Compound Interest Rate Calculation

This section details the calculation for finding the annual compound interest rate given the principal, final amount, and time period.

Problem Statement Details

• Principal Amount ($P$) = ₹15,625
• Final Amount ($A$) after 3 years = ₹24,389
• Time period ($n$) = 3 years
• Interest compounded annually.

Compound Interest Formula

The standard formula for the amount ($A$) accumulated with compound interest is:

$ A = P \left(1 + \frac{R}{100}\right)^n $

Where:

• $A$ is the final amount
• $P$ is the principal amount
• $R$ is the annual interest rate (in percent)
• $n$ is the number of years

Step-by-Step Calculation

1. Substitute the known values ($P=15625$, $A=24389$, $n=3$) into the formula:

   $ 24389 = 15625 \left(1 + \frac{R}{100}\right)^3 $

2. Isolate the term $\left(1 + \frac{R}{100}\right)^3$:

   $ \left(1 + \frac{R}{100}\right)^3 = \frac{24389}{15625} $

3. Calculate the cube root of both sides. Note that $15625 = 25^3$ and $24389 = 29^3$:

   $ 1 + \frac{R}{100} = \sqrt[3]{\frac{29^3}{25^3}} $

   $ 1 + \frac{R}{100} = \frac{29}{25} $

4. Solve for $\frac{R}{100}$:

   $ \frac{R}{100} = \frac{29}{25} - 1 $

   $ \frac{R}{100} = \frac{29 - 25}{25} = \frac{4}{25} $

5. Determine the rate $R$:

   $ R = \frac{4}{25} \times 100 $

   $ R = 16 $

Result

The annual rate of interest is 16%.