Calculating Compound Interest Quarterly

This problem asks us to find the compound interest on a principal amount when the interest is calculated quarterly. Understanding how to adjust the annual rate and time period for quarterly compounding is key to solving this problem.

Understanding the Key Terms

• Principal (P): The initial amount of money, which is ₹64000.
• Rate (R): The annual interest rate, given as 10% per annum.
• Time (T): The duration for which the money is invested or borrowed, which is 9 months.
• Compounding Frequency: How often the interest is added to the principal. Here, it's compounded quarterly.

Adjusting Rate and Time for Quarterly Compounding

When interest is compounded quarterly, it means the interest is calculated and added to the principal four times a year (every 3 months). We need to adjust the annual rate and the total time period accordingly.

• Quarterly Rate: The annual rate is divided by the number of compounding periods in a year (which is 4 for quarterly).
  Rate per quarter $= \frac{\text{Annual Rate}}{\text{Number of Quarters in a Year}}$
  Rate per quarter $= \frac{10\%}{4} = 2.5\%$
• Number of Compounding Periods: The total time period is converted into the number of quarters.
  Time $= 9$ months
  Number of quarters in 9 months $= \frac{9 \text{ months}}{3 \text{ months/quarter}} = 3$ quarters

So, we will use a rate of 2.5% per period for 3 periods.

Step-by-Step Compound Interest Calculation

The formula for the amount (A) when interest is compounded is:

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

Where:

• \( A \) is the amount after \( n \) periods.
• \( P \) is the principal amount (₹64000).
• \( R' \) is the rate per compounding period (2.5%).
• \( n \) is the number of compounding periods (3).

Let's plug in the values:

\( A = 64000 \left(1 + \frac{2.5}{100}\right)^{3} \)

\( A = 64000 \left(1 + 0.025\right)^{3} \)

\( A = 64000 \left(1.025\right)^{3} \)

Now, calculate \((1.025)^3\):

\( (1.025)^2 = 1.025 \times 1.025 = 1.050625 \)

\( (1.025)^3 = 1.050625 \times 1.025 = 1.076890625 \)

Now, calculate the amount \( A \):

\( A = 64000 \times 1.076890625 \)

\( A = 68920.90625 \)

Calculating the Compound Interest

The compound interest (CI) is the difference between the final amount (A) and the principal (P).

\( CI = A - P \)

\( CI = 68920.90625 - 64000 \)

\( CI = 4920.90625 \)

Rounding to the nearest whole rupee, the compound interest is approximately ₹4921.

Final Answer Summary

The calculated compound interest on ₹64000 at 10% per annum for 9 months, compounded quarterly, is ₹4921 (approximately).

Compound Interest Calculation Revision Table

Additional Information on Compound Interest

Compound interest is interest calculated on the initial principal and also on all the accumulated interest from previous periods. It's often referred to as "interest on interest."

Different compounding frequencies impact the final interest earned:

• Annually: Interest compounded once a year.
• Semi-annually: Interest compounded twice a year (every 6 months). Rate is R/2, time is 2T (if T is in years).
• Quarterly: Interest compounded four times a year (every 3 months). Rate is R/4, time is 4T (if T is in years).
• Monthly: Interest compounded twelve times a year. Rate is R/12, time is 12T (if T is in years).
• Daily: Interest compounded 365 times a year. Rate is R/365, time is 365T (if T is in years).

The more frequent the compounding, the higher the compound interest earned over the same period, assuming the same nominal annual rate.