Compound Interest Calculation: Finding the Time Period

This problem asks us to determine the time it takes for a principal sum to grow to a specific amount under compound interest, compounded half-yearly. Understanding the formula for compound interest and how to apply it for half-yearly compounding is crucial here.

Understanding the Key Terms

• Principal Amount (P): This is the initial sum of money deposited or invested. In this question, the principal amount is Rs. 1,25,000.
• Amount (A): This is the total sum after the interest has been added to the principal. Here, the final amount is Rs. 1,57,464.
• Interest Rate (R): This is the rate at which interest is charged or earned annually. The given annual interest rate is 16% per annum.
• Compounding Frequency: This indicates how often the interest is calculated and added to the principal. In this problem, the interest is compounded half-yearly, meaning twice a year.
• Time (t): This is the duration for which the money is invested or borrowed, which we need to find in years.

Formula for Half-Yearly Compounding

When interest is compounded half-yearly, the annual interest rate is halved, and the number of compounding periods is doubled. The formula used is:

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

Where:

• \(A\) = Final Amount
• \(P\) = Principal Amount
• \(R\) = Annual Interest Rate (in %)
• \(t\) = Time (in years)
• \(R/2\) = Half-yearly interest rate
• \(2t\) = Total number of half-years (compounding periods)

Step-by-Step Solution

Let's substitute the given values into the compound interest formula:

Given:

• Principal (P) = Rs. 1,25,000
• Amount (A) = Rs. 1,57,464
• Annual Rate (R) = 16%

Since the interest is compounded half-yearly:

• Half-yearly Rate = \(R/2 = 16\% / 2 = 8\%\)

Now, apply the formula:

\[157464 = 125000 \left(1 + \frac{8}{100}\right)^{2t}\]

Simplify the term inside the parenthesis:

\[157464 = 125000 \left(1 + 0.08\right)^{2t}\]

\[157464 = 125000 \left(1.08\right)^{2t}\]

To isolate the term with the exponent, divide both sides by the principal amount (1,25,000):

\[\frac{157464}{125000} = (1.08)^{2t}\]

Perform the division:

\[1.259712 = (1.08)^{2t}\]

Now, we need to find what power of 1.08 equals 1.259712. We can do this by testing powers of 1.08:

• \(1.08^1 = 1.08\)
• \(1.08^2 = 1.08 \times 1.08 = 1.1664\)
• \(1.08^3 = 1.1664 \times 1.08 = 1.259712\)

From the above calculation, we see that \( (1.08)^3 = 1.259712 \). Therefore, we can equate the exponents:

\[2t = 3\]

Finally, solve for \(t\):

\[t = \frac{3}{2}\]

\[t = 1.5 \text{ years}\]

Summary of Values

Thus, it will take 1.5 years for a sum of Rs. 1,25,000 to become Rs. 1,57,464 at 16% per annum interest, compounded half-yearly.