Understanding Compound Interest with Half-Yearly Compounding

This problem involves calculating the initial sum (principal) invested under compound interest, where the interest is compounded half-yearly.

When interest is compounded half-yearly, the annual interest rate is divided by 2, and the time period is multiplied by 2 to get the number of compounding periods.

The formula for compound interest is:

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

Where:

• \(A\) is the Amount
• \(P\) is the Principal
• \(R\) is the Rate of interest per compounding period
• \(n\) is the number of compounding periods

Applying the Formula for Half-Yearly Compounding

Given:

• Annual rate = 40% p.a.
• Compounding frequency = Half-yearly
• Time = One and a half years (1.5 years)
• Amount at the end of 1.5 years = Rs. 2,16,000

Let's adjust the rate and time for half-yearly compounding:

• Rate per half-year (\(R\)): The annual rate is 40%. For half-yearly compounding, the rate per period is \(40\% / 2 = 20\%\). As a decimal, \(R = 20/100 = 0.20\). This can also be written as a fraction \(1/5\).
• Number of compounding periods (\(n\)): The time is 1.5 years. Since interest is compounded half-yearly, there are 2 compounding periods in a year. So, the total number of periods is \(1.5 \text{ years} \times 2 \text{ periods/year} = 3\) periods.

Now, we can plug these values into the compound interest formula:

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

\(2,16,000 = P \left(1 + \frac{20}{100}\right)^3\)

\(2,16,000 = P \left(1 + \frac{1}{5}\right)^3\)

\(2,16,000 = P \left(\frac{5+1}{5}\right)^3\)

\(2,16,000 = P \left(\frac{6}{5}\right)^3\)

\(2,16,000 = P \times \left(\frac{6 \times 6 \times 6}{5 \times 5 \times 5}\right)\)

\(2,16,000 = P \times \left(\frac{216}{125}\right)\)

Calculating the Principal Sum

To find the principal \(P\), we need to rearrange the equation:

\(P = 2,16,000 \times \left(\frac{125}{216}\right)\)

We can see that 2,16,000 is \(1000 \times 216\). Substitute this into the equation:

\(P = 1000 \times 216 \times \left(\frac{125}{216}\right)\)

Cancel out 216 from the numerator and denominator:

\(P = 1000 \times 125\)

\(P = 1,25,000\)

So, the initial sum (principal) invested was Rs. 1,25,000.

Final Answer

The principal sum is Rs. 1,25,000.

Revision Table: Compound Interest Concepts

Additional Information: Different Compounding Frequencies

Compound interest can be calculated based on different frequencies. Here are a few common ones and how they affect the rate and number of periods:

• Annually: \(R = \text{Annual Rate}\), \(n = \text{Time in Years}\)
• Half-Yearly: \(R = \text{Annual Rate} / 2\), \(n = \text{Time in Years} \times 2\)
• Quarterly: \(R = \text{Annual Rate} / 4\), \(n = \text{Time in Years} \times 4\)
• Monthly: \(R = \text{Annual Rate} / 12\), \(n = \text{Time in Years} \times 12\)

Understanding the compounding frequency is crucial for correctly calculating compound interest and finding either the amount or the principal.