Accumulation with Simple and Compound Interest
The Core Intuition
Imagine you lend a friend ₹100. A year later, they return ₹110. That extra ₹10 is the interest — the price of using your money. Now, the question is: should next year's interest be calculated on the original ₹100, or on the ₹110 they now owe you?
That single choice is the entire difference between simple interest and compound interest.
- Simple interest says: "I only care about the original amount. Every year, you pay me interest on that same ₹100." So year after year, the interest is constant: ₹10, ₹10, ₹10...
- Compound interest says: "Last year you owed me ₹110. This year, I'll charge interest on that ₹110." So the interest grows each year: ₹10, then ₹11, then ₹12.10...
Simple interest is linear — a straight line. Compound interest is exponential — a curve that gets steeper over time.
The Precise Statement
Let:
- P = principal (the initial amount)
- r = annual interest rate (as a decimal, e.g., 10% = 0.10)
- t = time in years
- A = accumulated amount after t years
Simple Interest
Interest is earned only on the original principal. The interest per year is P×r, so after t years:
A=P+(P×r×t)
A=P(1+rt)
Asimple=P(1+rt)
Compound Interest
Interest is earned on the principal plus all previously earned interest. If interest is compounded once per year, after the first year you have P(1+r). That becomes the new principal for the second year:
A=P(1+r)(1+r)⋯(1+r)=P(1+r)t
Acompound=P(1+r)t
Why the Difference Matters
Compare ₹100 at 10% per year for 10 years:
| Year | Simple Interest (₹) | Compound Interest (₹) |
|---|
| 0 | 100 | 100 |
| 5 | 150 | 161.05 |
| 10 | 200 | 259.37 |
After 10 years, compound interest gives you ₹59.37 more — nearly 30% extra — for doing nothing different except letting your interest earn interest.
A common mistake is to think compound interest always "doubles" or "triples" the simple interest. It doesn't — the gap grows slowly at first, then accelerates. For short periods (1–2 years), the difference is tiny. For long periods (20+ years), it becomes enormous.
The Deeper "Why"
Simple interest treats money as a static resource — you lend it, you get a fixed rent. Compound interest treats money as self-reproducing — the interest itself becomes capital that earns more interest. This is why Albert Einstein reportedly called compound interest "the eighth wonder of the world." It's not magic; it's just repeated multiplication.
For quick mental estimates, use the Rule of 72: divide 72 by the annual interest rate (as a percentage) to approximate how many years it takes for money to double under compound interest. At 10%, it doubles in about 72/10=7.2 years. Under simple interest, doubling at 10% takes 10 years.
One More Layer: Compounding Frequency
In real life, interest is often compounded more than once a year — monthly, quarterly, or daily. If interest is compounded n times per year, the formula becomes:
A=P(1+nr)nt
As n increases (daily, hourly, every second), the amount approaches a limit called continuous compounding:
where e≈2.71828. This is the mathematical ceiling of how fast money can grow at a given rate.
For Indian exams (CBSE, ICSE, competitive), the standard compound interest formula is A=P(1+100r)t when r is given as a percentage. The logic is identical — just replace r with 100R where R is the rate in percent.
The Takeaway
Simple interest is arithmetic growth — add the same amount each period. Compound interest is geometric growth — multiply by the same factor each period. The first is predictable and flat; the second is powerful and accelerating. Understanding this difference is not just a formula — it's a lens for seeing how money, population, and many natural processes behave over time.