Proportion
Understanding Proportion

Proportion: Is a relationship showing how two or more amounts compare to each other. It can be expressed as two ratios or fractions that are equal to each other.

Direct Proportion: This occurs when two quantities increase or decrease at the same rate. It can be expressed as y ∝ x (y is directly proportional to x) or y = kx (where k is the constant of proportionality).
Direct and Inverse Proportions

Inverse Proportion: Conversely, this is a relationship where an increase in one quantity corresponds to a decrease in another. Expressed as y ∝ 1/x (y is inversely proportional to x) or y = k/x.

Recognising proportionality: In cases of direct proportion, graphs through the origin will have a straight line, while inverse proportion graphs will have a hyperbolic curve.
Solving Proportional Problems

Finding the constant of proportionality: To determine the constant (k), use known values of x and y and solve the equation.

Solving for unknowns: When given a proportional equation and a set of values, substitution will help solve for unknowns.

Unitary method: This technique involves first finding the value of a single unit, which can then be used to find the values of the multiples.
Applying Proportional Relationships

Direct variations: This is when the ratio of two variables is constant. For instance, if speed is directly proportional to time, then doubling the time will also double the speed.

Inverse variations: In this scenario, the product of the two variables is constant. As an example, if speed is inversely proportional to time, then doubling the time will halve the speed.

Combining types of proportionality: A combination of direct and inverse proportionality can also occur, for example, y ∝ x/z (y is directly proportional to x and inversely proportional to z).
Practising Your Skills

Checking your solutions: Always verify your answer by substituting the computed value back into the initial equation.

Understanding proportionality: Regular practice of problems involving proportion will enhance understanding and application skills. Understanding the theory behind direct and inverse relationships is essential to successfully solving proportional equations.