Compound Interest - Algebra-1

1. Fundamental Concepts

Compound interest refers to the calculation of interest where not only the principal generates interest, but also the accumulated interest from previous periods is added to the principal to calculate new interest, which is often called "interest on interest". Its core lies in the reinvestment of interest, making the growth rate of funds accelerate over time, and it is a typical application of exponential growth.

Formula: The general formula for compound interest is:$P = A \cdot \left(1 + \frac{r}{n}\right)^{nt}$ Where:
- P represents the final amount (future value) including principal and interest;
- A is the initial principal;
- r is the annual interest rate (converted to a decimal, e.g., 5% is 0.05);
- n is the number of times interest is compounded per year (e.g., for monthly compounding, for quarterly compounding);
- t is the time the money is invested for, in years.
Key Requirement: The interest rate r and time t must share the same time unit (usually based on years).

Easy Level: If the initial principal dollars, the annual interest rate (i.e., 0.05), and interest is compounded once a year (), find the amount P after 3 years.
Solution: Substitute into the formula
Medium Level: The initial principal dollars, the annual interest rate , and interest is compounded semi-annually (). Find the amount P after 5 years.
Solution: The semi-annual interest rate is , and the total number of compounding periods . Then
Hard Level: The initial principal dollars, the annual interest rate , and interest is compounded monthly (). Find the amount P after 2 years and 3 months.
Solution: The monthly interest rate is , the time years, and the total number of compounding periods . Then

Clarify Variable Meanings: First, distinguish between the principal A, interest rate r, number of compounding periods n, and time t, ensuring uniform units (e.g., when the interest rate is an annual rate, time should be in years).
Split Complex Time: If the time includes mixed units such as years and months (e.g., 2 years and 3 months), convert it to a decimal in years (e.g., 2.25 years).
Calculate Exponents Step by Step: For high powers (e.g., ), you can use a calculator for step-by-step calculation or simplify using the properties of exponential operations.
Verify Logical Reasonableness: The result of compound interest should be greater than the principal (due to exponential growth). If the result is abnormal, check whether the interest rate is converted to a decimal, whether the number of compounding periods is correct, etc.