Episode 8 — Aptitude and Reasoning / 8.4 — Compound Interest

8.4.b Compound Interest -- Tips, Tricks, and Shortcuts

1. The CI-SI Difference Shortcut (Most Important!)

This shortcut alone can save 2-3 minutes in exams.

For 2 Years

CI - SI = P (R/100)^2

This is simply: SI for 1 year x R/100, or equivalently, the interest on one year's interest.

Quick way to remember: The difference is the interest on the first year's interest.

Example: P = 5000, R = 10%, T = 2 years

  SI for 1 year = 500
  CI - SI = 500 x 10/100 = Rs. 50

  OR directly: 5000 x (10/100)^2 = 5000 x 0.01 = Rs. 50

For 3 Years

CI - SI = P (R/100)^2 (3 + R/100)

Example: P = 10000, R = 10%, T = 3 years

  CI - SI = 10000 x (0.1)^2 x (3 + 0.1)
          = 10000 x 0.01 x 3.1
          = Rs. 310

2. Successive Compounding -- The Multiplier Method

Instead of memorizing formulas, think of CI as successive percentage increases.

Rate = 10%  -->  Multiplier = 1.10
Rate = 20%  -->  Multiplier = 1.20
Rate = 5%   -->  Multiplier = 1.05

For T years at R%: multiply the principal by the multiplier T times.

Example: Rs. 5000 at 10% for 3 years

  After Year 1: 5000 x 1.1 = 5500
  After Year 2: 5500 x 1.1 = 6050
  After Year 3: 6050 x 1.1 = 6655

  CI = 6655 - 5000 = Rs. 1655

This is especially useful when the rate changes each year.

3. Power Values to Memorize

Memorizing these will save enormous time in exams:

Powers of Common Multipliers

(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

Amounts on Rs. 100 (Quick Reference)

Rate	After 2 Years	After 3 Years	CI (2 yr)	CI (3 yr)
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

How to use this table:

To find CI on Rs. 8000 at 10% for 2 years:

  CI on Rs. 100 at 10% for 2 years = Rs. 21
  CI on Rs. 8000 = 21 x (8000/100) = 21 x 80 = Rs. 1680

4. Fraction-Based Rate Shortcuts

Many common rates have clean fraction equivalents. Using fractions is often faster than decimals.

Rate    Fraction    Multiplier (as fraction)
 5%      1/20       21/20
 8%      2/25       27/25  (approx; exact = 27/25 for quick calc)
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

Example: Rs. 3200 at 25% CI for 2 years

  After Year 1: 3200 x 5/4 = 4000
  After Year 2: 4000 x 5/4 = 5000
  CI = 5000 - 3200 = Rs. 1800

No calculator needed!

5. Population Growth / Depreciation Tricks

Trick 1: If Population Doubles

If a population grows at R% and you need to find when it doubles:

Doubling time (approx) = 72 / R    (Rule of 72)

Example: At 8% growth, population doubles in approximately 72/8 = 9 years.

Trick 2: Net Change After Growth and Decline

If a value increases by R% and then decreases by R%:

Net change = -R^2/100 %   (always a decrease!)

Example: If population grows by 10% then falls by 10%:

  Net effect = -(10)^2 / 100 = -1%
  So the population actually decreases by 1%.

Trick 3: Same Rate Growth for Multiple Years

For population or depreciation problems, just use the multiplier method:

Growth:       Final = Initial x (1 + R/100)^T
Depreciation: Final = Initial x (1 - R/100)^T

6. When to Use Which Formula

Situation	Formula to Use
"Compounded annually"	A = P(1 + R/100)^T
"Compounded half-yearly / semi-annually"	A = P(1 + R/200)^(2T)
"Compounded quarterly"	A = P(1 + R/400)^(4T)
"Find the difference between CI and SI" (2 yr)	P(R/100)^2
"Find the difference between CI and SI" (3 yr)	P(R/100)^2(3 + R/100)
"Population increases/grows"	P(1 + R/100)^T
"Value depreciates/decreases"	V(1 - R/100)^T
"Different rate each year"	P(1+R1/100)(1+R2/100)...
"Find the effective rate"	(1 + R/(n*100))^n - 1
"Equal annual installments"	Present value of annuity

7. Reverse Calculation Tricks

Finding Principal When CI Is Given

If CI for 2 years is given:
  CI = P[(1 + R/100)^2 - 1]
  P = CI / [(1 + R/100)^2 - 1]

Finding Rate When CI for Two Successive Years Is Given

If CI for the 1st year = I1, and CI for the 2nd year = I2:

  Rate = [(I2 - I1) / I1] x 100

Why? The extra interest in the 2nd year is the interest earned on the 1st year's interest.

