Compound Interest

• Exponential Growth: Unlike simple interest which grows by a fixed amount each period, compound interest grows by a fixed percentage. This means the absolute amount of interest earned increases every period because the base (principal + previous interest) is larger.

• The Compounding Effect: This principle describes how reinvesting earnings generates its own earnings. Over long durations, the 'interest on interest' component can eventually exceed the original principal in value.

• Mathematical Foundation: The relationship is modeled by the formula $A = P(1 + \frac{r}{100})^n$. The term $(1 + \frac{r}{100})$ is the growth factor, and raising it to the power of $n$ accounts for the repeated multiplication over time.

Forward Calculation

• Step 1: Identify the principal, the interest rate as a decimal, and the number of periods. Ensure the rate and the time periods are consistent (e.g., annual rate for annual periods).
• Step 2: Calculate the multiplier by adding the decimal rate to $1$. For a $4\%$ increase, the multiplier is $1.04$.
• Step 3: Apply the formula $A = P \times \text{multiplier}^n$ to find the final balance.

Reverse Calculation

• Finding the Principal: If the final amount is known, the original principal can be found by dividing the final amount by the growth factor raised to the power of $n$. The formula is $P = \frac{A}{\text{multiplier}^n}$.

Varying Interest Rates

• Sequential Multipliers: When interest rates change over time, calculate the balance for each segment separately or multiply the principal by a string of different multipliers. For example, $A = P \times m_1^{n_1} \times m_2^{n_2}$.

• Appreciation vs. Depreciation: Appreciation follows the standard compound interest model where the multiplier is greater than $1$. Depreciation uses the same logic but with a multiplier less than $1$ (e.g., $0.85$ for a $15\%$ decrease), resulting in an exponential decay of value.

• Formula Memorization: The compound interest formula is rarely provided in exam formula sheets. Students must memorize $A = P(1 + \frac{r}{100})^n$ and understand how to manipulate it for $P$.

• Multiplier Accuracy: Always use the full multiplier in your calculator rather than rounding intermediate steps. Small rounding errors in the multiplier can lead to large discrepancies when raised to a high power.

• Sanity Checks: For appreciation, the final answer must be larger than the principal. For depreciation, it must be smaller. If your answer for a $5\%$ increase over $10$ years is less than the starting value, check if you accidentally subtracted the rate from $1$.

• Read the Question Carefully: Distinguish between 'find the total value' and 'find the interest earned'. To find the interest earned, you must subtract the original principal from the final calculated amount ($I = A - P$).

• Confusing $n$ and $t$: Students often mistake the number of years for the number of compounding periods. If interest is compounded monthly for $2$ years, $n$ should be $24$, not $2$.

• Incorrect Multiplier Construction: A common error is using $1.5$ for a $5\%$ increase instead of $1.05$. Always divide the percentage by $100$ before adding it to $1$.

• Linear Thinking: Many students intuitively expect the interest in year $10$ to be the same as year $1$. It is vital to remember that in compound interest, the amount of interest added grows every single period.