Answer :
To find the annual interest rate \( R \) in compound interest, you typically need to know the following information:
1. **Initial Principal (P)**: The initial amount of money invested or borrowed.
2. **Final Amount (A)**: The amount of money accumulated after interest over a specified time period.
3. **Number of Compounding Periods (n)**: How many times interest is compounded per year.
4. **Time (t)**: The time period over which the interest is applied, usually in years.
The formula for compound interest can be expressed as:
\[ A = P \left(1 + \frac{R}{n}\right)^{nt} \]
Here,
- \( A \) is the final amount (including principal and interest).
- \( P \) is the principal amount (initial investment or loan amount).
- \( R \) is the annual interest rate (what we want to find).
- \( n \) is the number of times interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed for.
To find \( R \) from this equation, you typically need to rearrange it algebraically. Here's how you can find \( R \):
1. **Isolate \( \left(1 + \frac{R}{n}\right)^{nt} \) on one side of the equation**:
\[ \left(1 + \frac{R}{n}\right)^{nt} = \frac{A}{P} \]
2. **Take the natural logarithm (ln) of both sides** to simplify the equation:
\[ \ln\left(\left(1 + \frac{R}{n}\right)^{nt}\right) = \ln\left(\frac{A}{P}\right) \]
3. **Apply the power rule of logarithms** \( \ln(a^b) = b \ln(a) \):
\[ nt \ln\left(1 + \frac{R}{n}\right) = \ln\left(\frac{A}{P}\right) \]
4. **Divide both sides by \( nt \)** to solve for \( \ln\left(1 + \frac{R}{n}\right) \):
\[ \ln\left(1 + \frac{R}{n}\right) = \frac{\ln\left(\frac{A}{P}\right)}{nt} \]
5. **Exponentiate both sides** to solve for \( 1 + \frac{R}{n} \):
\[ 1 + \frac{R}{n} = e^{\frac{\ln\left(\frac{A}{P}\right)}{nt}} \]
6. **Subtract 1 and solve for \( R \)**:
\[ \frac{R}{n} = e^{\frac{\ln\left(\frac{A}{P}\right)}{nt}} - 1 \]
\[ R = n \left( e^{\frac{\ln\left(\frac{A}{P}\right)}{nt}} - 1 \right) \]
Thus, \( R \) can be found using the above formula, provided you know \( P \), \( A \), \( n \), and \( t \).
**Note:** Sometimes, if \( n = 1 \) (interest is compounded annually), the formula simplifies to a more straightforward form:
\[ R = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1 \]
This is derived by isolating \( \left(\frac{A}{P}\right)^{\frac{1}{t}} \) and subtracting 1 from both sides.
