Understanding Compound Interest Problems

This problem asks us to find the original sum of money (Principal) that grew to a specific amount over a certain period at a given compound interest rate. We are given the final amount, the time period, and the annual interest rate. The interest is compounded annually.

Key Concepts: Compound Interest

Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. This means the interest earns interest. The formula for the final amount (A) when the principal (P) is invested at an annual interest rate (r) compounded annually for n years is:

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

In this question, we are given:

• Amount (A) = Rs. 32490
• Time period (n) = 2 years
• Rate of interest (r) = 14% per annum

We need to find the Principal (P).

Step-by-Step Solution to Find the Sum of Money

We need to rearrange the compound interest formula to solve for P:

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

Now, substitute the given values into the formula:

\( P = \frac{32490}{\left(1 + \frac{14}{100}\right)^2} \)

Let's simplify the term inside the parenthesis:

\( 1 + \frac{14}{100} = 1 + 0.14 = 1.14 \)

Now, calculate the term in the denominator:

\( (1.14)^2 = 1.14 \times 1.14 \)

Let's calculate \(1.14 \times 1.14\):

So, \( (1.14)^2 = 1.2996 \).

Now, substitute this back into the formula for P:

\( P = \frac{32490}{1.2996} \)

To perform this division, we can multiply both the numerator and denominator by 10000 to remove the decimal from the denominator:

\( P = \frac{32490 \times 10000}{1.2996 \times 10000} = \frac{324900000}{12996} \)

Performing the division:

\( \frac{324900000}{12996} \approx 25000 \)

So, the original sum of money (Principal) is Rs. 25000.

Verification

Let's check if Rs. 25000 grows to Rs. 32490 in 2 years at 14% per annum compounded annually:

\( A = 25000 \left(1 + \frac{14}{100}\right)^2 \)

\( A = 25000 (1.14)^2 \)

\( A = 25000 \times 1.2996 \)

\( A = 32490 \)

The calculated amount matches the given amount, confirming our Principal calculation is correct.

The sum of money is Rs. 25000.

Revision Table: Compound Interest Terms

Additional Information on Compound Interest Calculation

The frequency of compounding can affect the final amount. If interest is compounded more than once a year (e.g., half-yearly, quarterly, monthly), the formula changes slightly:

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

Where:

• \(r\) is the annual interest rate.
• \(k\) is the number of times interest is compounded per year.
• \(n\) is the number of years.

For example, if compounded half-yearly, \(k=2\); if quarterly, \(k=4\); if monthly, \(k=12\).

In this specific problem, compounding is annual, so \(k=1\), and the formula simplifies back to \( A = P \left(1 + \frac{r}{100}\right)^n \).

Understanding the compounding frequency is crucial for accurate compound interest calculations.