The Master Formula (EMI)
$$E = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}$$
Where:
- P: Loan Principal (Amount borrowed)
- r: Monthly Interest Rate (Annual Rate / 12 / 100)
- n: Loan tenure in months
Dimensional Analysis
The term $\frac{r(1+r)^n}{(1+r)^n - 1}$ has the unit of $r$ (which is $T^{-1}$) because the other parts are dimensionless ratios. Therefore, the unit of $E$ (Payment per month) matches the units of $P \times \text{Rate}$, confirming the formula is dimensionally consistent with a "rate of flow" of money.
Variations: Finding the Loan Amount
If you know how much EMI you can afford ($E$), how much loan can you get?$$P = E \cdot \frac{(1+r)^n - 1}{r(1+r)^n}$$This is the Present Value of an Annuity formula.
Shortcuts & Mnemonics
- The "One-Plus" Rule: Always calculate $(1+r)^n$ first. It is the "engine" of the formula.
- Flat Rate vs. Reducing Rate: A "Flat Rate" of 7% is often more expensive than a "Reducing Rate" of 10%. Never compare them directly!
- Mnemonic: "Principal times Rate times Growth over Growth minus One."
Edge Cases
- Zero Interest: If $r = 0$, the formula becomes undefined ($0/0$). Using L'Hopital's Rule, the limit is simply $P/n$. (Simple division).
- Infinite Tenure ($n \to \infty$): The EMI $E$ approaches $P \times r$. This is called a Perpetuity—you only pay the interest, and the principal is never cleared.
- Negative Interest: Mathematically possible (seen in some European economies), where the bank effectively pays you to borrow!
