Study Notes – Simple & Compound Interest
(For LIC / GIC Quantitative Aptitude)
1. Simple Explanation
| Concept | What it is | When it is used |
|---|---|---|
| Simple Interest (SI) | Interest calculated only on the principal amount for the whole period. | Short‐term loans, savings accounts, insurance premiums that are paid once. |
| Compound Interest (CI) | Interest is calculated on the principal and on any interest that has already accumulated. | Long‑term investments, annuity contracts, certain insurance products, loans with periodic compounding. |
Key Idea – Simple → one level of calculation. Compound → interest itself earns interest.
2. Key Points & Formulas
| Item | Formula | Notes |
|---|---|---|
| Simple Interest | ( SI = P \times r \times t ) | (P) = principal, (r) = annual rate (in decimal), (t) = time in years. |
| Total Amount (A) | ( A = P + SI = P(1 + rt) ) | “Total” or “Future Value” of the investment. |
| Compound Interest (annual compounding) | ( A = P(1 + r)^t ) | (r) in decimal. |
| Compound Interest (m times per year) | ( A = P\left(1 + \frac{r}{m}\right)^{mt} ) | (m) = compounding frequency (e.g., 4 for quarterly). |
| Effective Annual Rate (EAR) | ( EAR = \left(1 + \frac{r}{m}\right)^m - 1 ) | Useful when comparing different compounding frequencies. |
| Number of Years from Two Amounts (SI) | ( t = \frac{SI}{P \cdot r} ) | Rearrange SI formula. |
| Inverse for Principal (SI) | ( P = \frac{SI}{r \cdot t} ) | |
| Rate from Two Amounts (SI) | ( r = \frac{SI}{P \cdot t} ) | |
| Effective Rate from Two Amounts (CI) | ( r_{\text{eff}} = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1 ) | |
| Time from Two Amounts (CI) | ( t = \frac{\ln(A/P)}{\ln(1+r)} ) | Requires log (natural or base‑10). |
Quick “Pay‑off” Rules (Useful for Mental Math)
| Situation | Shortcut |
|---|---|
| SI for 10 % per annum, 3 years | (SI = P \times 0.10 \times 3 = 0.30P) → 30% of principal. |
| CI for 12 % per annum, 1 year | (A = P(1+0.12) = 1.12P) → 12% growth. |
| CI for 12 % per annum, 2 years | (A = P(1.12)^2 \approx P(1.2544)) → 25.44% growth. |
| CI for 12 % per annum, 3 years | (A = P(1.12)^3 \approx P(1.404928)) → 40.49% growth. |
(Use a calculator or pre‑memorised values: 1.12² = 1.2544, 1.12³ = 1.404928, etc.)
3. Worked Example 1 – Simple Interest
Problem
A LIC policyholder lends ₹50,000 to a friend for 4 years at an annual simple interest rate of 8 %.
What is the total amount payable at the end of the period?
Solution
-
Identify variables:
(P = ₹50,000)
(r = 8% = 0.08)
(t = 4) years -
Compute SI:
[ SI = P \times r \times t = 50{,}000 \times 0.08 \times 4 = 50{,}000 \times 0.32 = ₹16{,}000 ] -
Compute Total Amount (A):
[ A = P + SI = 50{,}000 + 16{,}000 = ₹66{,}000 ]
Answer – ₹66,000 will be payable after 4 years.
4. Worked Example 2 – Compound Interest
Problem
An LIC investment grows to ₹1,20,000 after 5 years.
The investment was made at an annual compound interest rate of 10 % compounded quarterly.
What was the original principal?
Solution
-
Variables:
(A = ₹1,20,000)
(r = 10% = 0.10)
(m = 4) (quarterly)
(t = 5) years -
Formula for (A):
[ A = P\left(1 + \frac{r}{m}\right)^{mt} ] Plug in the numbers:
[ 120{,}000 = P\left(1 + \frac{0.10}{4}\right)^{4 \times 5} = P(1.025)^{20} ] -
Compute the compound factor:
[ (1.025)^{20} \approx 1.63862 ] (Use a calculator or logarithms.) -
Solve for (P):
[ P = \frac{120{,}000}{1.63862} \approx ₹73{,}332 ]
Answer – The original principal was approximately ₹73,332.