Compound Interest

• Compound interest is interest calculated on the running total, not just on the original amount. This means each period's interest becomes part of the balance for the next period, so future interest is earned on earlier interest as well.
• The main quantities are principal $P$, rate $r\%$, multiplier $1+\frac{r}{100}$, and number of periods $n$. These variables work together because the same percentage growth is repeatedly applied over equal time intervals.
• The final amount after compound growth is found using repeated multiplication, not repeated addition. That is why compound interest follows an exponential pattern rather than a linear one.

Key formula: $A = P\left(1 + \frac{r}{100}\right)^n$

• In this formula, $A$ is the final balance, $P$ is the starting amount, $r$ is the percentage increase per compounding period, and $n$ is the number of periods. The formula applies when the percentage rate stays constant and is applied at regular intervals.

• Interest earned and final balance are different outputs, and exam questions often switch between them. The final balance is $A$, while the total interest earned is $A-P$, so you must identify exactly what the question is asking for before calculating.

• The compounding period matters just as much as the percentage rate. A rate quoted per year should be compounded yearly unless the problem states monthly, quarterly, or another interval, because the number of multiplier applications depends on that timing.

• The reason compound interest grows faster over time is that each increase is a percentage of a larger balance than before. Because the base amount keeps changing, the growth added in later periods is larger than the growth added in earlier periods.

• A percentage increase can always be written as a multiplier. For example, increasing by $r\%$ means keeping $100\%$ of the current amount and adding $r\%$, so the total multiplier becomes $1+\frac{r}{100}$.

Multiplier principle: increase by $r\% \Rightarrow$ multiply by $1+\frac{r}{100}$

• Repeating the same percentage change over $n$ periods means multiplying by the same multiplier $n$ times. This is why powers appear in the formula: $A = Pm^n$ where $m = 1+\frac{r}{100}$.

• Compound interest is a special case of exponential growth. The balance does not rise by equal amounts each period; instead, it rises by equal proportions, which is the defining feature of exponential change.

• If the interest rate changes from one period to another, a single power formula is no longer sufficient. In that case, you multiply by each period's separate multiplier in sequence, because the growth process is still multiplicative even though the rate is not constant.

Finding the final balance

• Start by identifying the initial amount, the percentage rate, and the number of compounding periods. Then convert the percentage rate to a multiplier using $1+\frac{r}{100}$ and raise that multiplier to the power of the number of periods.
• The full method is $A = P\left(1+\frac{r}{100}\right)^n$. This is the most efficient method when the rate and compounding interval remain constant throughout the problem.

Finding the interest earned

• If the question asks for the interest, do not stop at the final balance. First calculate $A$, then subtract the original principal: $\text{interest earned} = A - P$
• This works because compound interest questions usually model the total accumulated amount, while the actual interest is only the extra growth above the starting sum.

Working backwards

• If you know the final amount and need the starting amount, divide by the compound multiplier factor. Rearranging gives $P = \frac{A}{\left(1+\frac{r}{100}\right)^n}$, which is useful for reverse problems involving an unknown initial investment.
• If you need the rate and know the other values, isolate the multiplier first and then interpret it as a percentage increase. This often requires roots, because the multiplier is raised to the power $n$.
• If you need the number of periods, you compare repeated multiplier growth to the final balance. In more advanced settings this uses logarithms, but at an introductory level it may be found by testing powers or recognizing patterns.

• The most important comparison is between simple interest and compound interest. Simple interest adds the same amount each period because it is always based on the original amount, while compound interest adds changing amounts because it is based on the current balance.

• A second distinction is between final balance and interest earned. Final balance includes the original principal and all accumulated interest, whereas interest earned measures only the growth portion.

• Students must also distinguish between a percentage rate and a multiplier. The rate is written as a percent such as $6\%$, but the multiplier is the decimal factor used in calculation, such as $1.06$.

• Another key distinction is between compounding frequency and duration. A rate applied monthly for 24 months is not handled the same way as a rate applied yearly for 2 years, because the number of times the multiplier is applied changes.

• A common misconception is thinking that compound interest adds the same amount every period. That idea belongs to simple interest; under compound interest the added amount changes because the balance being multiplied changes.

• Another frequent error is using $\frac{r}{100}$ as the multiplier instead of $1+\frac{r}{100}$. Multiplying by only the rate finds the interest for one period, not the new total after that period.

• Students often confuse the number of years with the number of compounding periods. If compounding happens monthly, then 3 years means 36 applications of the multiplier, not 3.

• Some answers lose marks because the student calculates the final balance correctly but gives it when the question asked for the interest earned. Always compare your result to the wording of the task before finalizing the answer.

• Compound interest is one of the clearest practical examples of exponential growth. The same structure appears in population models, repeated percentage increases in prices, and many scientific processes involving constant proportional change.

• The idea also connects directly to depreciation and other repeated percentage decreases. The only structural change is the multiplier: growth uses $1+\frac{r}{100}$, while decrease uses $1-\frac{r}{100}$.

• In finance, compound interest helps explain why time matters so strongly in long-term saving and borrowing. Small differences in interest rate or duration can lead to large differences in final amount because powers magnify repeated percentage effects.