Compound Interest (ICSE 9)

Cheatsheet Content

### Definition of Compound Interest Compound interest is the interest calculated on the initial principal and also on the accumulated interest from previous periods. It's often referred to as "interest on interest." This means that over time, your money grows at an accelerating rate. **Simple Interest vs. Compound Interest:** * **Simple Interest:** Calculated only on the original principal amount. The interest earned each period remains constant. * **Compound Interest:** Calculated on the original principal **plus** all accumulated interest from prior periods. The interest earned each period increases. ### Compound Interest Formula The formula for calculating the amount (A) when interest is compounded annually is: $$A = P \left(1 + \frac{R}{100}\right)^n$$ Where: * **A** = Amount (total money after interest) * **P** = Principal (original sum of money invested or borrowed) * **R** = Rate of interest per annum (per year) * **n** = Number of conversion periods (usually years) **Compound Interest (C.I.) Calculation:** To find the actual compound interest earned, use: $$C.I. = A - P$$ or $$C.I. = P \left[\left(1 + \frac{R}{100}\right)^n - 1\right]$$ ### Compounding Frequency Interest can be compounded more frequently than annually. The formula adjusts based on the compounding period: 1. **Compounded Half-Yearly (Semi-Annually):** * Rate becomes $\frac{R}{2}$ * Time becomes $2n$ * Formula: $$A = P \left(1 + \frac{R}{2 \times 100}\right)^{2n}$$ 2. **Compounded Quarterly:** * Rate becomes $\frac{R}{4}$ * Time becomes $4n$ * Formula: $$A = P \left(1 + \frac{R}{4 \times 100}\right)^{4n}$$ 3. **Compounded Monthly:** * Rate becomes $\frac{R}{12}$ * Time becomes $12n$ * Formula: $$A = P \left(1 + \frac{R}{12 \times 100}\right)^{12n}$$ ### Important Points to Remember * **Units:** Ensure that the rate (R) and time (n) are consistent with the compounding period. If R is annual, and compounding is half-yearly, adjust R and n accordingly. * **Percentage Rate:** The rate 'R' in the formula is always taken as a percentage (e.g., for 10%, use 10, not 0.10). The division by 100 in the formula handles the conversion. * **Time Period:** 'n' represents the number of times interest is compounded. For annual compounding, it's the number of years. For half-yearly, it's twice the number of years. * **Growth:** Compound interest leads to exponential growth of money, which is why it's powerful for investments but also costly for loans. * **Fractional Years:** If the time period is not a whole number of years (e.g., 2.5 years), calculate the amount for the whole years using the compound interest formula, and then calculate simple interest on that amount for the fractional part. * Example: For $2\frac{1}{2}$ years compounded annually: $$A = P \left(1 + \frac{R}{100}\right)^2 \left(1 + \frac{\frac{1}{2}R}{100}\right)$$