Episode 8 — Aptitude and Reasoning / 8.4 — Compound Interest
8.4 Compound Interest -- Quick Revision Sheet
Use this sheet for a rapid review before exams. Every formula you need is here.
1. Core Formulas
Amount (Annual) A = P (1 + R/100)^T
Compound Interest CI = A - P = P [(1 + R/100)^T - 1]
2. Compounding Frequency Formulas
+-------------------+------------+-------------------+----------------------------------+
| Compounding | n | Rate per Period | Formula |
+-------------------+------------+-------------------+----------------------------------+
| Annual | 1 | R | A = P(1 + R/100)^T |
| Semi-Annual | 2 | R/2 | A = P(1 + R/200)^(2T) |
| Quarterly | 4 | R/4 | A = P(1 + R/400)^(4T) |
| Monthly | 12 | R/12 | A = P(1 + R/1200)^(12T) |
+-------------------+------------+-------------------+----------------------------------+
Rule: Divide rate by n, multiply time by n.
3. SI vs CI -- Comparison
+----------------------------+---------------------------+----------------------------+
| Feature | Simple Interest | Compound Interest |
+----------------------------+---------------------------+----------------------------+
| Interest on | Original principal only | Principal + past interest |
| Growth | Linear | Exponential |
| Formula | SI = PRT/100 | CI = P[(1+R/100)^T - 1] |
| For T = 1 year | SI = CI | SI = CI |
| For T > 1 year | SI < CI | CI > SI |
| Each year's interest | Constant | Increasing |
+----------------------------+---------------------------+----------------------------+
4. CI - SI Difference Shortcuts
For 2 Years:
CI - SI = P (R/100)^2
Alternatively: CI - SI = (SI for 1 year) x R/100
For 3 Years:
CI - SI = P (R/100)^2 (3 + R/100)
Alternatively: CI - SI = P R^2 (300 + R) / 100^3
5. Net CI Rate (on Rs. 100)
For quick calculations, use the net rate directly:
2 Years: Net rate = 2R + R^2/100
3 Years: Net rate = 3R + 3R^2/100 + R^3/10000
Pre-Computed Table (Amount on Rs. 100):
Rate 2 Years 3 Years CI (2yr) CI (3yr)
5% 110.25 115.76 10.25 15.76
8% 116.64 125.97 16.64 25.97
10% 121.00 133.10 21.00 33.10
12% 125.44 140.49 25.44 40.49
15% 132.25 152.09 32.25 52.09
20% 144.00 172.80 44.00 72.80
25% 156.25 195.31 56.25 95.31
Usage: CI on Rs. P = (CI on Rs. 100) x P/100
6. Powers to Memorize
(1.05)^2 = 1.1025 (1.05)^3 = 1.157625
(1.08)^2 = 1.1664 (1.08)^3 = 1.259712
(1.10)^2 = 1.21 (1.10)^3 = 1.331
(1.12)^2 = 1.2544 (1.12)^3 = 1.404928
(1.15)^2 = 1.3225 (1.15)^3 = 1.520875
(1.20)^2 = 1.44 (1.20)^3 = 1.728
(1.25)^2 = 1.5625 (1.25)^3 = 1.953125
7. Fraction Multipliers
Rate Fraction Multiplier
5% 1/20 21/20
10% 1/10 11/10
12.5% 1/8 9/8
15% 3/20 23/20
20% 1/5 6/5
25% 1/4 5/4
33.33% 1/3 4/3
50% 1/2 3/2
8. Population and Depreciation
Population Growth: P_future = P_now (1 + R/100)^T
Population Decline: P_future = P_now (1 - R/100)^T
Depreciation: V_future = V_now (1 - R/100)^T
Variable Rates: Final = Initial x (1 +/- R1/100)(1 +/- R2/100)...
9. Effective Rate of Interest
E = (1 + R/(n x 100))^n - 1
Common values:
10% semi-annual --> 10.25% effective
10% quarterly --> 10.38% effective
12% semi-annual --> 12.36% effective
12% quarterly --> 12.55% effective
20% semi-annual --> 21.00% effective
20% quarterly --> 21.55% effective
10. Finding Unknowns
Finding Rate (from successive year CI):
Rate = [(CI_year2 - CI_year1) / CI_year1] x 100
Finding Time (amount given):
(1 + R/100)^T = A/P
Match with known powers of (1 + R/100)
Finding Principal (from CI-SI difference):
2 years: P = (CI - SI) / (R/100)^2
3 years: P = (CI - SI) / [(R/100)^2 (3 + R/100)]
11. Rule of 72 (Doubling Time)
Years to double ≈ 72 / R
At 6%: ~12 years
At 8%: ~9 years
At 10%: ~7.2 years
At 12%: ~6 years
At 15%: ~4.8 years
At 20%: ~3.6 years
12. Fractional Time
For T = n + p/q years:
A = P(1 + R/100)^n x (1 + (p/q) x R/100)
|_______________| |___________________|
Whole years Fraction (use SI)
13. Installments (Equal Annual)
For n equal installments of X each:
P = X/(1+R/100) + X/(1+R/100)^2 + ... + X/(1+R/100)^n
14. Common Patterns to Recognize
Pattern 1: "Difference between CI and SI" --> Use CI-SI shortcut
Pattern 2: "Compounded half-yearly" --> Halve rate, double time
Pattern 3: "Population grows/increases" --> Use growth formula
Pattern 4: "Value depreciates/decreases" --> Use depreciation formula
Pattern 5: "Different rate each year" --> Multiply individual factors
Pattern 6: "CI for 2nd year" or "3rd year" --> R% of previous year's amount
Pattern 7: "Becomes n times in T years" --> (1+R/100)^T = n
Pattern 8: "Doubles in T years, when 4x?" --> 4x in 2T years, 8x in 3T
15. Common Exam Traps -- Checklist
[ ] Did I read "CI" vs "Amount"? (They ask different things!)
[ ] Is it annual, half-yearly, or quarterly compounding?
[ ] Is the rate "per annum" or "per period"?
[ ] Did I convert time to years if given in months?
[ ] For CI-SI difference: is T = 2 or T = 3? (Different formulas!)
[ ] For population: growth (+) or decline (-)?
[ ] For installments: did I discount each payment properly?
16. One-Minute Formula Lookup
Need Formula
---- -------
Amount (annual CI) P(1 + R/100)^T
CI only A - P
Semi-annual compounding P(1 + R/200)^(2T)
Quarterly compounding P(1 + R/400)^(4T)
CI - SI (2 years) P(R/100)^2
CI - SI (3 years) P(R/100)^2 (3 + R/100)
Population growth P(1 + R/100)^T
Depreciation V(1 - R/100)^T
Effective rate (1 + R/n*100)^n - 1
Doubling time 72 / R
Fractional time P(1+R/100)^n x (1 + fraction x R/100)
Rate from successive CI (I2 - I1)/I1 x 100