Ratio, proportion and rates of change Study Guide

Inverse proportion describes a relationship between two quantities where an increase in one quantity results in a decrease in the other quantity, and vice-versa, in such a way that their product remains constant. If quantity A is inversely proportional to quantity B, we write this as A $\propto$ 1/B. To convert this proportionality into an equation, we introduce a constant of proportionality, 'k'. So, A = k/B, or equivalently, AB = k.

Developing and applying higher methods for inverse proportion often involves more complex scenarios. For instance, one quantity might be inversely proportional to the square or cube of another quantity, or even to a combination of quantities. If A is inversely proportional to B squared, we write A $\propto$ 1/B$^2$, which translates to A = k/B$^2$ or AB$^2$ = k. Similarly, if A is inversely proportional to the square root of B, then A $\propto$ 1/$\sqrt{B}$, meaning A = k/$\sqrt{B}$.

The process for solving problems involving inverse proportion generally follows these steps:

1. Write down the proportionality statement (e.g., y $\propto$ 1/x$^n$).
2. Introduce the constant of proportionality, k, to form an equation (e.g., y = k/x$^n$).
3. Use a given pair of values for the quantities to substitute into the equation and solve for k.
4. Write the complete formula relating the two quantities.
5. Use this formula to solve for an unknown value of one quantity when given a value for the other.

For example, if y is inversely proportional to x and y = 10 when x = 2:

1. y $\propto$ 1/x
2. y = k/x
3. 10 = k/2 $\implies$ k = 20
4. y = 20/x
5. If we need to find y when x = 5, then y = 20/5 = 4.

Inverse proportion problems can also involve real-world contexts, requiring careful interpretation of the problem to set up the correct proportionality before applying the algebraic steps.

Growth and decay problems describe situations where a quantity increases or decreases by a certain percentage over regular time intervals. These are common in real-world scenarios such as compound interest, population changes, or value depreciation. The key is to understand that the change is applied to the *current* amount, not the original amount each time. This is why we use multiplying factors. For growth, the multiplying factor is (1 + percentage increase as a decimal), and for decay, it's (1 - percentage decrease as a decimal). The general formula is Final Amount = Original Amount × (Multiplying Factor)^Number of Time Periods. For example, if a population grows by 5% per year, the multiplying factor is 1 + 0.05 = 1.05. If an item depreciates by 10% per year, the multiplying factor is 1 - 0.10 = 0.90. This algebraic approach allows us to calculate future values without step-by-step calculations for each time period, making it efficient and suitable for longer periods.