Simple Interest vs Compound Interest — Formulas and Worked Examples
Simple interest is charged on the original principal, every year, forever. Compound interest is charged on the principal plus the interest already added. That one difference produces a gap that starts small and grows without limit — and it is the reason compound interest questions are worth more marks than simple interest ones. This guide gives both formulas, the shortcuts for the difference, and five worked examples with every step computed.
The formulas
What each symbol means
- P — principal, the amount invested or borrowed at the start.
- R — rate per annum, as a percentage number (write 8, not 0.08, in these formulas, because they already contain the ÷ 100).
- T — time in years. Months must be converted: 9 months = 0.75 year.
- k — compounding periods per year: 2 for half-yearly, 4 for quarterly, 12 for monthly.
- A — the amount, meaning principal plus interest. The compound formula gives you A, not the interest — subtract P to get CI. Forgetting this is the most common single error in the topic.
Worked example 1 — the two side by side
P = ₹10,000, R = 8% per annum, T = 3 years.
Simple interest:
SI = 10 000 × 8 × 3 ÷ 100 = 240 000 ÷ 100 = ₹2,400
Amount = 10 000 + 2400 = ₹12,400
Compound interest:
(1.08)³ = 1.08 × 1.08 × 1.08 = 1.259712
A = 10 000 × 1.259712 = ₹12,597.12
CI = 12 597.12 − 10 000 = ₹2,597.12
Difference = 2597.12 − 2400 = ₹197.12
Year by year, the compound interest is ₹800, then ₹864, then ₹933.12 — each year's interest slightly larger, because it is charged on a larger balance. Simple interest stays flat at ₹800 a year.
The difference shortcuts
Over 3 years: CI − SI = P × (R ÷ 100)² × (3 + R ÷ 100)
Check the 3-year formula against example 1:
10 000 × (0.08)² × (3 + 0.08) = 10 000 × 0.0064 × 3.08 = 64 × 3.08 =
₹197.12 ✓ — matches the long calculation exactly.
The 2-year case. P = ₹20,000, R = 10%, T = 2 years.
SI = 20 000 × 10 × 2 ÷ 100 = ₹4,000
A = 20 000 × (1.10)² = 20 000 × 1.21 = ₹24,200, so CI = ₹4,200
Difference = ₹200
Shortcut: 20 000 × (0.10)² = 20 000 × 0.01 = ₹200 ✓
These shortcuts are exam gold, because the usual question form is "the difference between CI and SI on a sum for 2 years at 10% is ₹200 — find the sum", which the shortcut answers in one line: P = 200 ÷ 0.01 = ₹20,000.
Worked example 2 — compounding more often than yearly
P = ₹50,000 at 12% per annum for 1 year, compounded quarterly.
Rate per quarter = 12 ÷ 4 = 3%, so the multiplier is 1.03
Number of periods = 4 × 1 = 4
A = 50 000 × (1.03)⁴ = 50 000 × 1.12550881 = ₹56,275.44
CI = ₹6,275.44
Compare with annual compounding at the same 12%: 50 000 × 1.12 = ₹56,000, i.e. CI of ₹6,000. Quarterly compounding earned ₹275.44 more on the identical stated rate.
Effective annual rate = (1.03⁴ − 1) × 100 = 12.5509%. That is the number to compare between two offers — a nominal rate means little until you know how often it compounds.
Worked example 3 — how long to double your money
At 8% compounded annually, we need (1.08)T = 2.
T = ln 2 ÷ ln 1.08 = 0.693147 ÷ 0.0769610 = 9.006 years
The Rule of 72 estimates this as 72 ÷ 8 = 9 years — an excellent approximation, and one you can do in your head. It works well for rates roughly between 5% and 15%; at very high rates it drifts. At 12%, the rule says 6 years and the exact answer is 6.12 years; at 2% the rule says 36 and the exact answer is 35.0.
Worked example 4 — reading the question carefully
"Find the compound interest on ₹8,000 for 2 years at 5% per annum."
A = 8000 × (1.05)² = 8000 × 1.1025 = ₹8,820
CI = 8820 − 8000 = ₹420
Answering ₹8,820 would be answering a question about the amount. Read whether the paper asks for the amount or the interest, and underline it before you start.
Simple vs compound, at a glance
| Simple interest | Compound interest | |
|---|---|---|
| Interest charged on | Original principal only | Principal plus accumulated interest |
| Growth pattern | Linear — equal every year | Exponential — larger every year |
| Equal in the first year? | Yes, always identical for one year at annual compounding | |
| Typical use | Some short-term loans, certain government schemes | Bank deposits, most loans, mutual funds, credit cards |
| On ₹10,000 at 8% for 3 years | ₹2,400 | ₹2,597.12 |
The two are always equal for the first year at annual compounding, which is a useful sanity check: if your one-year figures differ, one of the calculations is wrong.
Common mistakes
- Reporting A as CI. The compound formula gives the amount. Subtract the principal.
- Writing R as a decimal in the /100 formulas. SI = P × R × T ÷ 100 already divides by 100 — feeding it 0.08 gives an answer 100 times too small.
- Using months as T. Convert to years, or convert the rate to match.
- Ignoring k. "12% compounded quarterly" means 3% four times, not 12% four times.
- Using the 2-year shortcut for 3 years. They are different formulas.
- Rounding the power term early. (1.03)⁴ to two decimals is 1.13, which shifts the answer by hundreds of rupees.
- Assuming banks use simple interest. Deposits and loans in India are overwhelmingly compounded; simple interest is the exception.
Where this maths appears
| Context | Use |
|---|---|
| School maths (Class 7–10) | Full chapter; difference-between-CI-and-SI problems are standard |
| Aptitude and banking exams | Finding P, R or T from a stated difference; effective annual rate |
| Commerce and economics | Present value, annuities, depreciation (compound decay) |
| Science, indirectly | Exponential growth and decay — bacterial growth, radioactive decay and first-order kinetics are the same equation |
That last row is not a stretch. A = P(1 + r)T and N = N₀e−λt are the same mathematics with the sign of the exponent flipped: money compounding upward, nuclei decaying downward. A student who is fluent in one has already done most of the work for the other.
Try the compounding frequency for yourself. The Finance and Everyday section of the calculator suite holds the EMI, GST, income tax and discount tools, and the scientific calculator handles the power terms — (1.03)⁴ and ln 2 ÷ ln 1.08 included — without rounding on you.
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