Comparing Quantities - Compound Interest

Compound Interest (CI) is the interest calculated on the previous period's principal plus the accumulated interest. Unlike Simple Interest, where interest is only earned on the initial sum, Compound Interest leads to 'interest on interest,' causing the amount to grow faster over time. Visually, if Simple Interest is represented by a straight line on a graph, Compound Interest is represented by an upward-curving line that gets steeper as time passes.

The Conversion Period is the fixed time interval after which the interest is added to the principal to form the new principal for the next period. Interest can be compounded annually (once a year), half-yearly (every 6 months), or quarterly (every 3 months). On a timeline, annual compounding has one jump per year, while half-yearly compounding shows two jumps per year, effectively splitting the interest calculation into two smaller steps.

Principal ($P$) is the initial amount of money borrowed or invested, and the Amount ($A$) is the total sum of money at the end of the time period. The difference between the Amount and the Principal ($A - P$) gives the Compound Interest. Imagine two bars in a chart: the first bar represents the Principal, and the second, taller bar represents the Amount; the extra height on the second bar represents the total Compound Interest accumulated.

When interest is compounded half-yearly, we must adjust the variables to match the conversion periods: the annual Rate of interest ($R$) is divided by $2$ ($\\frac{R}{2}\%$), and the Time ($n$) is multiplied by $2$ ($2n$). This reflects that the interest is calculated twice as often at half the rate.

The Compound Interest formula is also used for calculating Population Growth. If a population increases at a constant rate, the final population is calculated like the Amount ($A$). If the population decreases, the rate becomes negative. Visually, population growth mirrors the exponential curve of compound interest rising over time.

Depreciation refers to the reduction in the value of an item (like a car or machinery) over time due to use and age. In this case, the value decreases, so we use a negative sign in the formula: $A = P(1 - \\frac{R}{100})^n$. On a graph, this would be shown as a downward curve that flattens out as the value approaches zero but never quite reaches it.