Calculating Compound Interest with 5-Monthly Compounding

This problem asks us to find the compound interest on a given principal amount over a specific time period at a certain annual rate, with the interest being compounded every 5 months.

Understanding the Given Information

• Principal amount (P) = Rs. 8192
• Time period (T) = \(1 \frac{1}{4}\) years = \( \frac{5}{4} \) years
• Annual interest rate (R) = 15% per annum
• Compounding frequency: 5-monthly

Adjusting Rate and Time for Compounding Frequency

Since the interest is compounded 5-monthly, we need to find the rate per compounding period and the total number of compounding periods in the given time.

• A year has 12 months.
• Number of 5-month periods in a year = \( \frac{12}{5} \)
• Rate per 5-month period (r) = Annual rate × \( \frac{5}{12} \) = 15% × \( \frac{5}{12} \) = \( \frac{15 \times 5}{12} \) % = \( \frac{75}{12} \) % = \( \frac{25}{4} \) %
• Convert the rate to a decimal: \( r = \frac{25}{400} = \frac{1}{16} \)

Now, let's find the total number of compounding periods in the given time:

• Time in years = \( \frac{5}{4} \) years
• Time in months = \( \frac{5}{4} \) × 12 months = 15 months
• Number of 5-month periods (n) = Total months / Months per period = 15 months / 5 months = 3 periods

Calculating the Amount Using the Compound Interest Formula

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

\( A = P(1 + r)^n \)

Where:

• P = Principal amount
• r = Rate per compounding period
• n = Number of compounding periods

Substitute the values we found:

\( A = 8192 \left(1 + \frac{1}{16}\right)^3 \)

\( A = 8192 \left(\frac{16 + 1}{16}\right)^3 \)

\( A = 8192 \left(\frac{17}{16}\right)^3 \)

\( A = 8192 \times \frac{17^3}{16^3} \)

\( A = 8192 \times \frac{4913}{4096} \)

Since \( 8192 = 2 \times 4096 \), we can simplify:

\( A = 2 \times 4913 \)

\( A = 9826 \)

The amount after \(1 \frac{1}{4}\) years, with interest compounded 5-monthly, is Rs. 9826.

Calculating the Compound Interest

The compound interest (CI) is the difference between the amount and the principal:

\( CI = A - P \)

\( CI = 9826 - 8192 \)

\( CI = 1634 \)

The compound interest is Rs. 1634.

Let's summarise the key values:

The calculated compound interest is Rs. 1634, which matches one of the given options.

Revision Table: Key Concepts in Compound Interest

Additional Information: Different Compounding Frequencies

The way interest is compounded significantly affects the final amount. Here are common compounding frequencies and how they affect the rate and time conversion:

• Annually: Interest is compounded once a year. Rate per period = Annual Rate. Number of periods = Time in years.
• Semi-Annually: Interest is compounded twice a year (every 6 months). Rate per period = Annual Rate / 2. Number of periods = Time in years × 2.
• Quarterly: Interest is compounded four times a year (every 3 months). Rate per period = Annual Rate / 4. Number of periods = Time in years × 4.
• Monthly: Interest is compounded twelve times a year (every month). Rate per period = Annual Rate / 12. Number of periods = Time in years × 12.
• n-Monthly: Interest is compounded every 'n' months. Number of periods per year = \( \frac{12}{n} \). Rate per period = Annual Rate × \( \frac{n}{12} \). Number of periods = Time in years × \( \frac{12}{n} \). In our problem, n = 5.

Understanding these conversions is crucial for solving compound interest problems with varying compounding frequencies.