This problem asks us to calculate the total amount accumulated when a principal sum is invested at a certain rate of interest, compounded quarterly for a specific time period. Understanding the concepts of compound interest and how compounding frequency affects the calculation is crucial here.

Understanding the Given Financial Data

Let's identify the key information provided in the question:

• Principal (P): The initial sum of money is ₹32,000. This is the amount on which interest is calculated.
• Annual Interest Rate (R): The interest rate is given as 20% per annum.
• Time Period (T): The investment duration is 9 months.
• Compounding Frequency: The interest is compounded quarterly. This means interest is calculated and added to the principal every three months.

Adjusting Rate and Time for Quarterly Compounding

When interest is compounded quarterly, the annual interest rate and the total time period need to be adjusted to match the compounding frequency. There are 4 quarters in a year.

• Rate per Quarter (r): The annual rate is 20%. For quarterly compounding, the rate applied each quarter is the annual rate divided by the number of quarters in a year.
  \(r = \frac{\text{Annual Rate}}{\text{Number of Quarters in a Year}} = \frac{20\%}{4} = 5\%\) per quarter.
• Number of Compounding Periods (n): The total time period is 9 months. Since each compounding period is one quarter (3 months), the total number of compounding periods is the total time in months divided by the length of one quarter in months.
  \(n = \frac{\text{Total Time in Months}}{\text{Months per Quarter}} = \frac{9}{3} = 3\) quarters.

Calculating the Compound Amount

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

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

• \(P\) is the Principal amount
• \(r\) is the interest rate per compounding period
• \(n\) is the total number of compounding periods

Now, let's plug in the values we have:

• \(P = 32000\)
• \(r = 5\) (which is 5% per quarter)
• \(n = 3\) (quarters)

So, the amount will be:

\[ A = 32000 \left(1 + \frac{5}{100}\right)^3 \]

Let's simplify the expression inside the parentheses:

\[ 1 + \frac{5}{100} = 1 + 0.05 = 1.05 \]

Now substitute this back into the formula:

\[ A = 32000 (1.05)^3 \]

Next, calculate \((1.05)^3\):

\[ (1.05)^3 = 1.05 \times 1.05 \times 1.05 = 1.1025 \times 1.05 = 1.157625 \]

Finally, multiply this by the principal amount:

\[ A = 32000 \times 1.157625 \]

\[ A = 37044 \]

The amount after 9 months, compounded quarterly, is ₹37,044.

Revision Table: Compound Interest Basics

Additional Information on Compounding Frequency

The frequency at which interest is compounded significantly impacts the total amount earned. The more frequently interest is compounded, the higher the final amount will be, assuming the same annual rate.

• Annually: Compounded once a year (\(n=T\), rate=\(R\%\)).
• Semi-annually: Compounded twice a year (\(n=2T\), rate=\(R/2\%\)).
• Quarterly: Compounded four times a year (\(n=4T\), rate=\(R/4\%\)). This was used in our problem.
• Monthly: Compounded twelve times a year (\(n=12T\), rate=\(R/12\%\)).
• Daily: Compounded 365 times a year (\(n=365T\), rate=\(R/365\%\)).

For a given annual rate, the effective annual rate (EAR) increases with the frequency of compounding. EAR is the actual rate of return earned in one year, taking into account the effect of compounding.