Simple & Compound Interest

Cheatsheet Content

### Simple Interest (SI) Simple interest is calculated only on the principal amount, or on the portion of the principal that remains unpaid. It does not compound. #### Formula $$SI = \frac{P \times R \times T}{100}$$ Where: - $P$ = Principal amount (the initial amount borrowed or invested) - $R$ = Rate of interest per annum (in percentage) - $T$ = Time period (in years) #### Total Amount The total amount ($A$) payable at the end of the term is: $$A = P + SI = P + \frac{P \times R \times T}{100} = P \left(1 + \frac{R \times T}{100}\right)$$ #### Key Characteristics - Interest earned is constant each period. - Used for short-term loans, bonds, and some deposits. - Does not take into account the interest accumulated in previous periods. #### Example If $P = ₹1000$, $R = 5\%$ per annum, $T = 3$ years: $SI = \frac{1000 \times 5 \times 3}{100} = ₹150$ $A = 1000 + 150 = ₹1150$ ### Compound Interest (CI) Compound interest is calculated on the principal amount and also on the accumulated interest of previous periods. It's "interest on interest." #### Formula for Amount $$A = P \left(1 + \frac{R}{100}\right)^T$$ Where: - $A$ = Total amount after $T$ years - $P$ = Principal amount - $R$ = Annual rate of interest (in percentage) - $T$ = Time period (in years) #### Compound Interest (CI) Calculation $$CI = A - P = P \left(1 + \frac{R}{100}\right)^T - P$$ #### Compounding Frequency If interest is compounded more than once a year (e.g., half-yearly, quarterly): $$A = P \left(1 + \frac{R/n}{100}\right)^{n \times T}$$ Where: - $n$ = Number of times interest is compounded per year - Annually: $n=1$ - Half-yearly: $n=2$ - Quarterly: $n=4$ - Monthly: $n=12$ #### Key Characteristics - Interest earned grows exponentially over time. - Used for most investments, savings accounts, and long-term loans. - The more frequently interest is compounded, the higher the effective interest rate. #### Example If $P = ₹1000$, $R = 5\%$ per annum, $T = 3$ years, compounded annually: $A = 1000 \left(1 + \frac{5}{100}\right)^3 = 1000 (1.05)^3 = 1000 \times 1.157625 = ₹1157.63$ $CI = 1157.63 - 1000 = ₹157.63$ ### Difference Between SI and CI | Feature | Simple Interest (SI) | Compound Interest (CI) | | :----------- | :----------------------------------------------- | :------------------------------------------------------- | | **Calculation** | Only on principal | On principal + accumulated interest | | **Growth** | Linear | Exponential | | **Return** | Lower over long periods | Higher over long periods | | **Formula** | $SI = \frac{PRT}{100}$ | $A = P(1 + \frac{R}{100})^T$ | | **Use Case** | Short-term loans, bonds | Savings, investments, long-term loans | ### Effective Annual Rate (EAR) The actual rate of interest earned or paid over a year, considering the effect of compounding. #### Formula $$EAR = \left(1 + \frac{R/n}{100}\right)^n - 1$$ Where: - $R$ = Nominal annual interest rate - $n$ = Number of compounding periods per year #### Example Nominal rate $R = 10\%$, compounded half-yearly ($n=2$): $EAR = \left(1 + \frac{10/2}{100}\right)^2 - 1 = (1 + 0.05)^2 - 1 = (1.05)^2 - 1 = 1.1025 - 1 = 0.1025 = 10.25\%$ This means $10\%$ compounded half-yearly is effectively $10.25\%$ annually. ### Important Relations & Shortcuts #### Difference between CI and SI for 2 Years $$CI - SI = P \left(\frac{R}{100}\right)^2$$ #### Difference between CI and SI for 3 Years $$CI - SI = P \left(\frac{R}{100}\right)^2 \left(3 + \frac{R}{100}\right)$$ #### Doubling Time (Rule of 72 - Approximation) For an investment to double with compound interest, a rough estimate is: $$T \approx \frac{72}{R}$$ Where $R$ is the annual interest rate (as a percentage, e.g., for $8\%$, use $8$). #### Tripling Time (Rule of 114 - Approximation) For an investment to triple with compound interest, a rough estimate is: $$T \approx \frac{114}{R}$$ #### When CI = SI (Only for T=1 Year) For $T=1$ year, Simple Interest = Compound Interest. $SI_1 = \frac{P \times R \times 1}{100}$ $A_1 = P \left(1 + \frac{R}{100}\right)^1 \implies CI_1 = P \left(1 + \frac{R}{100}\right) - P = \frac{PR}{100}$ Thus, $SI_1 = CI_1$. #### Installment Problems Often involve calculating the present value of future payments. If an amount $A$ is due in $T$ years at $R\%$ CI, and is paid in $n$ equal annual installments, the value of each installment ($X$) can be complex. A common type of problem asks for the present value of an installment payment. For two equal annual installments ($X$) at $R\%$ CI: $$P = \frac{X}{(1 + \frac{R}{100})} + \frac{X}{(1 + \frac{R}{100})^2}$$ This can be generalized for more installments. ### Aptitude Tips & Tricks - **Read Carefully:** Identify if it's SI or CI and the compounding frequency. - **Convert Units:** Ensure rate ($R$) and time ($T$) are in consistent units (usually annual rate, years). - **Fractional Time Periods:** - For SI: $T = 2$ years $6$ months $= 2.5$ years. - For CI: If $T = 2$ years $6$ months, calculate for $2$ full years, then calculate SI for $6$ months on the amount obtained after $2$ years. Example: $P(1 + \frac{R}{100})^2 \times (1 + \frac{R \times \text{fractional time}}{100})$ For $2$ years $6$ months, it's $P(1 + \frac{R}{100})^2 \times (1 + \frac{R/2}{100})$ - **Percentage Method for CI:** For small years (2-3), you can use successive percentages. - For 2 years at $R\%$ CI: Effective rate $= (R + R + \frac{R \times R}{100})\%$ - For 3 years at $R\%$ CI: Apply the 2-year effective rate, then compound again with $R\%$. Or use $3R.3R^2R^3$ (e.g., for $10\%$, $3 \times 10 . 3 \times 10^2 . 10^3 = 30.300.1000 \implies 33.1\%$). - **Approximations:** For competitive exams, sometimes approximations (like Rule of 72) or estimation can save time. - **Work Backwards:** If the final amount is given, work backwards to find the principal. - **Identify Key Terms:** - "Amount" means Principal + Interest. - "Interest" is the extra money earned/paid. - "Per annum" means per year.