Why does money grow on its own?
Ever wondered why a tiny amount in a bank can turn into a much bigger sum without you doing anything? That's the magic of compound interest – the interest you earn on both the original money (principal) and the interest that’s already been added.
In simple words, compound interest is when your money earns interest, and that interest itself starts earning interest. Imagine planting a seed that grows a new branch every year, and each branch also sprouts leaves that keep growing.
What is Compound Interest?
Compound interest is the extra amount you get when interest is calculated on the principal (the original sum) plus any interest that’s already been added. Think of it like a snowball rolling downhill: it picks up more snow (interest) as it rolls, and the bigger it gets, the faster it gathers even more.
Compound Interest Formula Explained
The standard formula that the UP Board expects you to use is:
A = P (1 + r/n)^(n·t)
- A: Amount of money you’ll have after the time period (principal + interest).
- P: Principal, the original sum you start with.
- r: Annual interest rate expressed as a decimal (so 5% becomes 0.05).
- n: Number of times interest is added (compounded) each year. If it’s yearly, n=1; half‑yearly, n=2; quarterly, n=4; monthly, n=12.
- t: Time the money is left to grow, in years.
Notice how the term (1 + r/n) is raised to the power n·t. That power tells the computer (or your brain) how many times the interest‑adding step happens.
Step‑by‑Step Method to Solve Compound Interest Problems
When you see a question, follow this simple checklist:
- Read the problem carefully and write down the values of P, r, n, and t.
- Convert the percentage rate into a decimal.
- Plug the numbers into the formula.
- Do the arithmetic inside the parentheses first, then raise to the power.
- Round the final answer to the required number of decimal places.
Worked Example 1 – Yearly Compounding
Problem: A student deposits ₹5,000 in a bank that offers 5% interest per year, compounded annually. How much will be in the account after 3 years?
Solution:
- P = 5,000
- r = 5% = 0.05
- n = 1 (interest added once a year)
- t = 3 years
Plug into the formula:
A = 5,000 × (1 + 0.05/1)^(1 × 3) = 5,000 × (1.05)^3.
Calculate (1.05)^3 ≈ 1.157625.
Finally, A ≈ 5,000 × 1.157625 = ₹5,788.13.
So after three years the amount becomes about ₹5,788.
Worked Example 2 – Half‑Yearly Compounding
Problem: Rahul saves ₹2,000 at an 8% annual rate, compounded half‑yearly. Find the amount after 2 years.
Solution:
- P = 2,000
- r = 8% = 0.08
- n = 2 (twice a year)
- t = 2 years
Formula:
A = 2,000 × (1 + 0.08/2)^(2 × 2) = 2,000 × (1 + 0.04)^4 = 2,000 × (1.04)^4.
(1.04)^4 ≈ 1.169859.
A ≈ 2,000 × 1.169859 = ₹2,339.72.
Hence Rahul will have roughly ₹2,340 after two years.
Quick Summary Table
| Symbol | Meaning | Example Value |
|---|---|---|
| P | Principal (initial amount) | ₹5,000 |
| r | Annual interest rate (decimal) | 0.05 for 5% |
| n | Number of compounding periods per year | 1 for yearly, 2 for half‑yearly |
| t | Time in years | 3 years |
| A | Amount after t years | ₹5,788.13 |
Common Mistakes to Avoid
- Forgetting to change the percent into a decimal.
- Mixing up the compounding frequency (n) with the time period.
- Skipping the parentheses – always compute
1 + r/nbefore raising to the power.
📝 Likely Exam Questions
- Question: A sum of ₹10,000 is invested at 6% per annum, compounded quarterly. Find the amount after 1.5 years.
Answer: n=4, r=0.06, t=1.5 → A = 10,000(1+0.06/4)^(4×1.5) = 10,000(1.015)^6 ≈ ₹10,938. - Question: If ₹7,500 grows to ₹9,000 in 2 years at a rate compounded annually, what is the rate?
Answer: Use A = P(1+r)^t → 9,000 = 7,500(1+r)^2 → (1+r)^2 = 1.2 → 1+r = √1.2 ≈ 1.095 → r ≈ 9.5% per annum. - Question: Find the compound interest earned on ₹4,000 at 10% per annum, compounded half‑yearly, for 3 years.
Answer: n=2, r=0.10, t=3 → A = 4,000(1+0.10/2)^(2×3)=4,000(1.05)^6≈4,000×1.3401=₹5,360.4. Interest = A‑P = ₹1,360.4.