Example: CI in 1st year = Rs. 400, CI in 2nd year = Rs. 440

  Rate = [(440 - 400) / 400] x 100 = (40/400) x 100 = 10%

Finding Principal from CI for Successive Years

  Principal = I1 x (100/R)

  Where I1 is the interest for the first year.

Example: CI in 1st year = 400, Rate = 10%

  P = 400 x 100/10 = Rs. 4000

8. The "Interest Table" Shortcut for 2-3 Year Problems

For problems with small time periods, build an interest table year by year. This avoids exponent calculations entirely.

Example: Rs. 12,000 at 15% for 3 years, compounded annually

  Year 1: Interest = 15% of 12000 = 1800   | Balance = 13800
  Year 2: Interest = 15% of 13800 = 2070   | Balance = 15870
  Year 3: Interest = 15% of 15870 = 2380.5 | Balance = 18250.5

  Total CI = 1800 + 2070 + 2380.5 = Rs. 6250.5

This is often faster than computing (1.15)^3 without a calculator.

9. Common Exam Traps to Watch Out For

Trap 1: "Compounded Half-Yearly" vs "Compounded Annually"

Many students use R and T directly even when the problem says "half-yearly." Always halve the rate and double the time.

WRONG: A = P(1 + 12/100)^2       (for 12% half-yearly, 1 year)
RIGHT: A = P(1 + 6/100)^2        (rate halved, periods = 2)

Trap 2: CI vs Amount

The question might ask for the compound interest (CI = A - P) or the total amount (A). Read carefully!

Trap 3: Rate Per Annum vs Rate Per Period

If a problem says "5% per half-year compounded half-yearly," the rate per period is 5% (not 2.5%).

"12% per annum compounded half-yearly" --> rate per period = 6%
"5% per half-year compounded half-yearly" --> rate per period = 5%

Trap 4: Time Units Mismatch

If the rate is per annum but time is given in months, convert time to years first (or adjust the formula).

Trap 5: Difference Question Without Specifying Years

When a problem says "find the difference between CI and SI," make sure you know T. The shortcut formulas only work for exactly 2 or 3 years.

Trap 6: "Payable at the End" vs "Compounded"

If interest is "payable at the end of each year," it usually means simple interest. "Compounded" is the keyword for CI.

10. Speed Practice -- Mental Math Patterns

Pattern 1: CI on Rs. 1000 at 10%

  1 year: CI = 100
  2 years: CI = 210      (100 + 110)
  3 years: CI = 331      (100 + 110 + 121)
  4 years: CI = 464.1    (100 + 110 + 121 + 133.1)

Pattern 2: The 20% Shortcut

At 20%, each year's amount is previous x 6/5:

  1000 -> 1200 -> 1440 -> 1728

Pattern 3: Doubling and Tripling (Approximate)

  At 10%, money doubles in ~7.2 years  (72/10)
  At 12%, money doubles in ~6 years    (72/12)
  At 15%, money doubles in ~4.8 years  (72/15)
  At 20%, money doubles in ~3.6 years  (72/20)

Pattern 4: Quick Check Using "Net CI Rate"

For 2 years at R%, the net CI rate on the principal is:

  Net CI rate (2 years) = 2R + R^2/100

  Example: R = 10%
  Net rate = 20 + 1 = 21%
  So CI on Rs. 5000 at 10% for 2 years = 21% of 5000 = Rs. 1050

For 3 years at R%:

  Net CI rate (3 years) = 3R + 3R^2/100 + R^3/10000

  Example: R = 10%
  Net rate = 30 + 3 + 0.1 = 33.1%
  So CI on Rs. 5000 at 10% for 3 years = 33.1% of 5000 = Rs. 1655