Ratio, proportion and rates of change Study Guide

Growth and decay problems describe how a quantity changes over time, either increasing (growth) or decreasing (decay) at a particular rate. These problems are often modelled using a multiplier. For growth, the multiplier is greater than 1, and for decay, the multiplier is less than 1. The general formula for these situations is often expressed as: Final Amount = Initial Amount × (Multiplier)^Number of Periods. For a percentage increase of 'p%', the multiplier is (1 + p/100). For example, a 5% increase means a multiplier of (1 + 5/100) = 1.05. For a percentage decrease of 'p%', the multiplier is (1 - p/100). For example, a 10% decrease means a multiplier of (1 - 10/100) = 0.90. The 'Number of Periods' refers to how many times the growth or decay rate is applied. This could be years, months, or any other defined time interval. Being able to set up and solve these equations is crucial for real-world applications, such as calculating compound interest, population changes, or depreciation of assets. Developing a strong understanding of how to determine the correct multiplier and apply it iteratively is key to mastering these types of algebraic problems.

Growth and decay problems in algebra often involve situations where a quantity increases or decreases by a fixed percentage over a period. These can be modelled using exponential functions. The general form for growth is $A = P(1 + r)^n$ and for decay is $A = P(1 - r)^n$, where $A$ is the final amount, $P$ is the initial amount, $r$ is the rate of growth or decay (as a decimal), and $n$ is the number of periods. Higher methods involve not just calculating specific values but also working with these formulas to find unknown rates, initial amounts, or the number of periods, which may require logarithms. For example, if you need to find the number of periods when the final and initial amounts, and the rate, are known, you would rearrange the formula and use logarithms to solve for $n$. Similarly, finding an unknown rate might involve taking roots. Compound growth or decay can be applied to investments, population changes, depreciation, or radioactive decay. Understanding how to manipulate these exponential equations is key to solving more complex problems within this topic, including situations where the growth or decay happens over fractional or continuous periods, though for GCSE, focus remains on discrete periods. Real-world scenarios often require careful interpretation of the problem to set up the correct model and ensure that the rate $r$ is converted correctly from a percentage to a decimal.