The Mathematics of Compound Interest
In finance and algebra, compound interest is often described as "interest on interest." Unlike simple interest, which strictly calculates returns solely on the initial amount invested, compound interest recalculates the principal at each specified compounding period. This means the interest you earned in previous periods is added to your base balance, resulting in an accelerated, exponential growth curve over time.
The Core Financial Formula
To mathematically project the future value of an investment or loan, the universally accepted algebraic equation is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Understanding each variable is critical to setting up the word problem correctly:
- \( P \) (Principal): The initial starting amount of money deposited or borrowed.
- \( r \) (Annual Interest Rate): The yearly percentage rate, which must strictly be converted to a decimal before calculation (e.g., 5% becomes 0.05).
- \( n \) (Compounding Frequency): The number of times the interest is calculated and added back into the account per year. For monthly compounding, \( n = 12 \); for daily, \( n = 365 \).
- \( t \) (Time): The total duration the money is invested or borrowed, expressed entirely in years.
Calculating the Total Interest Earned
The variable \( A \) from the formula calculates the "Future Value," which is the grand total of your original principal plus all the accumulated interest. If you need to isolate strictly the interest earned over that period, you simply subtract your starting principal from the final amount: \( I = A - P \).