Calculating Compound Interest Quarterly

This problem asks us to find the compound interest paid on a principal amount when the interest is compounded quarterly. We are given the principal amount, the annual rate of interest, and the time period.

Understanding the Problem Parameters

• Principal (P) = Rs. 10,000
• Annual Rate of Interest (R) = 12% per annum
• Time Period (T) = 9 months
• Compounding Frequency: Quarterly

Since the interest is compounded quarterly, we need to adjust the annual rate and the time period according to the compounding frequency.

Adjusting Rate and Time for Quarterly Compounding

When interest is compounded quarterly, the interest is calculated four times a year. This means:

1. The rate per compounding period is the annual rate divided by 4.
2. The number of compounding periods is the total time in years multiplied by 4 (or total time in months divided by 3).

Let's calculate the quarterly rate and the number of quarters:

• Quarterly Rate (r) = Annual Rate / 4 = 12% / 4 = 3% per quarter.
• Time Period in Quarters (n) = 9 months = \( \frac{9}{3} \) quarters = 3 quarters.

Formula for Compound Amount

The formula for the compound amount (A) when the interest is compounded 'n' times per year is:

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

In our case, compounding is quarterly, so we use the adjusted rate (r) and number of quarters (n):

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

Where:

• P = Principal amount = 10000
• r = Quarterly rate = 3%
• n = Number of quarters = 3

Step-by-Step Calculation

Now, let's substitute the values into the formula to find the compound amount after 9 months.

\( A = 10000 \left(1 + \frac{3}{100}\right)^3 \)

\( A = 10000 \left(1 + 0.03\right)^3 \)

\( A = 10000 (1.03)^3 \)

Let's calculate \( (1.03)^3 \):

\( (1.03)^2 = 1.03 \times 1.03 = 1.0609 \)

\( (1.03)^3 = 1.0609 \times 1.03 \)

\( (1.03)^3 = 1.092727 \)

Now, calculate the compound amount (A):

\( A = 10000 \times 1.092727 \)

\( A = 10927.27 \)

The compound amount after 9 months is Rs. 10,927.27.

Calculating the Compound Interest Paid

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

Compound Interest (CI) = A - P

\( CI = 10927.27 - 10000 \)

\( CI = 927.27 \)

The interest paid by the person after 9 months is Rs. 927.27.

This calculation shows that the total interest accumulated over 9 months with quarterly compounding at a 12% annual rate is Rs. 927.27.

Revision Table: Key Terms for Compound Interest

Additional Information on Compounding

Compounding is the process where the interest earned is added back to the principal, and future interest is calculated on this new, larger principal. The more frequently interest is compounded (like quarterly vs annually), the faster the amount grows.

• Annual Compounding: Interest added once a year.
• Semi-Annual Compounding: Interest added twice a year (rate/2, time*2 periods).
• Quarterly Compounding: Interest added four times a year (rate/4, time*4 periods).
• Monthly Compounding: Interest added twelve times a year (rate/12, time*12 periods).
• Daily Compounding: Interest added 365 times a year (rate/365, time*365 periods).

In this specific loan interest calculation problem, understanding quarterly compounding was crucial to correctly determine the rate and number of periods for the compound interest formula.