Calculating Compound Interest Rate Over 2 Years

This problem asks us to find the annual rate of interest when a sum of money grows to a certain multiple of itself in 2 years under compound interest. The core concept here is understanding the compound interest formula and how to manipulate it to solve for the interest rate.

The amount \(A\) after \(n\) years on a principal \(P\) at a compound interest rate of \(r\) percent per annum is given by the formula:

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

In this specific problem, we are given the following information:

• The sum (Principal) is \(P\).
• The amount after 2 years becomes \(\frac{36}{25}\) of the principal, so \(A = \frac{36}{25}P\).
• The number of years is \(n = 2\).

We need to find the rate of interest per annum, \(r\).

Step-by-Step Calculation of the Interest Rate

Let's substitute the given values into the compound interest formula:

\(\frac{36}{25}P = P\left(1 + \frac{r}{100}\right)^2\)

We can cancel the principal \(P\) from both sides of the equation, assuming \(P \neq 0\):

\(\frac{36}{25} = \left(1 + \frac{r}{100}\right)^2\)

To isolate the term involving \(r\), we need to take the square root of both sides of the equation:

\(\sqrt{\frac{36}{25}} = \sqrt{\left(1 + \frac{r}{100}\right)^2}\)

\(\frac{\sqrt{36}}{\sqrt{25}} = 1 + \frac{r}{100}\)

Calculate the square roots:

\(\frac{6}{5} = 1 + \frac{r}{100}\)

Now, we need to solve for \(r\). Subtract 1 from both sides:

\(\frac{6}{5} - 1 = \frac{r}{100}\)

\(\frac{6 - 5}{5} = \frac{r}{100}\)

\(\frac{1}{5} = \frac{r}{100}\)

To find \(r\), multiply both sides by 100:

\(r = \frac{1}{5} \times 100\)

\(r = \frac{100}{5}\)

\(r = 20\)

So, the rate of interest per annum is 20%.

Let's verify with the options provided:

• 15%
• 20%
• 25%
• 10%

Our calculated rate of 20% matches one of the options.

Verification of the Rate of Interest

If the rate of interest is 20% per annum compound interest, let's see if a sum \(P\) becomes \(\frac{36}{25}P\) after 2 years.

\(A = P\left(1 + \frac{20}{100}\right)^2\)

\(A = P\left(1 + \frac{1}{5}\right)^2\)

\(A = P\left(\frac{5+1}{5}\right)^2\)

\(A = P\left(\frac{6}{5}\right)^2\)

\(A = P\left(\frac{36}{25}\right)\)

\(A = \frac{36}{25}P\)

This matches the condition given in the problem. Therefore, the calculated rate of 20% is correct.

Revision Table: Compound Interest Concepts

Additional Information: Compound Interest Frequency

The problem mentions "compound interest" without specifying the frequency (e.g., annually, semi-annually, quarterly). When the frequency is not mentioned in such problems, it is generally assumed to be compounded annually. If the interest were compounded more frequently, the formula would be slightly different:

\(A = P\left(1 + \frac{r}{100k}\right)^{nk}\)

Where:

• \(k\) is the number of times interest is compounded per year.
• For annual compounding, \(k=1\).
• For semi-annual, \(k=2\).
• For quarterly, \(k=4\).
• For monthly, \(k=12\).

In our problem, since it is for 2 years and the context implies a standard scenario, we correctly assumed annual compounding, which means \(k=1\), simplifying the formula back to \(A = P\left(1 + \frac{r}{100}\right)^n\).

Understanding the difference between simple interest and compound interest is also crucial. Simple interest is calculated only on the principal amount, while compound interest is calculated on the principal plus accumulated interest, leading to faster growth.