Simple Interest and Compound Interest
Simple Interest and Compound Interest
Interest gives 2 questions in the Part-III paper, and the compound half is where the depth shows. Beyond the basic formulas, the examiner tests half-yearly compounding, the gap between the two systems, and the fact that first-year interest is identical under both.
Simple Interest
SI = (P × R × T) ÷ 100 | Amount = P + SI
Under simple interest the principal never changes, so the interest earned each year is identical. A sum doubles under simple interest when the interest equals the principal, which needs R × T = 100.
- Doubling needs R × T = 100; tripling needs R × T = 200.
- At 10 per cent, a sum doubles in 10 years and triples in 20 years under simple interest.
- If a sum becomes n times itself, then R × T = (n − 1) × 100.
Compound Interest
Amount = P (1 + R/100)ⁿ | CI = Amount − P
Here the interest is added to the principal at the end of each period, so the base grows. Two adjustments are asked regularly.
| Compounding | Rate becomes | Time becomes |
|---|---|---|
| Yearly | R | n |
| Half-yearly | R/2 | 2n |
| Quarterly | R/4 | 4n |
So a sum at 10 per cent compounded half-yearly for one year uses 5 per cent for 2 periods, not 10 per cent for 1. Candidates who halve the rate but forget to double the time get a predictable wrong answer.
Comparing the Two
| Point | Simple interest | Compound interest |
|---|---|---|
| Base | Principal fixed | Principal grows |
| Yearly interest | Equal every year | Increases every year |
| First year | Same as CI | Same as SI |
| Total after 2 years | Smaller | Larger |
First-year interest is always equal under both systems, because compounding has not yet had a chance to act. That fact converts several questions into one-line answers.
CI − SI for 2 years = P (R/100)²
CI − SI for 3 years = P (R/100)² × (3 + R/100)
The two-year difference is the one to memorise. If the difference between CI and SI on a sum for two years at 10 per cent is Rs 50, then P (0.1)² = 50, so the principal is Rs 5,000.
✗ At 10% for 2 years, CI and SI on Rs 5,000 differ by Rs 500 | ✓ The difference is P(R/100)² = 5000 × 0.01 = Rs 50
Instalments and Equal Payments
A sum borrowed and repaid in equal annual instalments is a standard harder variant. Each instalment is discounted back to the present.
P = X/(1 + R/100) + X/(1 + R/100)² + ... for X per instalment
For two instalments at 10 per cent, the borrowed sum equals X divided by 1.1 plus X divided by 1.21. You rarely need to go past two or three terms at this level, but recognising the structure is what the question tests.
पहले साल का ब्याज दोनों में बराबर होता है — इससे कई प्रश्न एक लाइन में हल हो जाते हैं।
TRE pointer: Half-yearly compounding is the single most reliable discriminator in this topic, because the adjustment has two halves and candidates apply only one. The CI minus SI for two years formula converts a three-step question into one line and appears almost every year. Remember that first-year interest is identical under both systems — that fact alone answers a whole class of questions, and with −1/3 negative marking, a one-line answer you can verify is worth more than a long one you cannot.
60-Second Recap
- SI = PRT/100; the principal never changes.
- A sum doubles under SI when R × T = 100; becomes n times when R × T = (n−1)100.
- CI amount = P(1 + R/100)ⁿ.
- Half-yearly: halve the rate and double the time. Quarterly: quarter the rate, four times the time.
- First-year interest is identical under SI and CI.
- CI − SI for two years equals P(R/100)².
Frequently Asked Questions
How do I handle half-yearly compounding?
Halve the rate and double the number of periods. Ten per cent per annum for one year compounded half-yearly means five per cent for two periods. Adjusting the rate but leaving the time unchanged is the error the options are built around.
Why is first-year interest the same under simple and compound interest?
Because compounding only takes effect when interest is added back to the principal, which first happens at the end of year one. During that first year both systems charge interest on the same original principal, so the amounts are identical.
What is the shortcut for the difference between CI and SI?
For two years it is the principal times the rate over a hundred, all squared. For three years it is that same quantity multiplied by three plus the rate over a hundred. The two-year version is the one that actually appears in exams.
In how many years does a sum double at simple interest?
When the rate multiplied by the time equals one hundred. At ten per cent that is ten years, and at eight per cent it is twelve and a half years. To become n times itself, the rate times the time must equal n minus one, times a hundred.