Simple interest and compound interest are among the most formula-heavy topics in competitive mathematics — or so most students believe. The reality is that both topics reduce to percentage calculations at their core. Once you understand the structural logic behind each type, the "formulas" become unnecessary — you derive the answer from first principles in the same time it would take to recall and apply a memorized formula.
This guide builds that structural understanding and then layers shortcut techniques on top. By the end, you will solve SI and CI problems — including the notoriously tricky CI-SI difference questions — faster than candidates who have memorized every formula in the textbook.
Part 1: Simple Interest — The Logic-First Approach
What Simple Interest Actually Is
Simple interest is interest calculated only on the original principal — never on previously accumulated interest. This means the interest amount is the same every year, making it a perfectly linear relationship.
The one formula you truly need:
SI = P × R × T / 100
Where:
- P = Principal (original amount)
- R = Rate of interest per year (%)
- T = Time (in years)
Amount = P + SI = P(1 + RT/100)
Why You Do Not Need to Memorize This
SI is simply "what percentage of P is earned per year, multiplied by number of years."
- R% per year for T years = RT% total interest
- SI = RT% of P
Example: P = ₹5,000, R = 8%, T = 3 years
- Total interest rate = 8 × 3 = 24%
- SI = 24% of 5,000 = ₹1,200
No formula application — just percentage calculation.
SI Shortcut 1: Finding Rate When SI, P, T Are Known
R = SI × 100 / (P × T)
Worked Example 1:
SI = ₹900, P = ₹5,000, T = 3 years. Find R.
- R = 900×100 / (5000×3) = 90,000/15,000 = 6%
Worked Example 2:
A sum doubles itself in 8 years at SI. Find R.
- SI = P (doubled means interest = principal)
- P = P × R × 8 / 100
- R = 100/8 = 12.5%
General rule: If a sum becomes n times itself at SI in T years:
- R = (n−1) × 100 / T
Examples:
- Doubles (n=2): R = 100/T
- Triples (n=3): R = 200/T
- Quadruples (n=4): R = 300/T
SI Shortcut 2: Finding Time When SI, P, R Are Known
T = SI × 100 / (P × R)
Worked Example 1:
P = ₹3,600, R = 5%, SI = ₹720. Find T.
- T = 720×100/(3600×5) = 72,000/18,000 = 4 years
Worked Example 2:
At what time will ₹2,500 at 4% per annum yield SI equal to ₹400?
- T = 400×100/(2500×4) = 40,000/10,000 = 4 years
SI Shortcut 3: Two Different Rates or Time Periods
When a sum is split and invested at different rates:
Worked Example:
₹8,000 split into two parts. One part at 6%, other at 9%. Total SI after 2 years = ₹1,140. Find each part.
- Let first part = x → second part = (8000−x)
- SI from first = x×6×2/100 = 12x/100
- SI from second = (8000−x)×9×2/100 = 18(8000−x)/100
- Total: 12x + 18(8000−x) = 1,14,000
- 12x + 1,44,000 − 18x = 1,14,000
- −6x = −30,000 → x = ₹5,000
- Second part = ₹3,000
Part 2: Compound Interest — Building From SI
The Core Concept
Compound interest is interest on interest. Each year, the interest earned is added to the principal — and the next year's interest is calculated on this new, larger amount.
Year-by-year logic (no formula needed):
P = ₹10,000, R = 10%, T = 3 years
| Year | Opening Balance | Interest (10%) | Closing Balance |
|---|---|---|---|
| 1 | 10,000 | 1,000 | 11,000 |
| 2 | 11,000 | 1,100 | 12,100 |
| 3 | 12,100 | 1,210 | 13,310 |
CI = 13,310 − 10,000 = ₹3,310
SI for same = 10,000 × 10 × 3/100 = ₹3,000
Difference = ₹310
The Compound Interest Formula (When Needed)
A = P(1 + R/100)ⁿ
CI = A − P = P[(1 + R/100)ⁿ − 1]
For 2 years expanded:
CI = P[2R/100 + (R/100)²] = 2PR/100 + PR²/10,000
For 3 years expanded:
CI = P[3R/100 + 3(R/100)² + (R/100)³]
CI Shortcut 1: The Multiplier Method
Instead of using the formula, express (1 + R/100) as a fraction and multiply.
Example 1: P = ₹8,000, R = 5%, T = 2 years
- Multiplier = 1.05 = 21/20
- After year 1: 8000 × 21/20 = 8,400
- After year 2: 8400 × 21/20 = ₹8,820
- CI = 8820 − 8000 = ₹820
Example 2: P = ₹12,000, R = 10%, T = 3 years
- Multiplier = 1.1 = 11/10
- Year 1: 12,000 × 11/10 = 13,200
- Year 2: 13,200 × 11/10 = 14,520
- Year 3: 14,520 × 11/10 = ₹15,972
- CI = ₹3,972
Example 3: P = ₹6,000, R = 20%, T = 2 years
- Multiplier = 6/5
- Year 1: 6000 × 6/5 = 7,200
- Year 2: 7200 × 6/5 = ₹8,640
- CI = ₹2,640
CI Shortcut 2: The 2-Year Expansion Formula
For exactly 2 years, CI can be expressed as:
CI (2 years) = 2 × SI (1 year) + Interest on first year's interest
Or more directly:
CI = SI + (SI for 1st year × R/100)
Example: P = ₹5,000, R = 8%, T = 2 years
- SI for 1 year = 5000×8/100 = 400
- CI = 2×400 + 400×8/100 = 800 + 32 = ₹832
- Verify: 5000×1.08² − 5000 = 5000×1.1664 − 5000 = 832 ✓
CI Shortcut 3: Half-Yearly and Quarterly Compounding
When interest is compounded more frequently:
Half-yearly: R becomes R/2, T becomes 2T (double the periods)
Quarterly: R becomes R/4, T becomes 4T (quadruple the periods)
Worked Example 1:
P = ₹10,000, R = 10% p.a., compounded half-yearly for 1 year.
