Q.In a bank, principal increases continuously at the rate of 5% per year. In how many years Rs 1000 double itself?

Concept: Exponential Growth Rate — when a quantity grows continuously at a fixed percentage rate, it follows $A = P e^{rt}$.

We are given $P = 1000$, $r = 5\% = 0.05$ per year, and we want $A = 2000$.

1. The continuous compounding formula is $A = P e^{rt}$.
2. Substitute: $2000 = 1000 \, e^{0.05 t}$.
3. Divide: $2 = e^{0.05 t}$. Take natural log: $\log 2 = 0.05 t$. …

The key here is the phrase "increases continuously." This is not the same as simple interest or even annual compounding. Continuous growth means the principal is being updated every instant, not just at the end of each year. The natural model for this is exponential growth, where the amount after time $t$ is given by $A = P e^{rt}$, with $r$ as the annual rate (as a decimal).

Why $e$? Because continuous compounding is the limit of compounding more and more frequently — daily, hourly, every second — and that limit is $e^{rt}$. So whenever a problem says "continuously," your first thought should be the exponential function.

Now let's work through the numbers.

1. Set up the equation. We start with principal $P = 1000$. The amount after time $t$ years is $A = 1000 e^{0.05t}$. We want this to be double the original, so $A = 2000$.

2. Simplify. Divide both sides by 1000:

3. Solve for $t$ using natural log. Take the natural logarithm of both sides. Since $\log(e^x) = x$, we get:

4. Isolate $t$.

5. Compute the value. $\log 2 \approx 0.693147$. Dividing by 0.05 gives:

So $t \approx 13.86$ years. …

Method: Continuous exponential growth/decay

Use this whenever a quantity is said to change "continuously" at a fixed percentage rate — growth of principal, population, or decay. The phrase "continuously" is the signal to model with $e$, not with simple or annual compounding.

Steps

Step 1: Set up the rate equation and its solution.

Continuous change at rate $r$ means the rate of increase is proportional to the current amount:

where $P_0$ is the starting amount and $r$ is the rate written as a decimal (5% becomes $0.05$).

Step 2: Impose the target condition. …

Common Mistakes

Mistake 1: Using simple interest or annual compounding.

Why it's wrong: "increases continuously" means the exponential model $P = P_0 e^{rt}$; the formulas $P_0(1+rt)$ or $P_0(1+r)^t$ give different (wrong) answers here. Correct approach: model with $e^{rt}$ whenever growth is continuous.

Mistake 2: Mishandling the percentage rate.

Why it's wrong: $5\%$ must enter as $r = 0.05$, not $5$; using $5$ makes $t$ a hundred times too small. Correct approach: convert the percentage to a decimal before substituting. …