Ratio, proportion and rates of change Study Guide

Inverse proportion describes a relationship between two quantities where if one quantity increases, the other quantity decreases, and vice-versa, in such a way that their product remains constant. This is different from direct proportion, where both quantities increase or decrease together. Mathematically, if 'y' is inversely proportional to 'x', we write this as y ∝ 1/x. To convert this proportionality into an equation, we introduce a constant of proportionality, 'k'. So, the relationship becomes y = k/x, or equivalently, xy = k. The constant 'k' can be found by substituting a known pair of values for 'x' and 'y' into this equation. Once 'k' is found, the equation can be used to determine unknown values of 'y' for given 'x', or unknown 'x' for given 'y'.

Inverse proportion describes a relationship between two quantities where an increase in one quantity results in a decrease in the other, and vice versa, in such a way that their product remains constant. This means that if quantity A is inversely proportional to quantity B, then A multiplied by B will always equal a fixed number, often called the constant of proportionality, 'k'. Mathematically, this is expressed as A ∝ 1/B, or A = k/B, which can be rearranged to A × B = k. For example, if it takes 3 workers 10 hours to complete a task, it would take 6 workers 5 hours to do the same task (3 × 10 = 30, and 6 × 5 = 30). The constant of proportionality, k, represents the total 'amount' of work or effect. When solving problems involving inverse proportion, you typically first use a given pair of values to find the constant 'k'. Once 'k' is known, you can use the relationship A = k/B (or B = k/A) to find unknown values for either quantity when the other is given. This concept is fundamental in many real-world scenarios, such as the relationship between speed and time for a fixed distance, or the number of people sharing a fixed amount of resources.