Calculating Compound Interest Rate (Half Yearly Compounding)

This question asks us to find the annual rate of interest at which a principal amount grows to a specific amount over one year, with interest compounded half yearly. Understanding the compound interest formula for half-yearly compounding is key to solving this problem.

Understanding Compound Interest Compounded Half Yearly

When interest is compounded half yearly, it means the interest is calculated and added to the principal twice a year. The annual rate is usually given, but for calculation, we use the half-yearly rate (which is half of the annual rate) and double the number of years for the number of compounding periods.

The formula for compound interest when compounded half yearly is:

\[ A = P\left(1 + \frac{R/2}{100}\right)^{2t} \] where:

• \(A\) is the final amount
• \(P\) is the principal amount
• \(R\) is the annual rate of interest in percent
• \(t\) is the time in years
• \(R/2\) is the half-yearly rate of interest
• \(2t\) is the number of compounding periods (twice per year for t years)

Step-by-Step Calculation

Let's identify the given values:

• Principal amount, \(P = \text{Rs. } 7200\)
• Final amount, \(A = \text{Rs. } 7938\)
• Time period, \(t = 1\) year
• Compounding frequency: Half yearly

We need to find the annual rate, \(R\). Let the half-yearly rate be \(r\%\). Then the annual rate \(R = 2r\).

The number of compounding periods in 1 year, compounded half yearly, is \(2 \times 1 = 2\).

Using the compound interest formula for half-yearly compounding:

\[ A = P\left(1 + \frac{r}{100}\right)^{2} \]

Substitute the given values:

\[ 7938 = 7200\left(1 + \frac{r}{100}\right)^{2} \]

Now, we need to solve for \(r\). Divide both sides by 7200:

\[ \frac{7938}{7200} = \left(1 + \frac{r}{100}\right)^{2} \]

Simplify the fraction \(\frac{7938}{7200}\). Both numbers are divisible by 18 (as 7938 = 18 * 441 and 7200 = 18 * 400):

\[ \frac{441}{400} = \left(1 + \frac{r}{100}\right)^{2} \]

Take the square root of both sides:

\[ \sqrt{\frac{441}{400}} = 1 + \frac{r}{100} \]

\[ \frac{21}{20} = 1 + \frac{r}{100} \]

Subtract 1 from both sides:

\[ \frac{21}{20} - 1 = \frac{r}{100} \]

\[ \frac{21 - 20}{20} = \frac{r}{100} \]

\[ \frac{1}{20} = \frac{r}{100} \]

Multiply both sides by 100 to find \(r\):

\[ r = \frac{1}{20} \times 100 \]

\[ r = 5 \]

So, the half-yearly rate (\(r\)) is 5%. The question asks for the annual rate (\(R\)).

Annual rate \(R = 2 \times \text{half-yearly rate}\)

\[ R = 2 \times 5\% = 10\% \]

The annual rate percent per annum is 10%.

Verification

Let's verify the result with an annual rate of 10% compounded half yearly.

• Principal \(P = 7200\)
• Annual rate \(R = 10\%\)
• Half-yearly rate \(r = 10\% / 2 = 5\%\)
• Time \(t = 1\) year
• Number of periods \(2t = 2\)

Amount \(A = P\left(1 + \frac{r}{100}\right)^{2}\)

\[ A = 7200\left(1 + \frac{5}{100}\right)^{2} \]

\[ A = 7200\left(1 + 0.05\right)^{2} \]

\[ A = 7200\left(1.05\right)^{2} \]

\[ A = 7200 \times 1.1025 \]

\[ A = 7938 \]

The calculated amount matches the given amount, confirming the annual rate is 10%.

Conclusion

The rate percent per annum at which Rs. 7200 will amount to Rs. 7938 in one year, compounded half yearly, is 10%.

Revision Table: Compound Interest Calculations

Additional Information on Compound Interest Rate

The compound interest rate significantly impacts the growth of an investment or loan over time. Unlike simple interest, compound interest earns interest on the accumulated interest from previous periods, leading to exponential growth.

• Effective Annual Rate (EAR): When interest is compounded more frequently than once a year (like half yearly), the actual annual return is slightly higher than the stated annual rate (R%). This is called the Effective Annual Rate. For half-yearly compounding at annual rate R, \(EAR = \left(1 + \frac{R/2}{100}\right)^{2} - 1\).
• Importance of Rate: A higher interest rate leads to faster growth in the amount. Even small differences in rates can result in significant differences in the final amount over long periods.
• Finding the Rate: As shown in this problem, finding the rate often involves solving an exponential equation, which can be done by taking roots (like square root for half-yearly) and algebraic manipulation.

Understanding how the compounding frequency affects the interest calculation is crucial for solving compound interest problems accurately.