: Simple quadratic equations can have zero, one, or two solutions (e.g., x² = 16 has solutions x = 4 and x = -4; x² = 0 has one solution x = 0; x² = -3 has no real solutions).
A ratio like 2:3 is always smaller than a ratio like 4:5.: Ratios must be compared after simplification or by converting them to a common format (e.g., fractions or decimals). 2:3 (or 0.66...) is greater than 4:5 (or 0.8).

Real World Examples

An orange costs 5c more than an apple, and together they cost $1.15.: Setting up and solving the equation 2x + 5 = 115 (where x is the cost of an apple) to find that apples cost 55 cents and oranges cost 60 cents.
A model car is 17 cm long, with a scale of 1:25.: Using the scale factor to calculate the actual car's length as 25 * 17 cm = 425 cm, or 4.25 m.
A truck uses 4 L of petrol to travel 36 km.: Using the unitary method to determine how far the truck will travel using 70 L of petrol (calculating 9 km per litre, then 9 * 70 = 630 km).

Statistics

Cost of an apple in a problem-solving example: 55 cents
Cost of an orange in a problem-solving example: 60 cents
Weight of 6 identical books: 1.2 kg
Average speed calculation example: 72 km/h
Petrol usage for a truck: 4L per 36 km