Q.What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

Compound Interest: The Snowball That Grows Itself

Imagine you plant a magic seed. The first year, it grows one new seed. Now you have two seeds. Next year, each of those two seeds grows one new seed — so you get two more, making four. The year after, each of the four seeds grows one more — you get four new seeds, making eight. The growth isn't just adding the same number each year; it's multiplying. That's compound interest.

The Core Intuition

Simple interest is like earning ₹100 every year on a ₹1000 deposit — you get the same ₹100 each time. Compound interest is different: after the first year, you earn interest on the interest you already earned. Your money starts working for you, and then the money your money earned also starts working.

The Precise Statement

Compound interest means that at the end of each compounding period (say, a year), the interest earned is added to the principal. The next period's interest is calculated on this new, larger principal.

Let's define the terms:

• $P$ = Principal (the initial amount you deposit or borrow)
• $r$ = Annual interest rate (as a decimal; e.g., 10% = 0.10)
• $n$ = Number of times interest is compounded per year (e.g., 1 for yearly, 12 for monthly, 365 for daily)
• $t$ = Time in years
• $A$ = Amount after time $t$ (principal + interest)

This is the compound interest formula. The exponent $nt$ is the total number of compounding periods. The fraction $\frac{r}{n}$ is the interest rate per period.

Why the Formula Works

Let's trace one year with annual compounding ($n=1$):

• Start: $P$
• After 1 year: $P + P \times r = P(1+r)$
• After 2 years: $P(1+r) + P(1+r) \times r = P(1+r)(1+r) = P(1+r)^2$
• After $t$ years: $P(1+r)^t$

For monthly compounding ($n=12$), the monthly rate is $\frac{r}{12}$, and in $t$ years there are $12t$ months. So after $t$ years: $P\left(1 + \frac{r}{12}\right)^{12t}$.

A Concrete Example

You deposit ₹10,000 at 8% per annum, compounded annually, for 3 years.

$P = 10000$, $r = 0.08$, $n = 1$, $t = 3$

$1.08^3 = 1.08 \times 1.08 \times 1.08 = 1.259712$

The compound interest earned = $A - P = ₹2,597.12$. …