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Simple & Compound Interest

Money makes money. Master the 'Effective Rate' method for CI ($x+y+xy/100$) and the 'Difference Formula' to solve 2-year and 3-year problems accurately.

1. SI vs CI (The Core Difference)

SI is Interest on Principal ONLY. CI is Interest on Principal + Interest on Interest.

SI: Rate $R%$ per year. Total Interest for $T$ years = $R \times T \%$ .
CI: Successive % Change. Use $x+y+xy/100$ for 2 years.

Example:

Q: Find CI on Rs 1000 for 2 years at 10%.

Solution: Effective Rate:

10+10+(10\times10)/100 = 21\%

21\%

of 1000 = 210.

2. The Difference Formulas (Golden Rules)

Direct shortcut for Difference ( $D$ ) between CI and SI.

2 Years: $D = P (R/100)^2$ .
3 Years: $D = P (R/100)^2 (3 + R/100)$ .

Example:

Q: Diff between CI and SI for 2 years at 5% is Rs 25. Find P.

Solution:

25 = P (5/100)^2

25 = P (1/400)

P = 10000

3. The 'Times' Concept (N Times)

Sum becomes N times in T years.

SI: Linear Growth. If it doubles (gain 100%) in 5 yrs, it triples (gain 200%) in 10 yrs.
CI: Exponential Powers. If it doubles ( $2^1$ ) in 5 yrs, it becomes 8 times ( $2^3$ ) in $5 \times 3 = 15$ yrs.

Example:

Q: Money doubles in 4 years at CI. When 8 times?

Solution:

2^1 \to 4

yrs.

8 = 2^3 \to 4 \times 3 = 12

yrs.

4. Effective Rate for CI (Successive)

Best for 2-3 years.

2 Years: $2R + R^2/100$ . Example 10% $\to$ 21%.
3 Years: $3R.3R^2 R^3$ (Pascal's Triangle 3:3:1 logic). Example 10% $\to$ 33.1%.

Example:

Q: CI on 5000 at 2% pa for 2 years.

Solution: Eff Rate:

2+2 + (4/100) = 4.04\%

4.04\%

of 5000 = 202.

5. Half-Yearly & Quarterly Compounding

Adjust Rate and Time.

Half-Yearly: Rate becomes $R/2$ , Time becomes $2T$ .
Quarterly: Rate becomes $R/4$ , Time becomes $4T$ .

Example:

Q: 20% pa compounded half-yearly for 1 year.

Solution: New Rate = 10%. New Time = 2 cycles.
Eff Rate =

10+10+1 = 21\%

6. Installments (EMI)

Present Value Concept.

SI Installment: Debt $A = x + (x + \text{int}) + ...$ (Reverse Order).
CI Installment: $P = \frac{I}{(1+r)} + \frac{I}{(1+r)^2}$ .

Example:

Q: Loan 2100 at 10% CI. 2 Annual Installments.

Solution: 10% is 1/10. Installment ratio:

10:11

and

100:121

.
Scale first to 121:

110:121

. Sum P = 210. Sum I = 242.

210 \to 2100

. Installment = 1210.