- Effective: R = 5%, T = 2 periods
- A = 10,000 × (1.05)² = 10,000 × 1.1025 = ₹11,025
- CI = ₹1,025 (vs ₹1,000 with annual compounding)
Worked Example 2:
P = ₹8,000, R = 20% p.a., compounded quarterly for 1 year.
- Effective: R = 5%, T = 4 periods
- A = 8,000 × (1.05)⁴ = 8,000 × 1.2155 ≈ ₹9,724
Part 3: The CI−SI Difference — The Most Tested Shortcut
The difference between CI and SI is one of the most frequently tested calculations in competitive exams. These two formulas eliminate all working:
For 2 years:
CI − SI = P(R/100)²
For 3 years:
CI − SI = P(R/100)²(3 + R/100)
Worked Examples
Example 1 — 2 Years:
P = ₹10,000, R = 8%, T = 2 years. Find CI−SI.
- CI−SI = 10,000 × (8/100)² = 10,000 × 64/10,000 = ₹64
Example 2 — 2 Years (Finding P):
CI−SI = ₹180, R = 6%, T = 2 years. Find P.
- 180 = P × (6/100)² = P × 36/10,000
- P = 180 × 10,000/36 = ₹50,000
Example 3 — 2 Years (Finding R):
CI−SI = ₹500, P = ₹50,000, T = 2 years. Find R.
- 500 = 50,000 × (R/100)²
- (R/100)² = 500/50,000 = 1/100
- R/100 = 1/10 → R = 10%
Example 4 — 3 Years:
P = ₹10,000, R = 10%, T = 3 years. Find CI−SI.
- CI−SI = 10,000 × (0.1)² × (3 + 0.1)
- = 10,000 × 0.01 × 3.1
- = 100 × 3.1 = ₹310
Part 4: The Rule of 72 — Fastest Doubling Time Estimate
The Rule of 72 is a mental shortcut for estimating how long it takes for money to double under compound interest.
Rule: Doubling time ≈ 72 / R
Examples:
| Rate | Doubling Time |
|---|---|
| 6% | 72/6 = 12 years |
| 8% | 72/8 = 9 years |
| 9% | 72/9 = 8 years |
| 10% | 72/10 = 7.2 years |
| 12% | 72/12 = 6 years |
| 18% | 72/18 = 4 years |
| 24% | 72/24 = 3 years |
Exam application: "At 12% CI, in how many years does ₹5,000 become ₹10,000?"
- 72/12 = 6 years (exact answer is also 6 years for this rate)
Part 5: Effective Annual Rate
When compounding is more frequent than annual, the effective annual rate (EAR) is higher than the stated rate.
Formula: EAR = (1 + R/n)ⁿ − 1
Where n = number of compounding periods per year.
Worked Example:
Nominal rate = 10% p.a., compounded quarterly. Effective annual rate?
- EAR = (1 + 0.025)⁴ − 1 = (1.025)⁴ − 1
- = 1.1038 − 1 = 10.38%
Practical exam use: Questions that ask "which investment is better — 10% compounded quarterly or 10.3% compounded annually?" — compare EAR of the quarterly option (10.38%) against 10.3% → quarterly compounding wins.
Part 6: Mixed and Advanced Problem Types
Type 1: Different Rates in Different Years
Worked Example:
P = ₹5,000. Rate = 8% in year 1, 10% in year 2. Find CI.
- After year 1: 5000 × 1.08 = 5,400
- After year 2: 5400 × 1.10 = ₹5,940
- CI = ₹940
Type 2: Present Value Problems
What amount invested today at CI will grow to ₹X in T years?
Formula: P = A / (1 + R/100)ⁿ
Worked Example:
What sum will grow to ₹14,641 at 10% CI in 4 years?
- P = 14,641 / (1.1)⁴ = 14,641 / 1.4641 = ₹10,000
Type 3: Installment Problems
Equal annual installments deposited at the end of each year — find total amount after n years.
Worked Example:
₹1,000 deposited annually at 10% CI for 3 years. Total at end?
- Year 1 deposit grows for 2 years: 1000 × (1.1)² = 1,210
- Year 2 deposit grows for 1 year: 1000 × 1.1 = 1,100
- Year 3 deposit (no growth): 1,000
- Total = ₹3,310
SI vs CI — Side-by-Side Comparison
| Feature | Simple Interest | Compound Interest |
|---|---|---|
| Base for interest | Always original principal | Principal + accumulated interest |
| Growth pattern | Linear | Exponential |
| Interest per year | Constant | Increases each year |
| Formula | SI = PRT/100 | A = P(1+R/100)ⁿ |
| After 2 years | Always less | Always more (same P, R, T) |
| Useful for | Short-term loans | Long-term investments |
| Exam frequency | Medium | High |
Quick Reference — All Key Shortcuts
| Situation | Shortcut |
|---|---|
| Sum doubles at SI | R = 100/T |
| Sum becomes n times at SI | R = (n−1)×100/T |
| CI for 2 years | 2×annual SI + interest on interest |
| CI−SI for 2 years | P(R/100)² |
| CI−SI for 3 years | P(R/100)²(3+R/100) |
| Doubling time at CI | ≈ 72/R (Rule of 72) |
| Half-yearly compounding | R÷2, T×2 |
| Quarterly compounding | R÷4, T×4 |