The Logic Behind the Math: Isolating 'n'

We start with the standard Annuity formula: $P = E \cdot [1 - (1+r)^{-n}] / r$.To find the payoff time, we must rearrange this to solve for $n$:

1. Multiply by $r$: $Pr = E[1 - (1+r)^{-n}]$
2. Divide by $E$: $Pr/E = 1 - (1+r)^{-n}$
3. Rearrange: $(1+r)^{-n} = 1 - (Pr/E)$
4. Take $\ln$ on both sides: $-n \ln(1+r) = \ln(1 - Pr/E)$
5. Final form: $n = -\frac{\ln(1 - Pr/E)}{\ln(1+r)}$

Step-by-Step Solved Example

Problem: You have a loan of ₹20,00,000 at 1% monthly interest. Your EMI is ₹30,000. How many months until it's paid off?

• Step 1: Identify Variables. $P = 2,000,000$, $r = 0.01$, $E = 30,000$.
• Step 2: Check Feasibility. Is $E > Pr$? Yes ($30,000 > 20,000$). If not, the loan grows forever!
• Step 3: Calculate the Ratio. $Pr/E = 20,000 / 30,000 = 2/3 \approx 0.666$.
• Step 4: The Numerator. $\ln(1 - 0.666) = \ln(0.334) \approx -1.096$.
• Step 5: The Denominator. $\ln(1.01) \approx 0.00995$.
• Step 6: Division. $n = -(-1.096) / 0.00995 \approx 110$ months.

Alternative Methods: The Iterative Approach

In exams without calculators, you can use Linear Approximation. If you know that at 100 months the balance is $X$, and at 120 months the balance is $Y$, you can use the Method of False Position (Regula Falsi) to estimate the zero-point between them. This is a common numerical method in JEE Advanced Math.

Exam Trap Alert: The "Infinite Loan" Paradox

In many competitive problems, a trap is set where the interest added ($Pr$) is greater than or equal to the payment ($E$).

Exam Warning: If the argument inside the Logarithm $(1 - Pr/E)$ is zero or negative, the loan will never be paid off. In Physics, this is like trying to reach a destination while moving slower than the treadmill moving backward. Always check $E > Pr$ before solving.

Practice Problem (JEE Calculus/Series)

Question: A debt is being paid off such that the rate of change of debt $D$ is given by $dD/dt = rD - k$, where $k$ is the constant repayment rate. Solve this differential equation to find the time $T$ when $D(T) = 0$.