Simple and Compound Interest Formulas

1. Simple Interest (SI)

Simple interest is calculated using the formula:

$SI = P \cdot R \cdot T$

Where:

• $P$ = Principal amount (initial investment or loan)
• $R$ = Annual interest rate (in decimal form, e.g., $5\% = 0.05$)
• $T$ = Time (in years)

Modified Versions:

• For time in months: $SI = P \cdot R \cdot \frac{T}{12}$
• To find $P$ (Principal): $P = \frac{SI}{R \cdot T}$
• To find $R$ (Rate): $R = \frac{SI}{P \cdot T}$
• To find $T$ (Time): $T = \frac{SI}{P \cdot R}$

Example:

If $P = 1000$, $R = 5\% = 0.05$, and $T = 3$ years: $SI = 1000 \cdot 0.05 \cdot 3 = 150$

2. Compound Interest (CI)

Compound interest is calculated using:

$A = P \cdot \left(1 + \frac{R}{n}\right)^{n \cdot T}$

Where:

• $A$ = Final amount (Principal + Interest)
• $P$ = Principal
• $R$ = Annual interest rate (in decimal form)
• $n$ = Number of times interest is compounded per year
• $T$ = Time (in years)

Compound Interest (CI) is then: $CI = A - P$

Modified Versions:

• For annual compounding ($n = 1$): $A = P \cdot (1 + R)^T$
• For semiannual compounding ($n = 2$): $A = P \cdot \left(1 + \frac{R}{2}\right)^{2 \cdot T}$
• For quarterly compounding ($n = 4$): $A = P \cdot \left(1 + \frac{R}{4}\right)^{4 \cdot T}$
• For continuous compounding: $A = P \cdot e^{R \cdot T}$

Example:

If $P = 1000$, $R = 5\% = 0.05$, $n = 4$ (quarterly), and $T = 3$ years: $A = 1000 \cdot \left(1 + \frac{0.05}{4}\right)^{4 \cdot 3} = 1000 \cdot \left(1.0125\right)^{12} \approx 1160.75$ $CI = A - P = 1160.75 - 1000 = 160.75$

Key Observations:

1. Simple Interest grows linearly, while Compound Interest grows exponentially.
2. For short durations or small interest rates, $SI \approx CI$.
3. Modified formulas are based on solving for one variable while keeping the others constant.

Practical Tips to Adapt to Modifications

• Identify which variable is unknown (e.g., $P$, $R$, $T$, etc.).
• Rewrite the formula by isolating the unknown.
• For compound interest, always note $n$ (number of compounding intervals) and $R$ (ensure it matches $n$).

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Questions for Practice:

1. What is the simple interest if $P = 2000$, $R = 3\%$, and $T = 5$ years?
2. Find the final amount after 2 years if $P = 1500$, $R = 8\%$, compounded quarterly.
3. How does the total interest differ for a principal of $5000$ at $6\%$ per annum under simple vs. compound interest for 3 years?
4. If compound interest is calculated continuously, how does the formula for $A$ change?
5. Derive the formula for $R$ in a simple interest context given $SI = 250$, $P = 1000$, and $T = 2$ years.

Tip: Always convert percentages to decimals and ensure consistent units for time!