The Master Formula
$$A = P \left(1 + \frac{R}{n \cdot 100}\right)^{nt}$$
Where:
- A: Final Amount
- P: Principal (Initial sum)
- R: Annual Interest Rate (as a percentage)
- n: Number of times interest is compounded per year
- t: Total time in years
Dimensional Analysis (Physics Check)
In Physics, we check units. Here, $R$ is $T^{-1}$ (per year) and $t$ is $T$ (years). Therefore, $nt$ is a dimensionless quantity. The term $(1 + R/n)$ is also dimensionless. Thus, $[A] = [P]$, which is exactly what we expect (Money = Money).
Variations for Quick Solving
- To find Principal: $P = A / (1+r)^n$
- To find Rate: $R = 100 * [(A/P)^{1/n} - 1]$
- Difference between CI and SI for 2 years: $D = P(R/100)^2$ (Very common JEE Main shortcut!)
Shortcuts & Mnemonics
The Rule of 72: To find how long it takes to double your money, divide 72 by the interest rate ($t \approx 72/R$).
Pascal’s Triangle for CI: For small years, the interest components follow Pascal’s coefficients:
- 2 Years: 2:1
- 3 Years: 3:3:1
- 4 Years: 4:6:4:1
Edge Cases
- Infinite Compounding ($n \to \infty$): The formula transforms into $A = Pe^{rt}$. This is the basis for most growth/decay models in Physics and Calculus.
- Zero Rate: If $R=0$, then $A=P$. The growth factor becomes 1.
- $t$ is not an integer: If $t = 2.5$ years, calculate 2 years with CI formula and the remaining 0.5 year using SI logic on the 2nd-year amount.
