This problem involves calculating the annual rate of interest when interest is compounded half-yearly.

Understanding the Compound Interest Formula

The formula for compound interest is:

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

Where:

• A is the future value of the investment/loan, including interest (Amount = ₹19,683).
• P is the principal investment amount (Principal = ₹15,625).
• r is the annual interest rate (what we need to find).
• n is the number of times that interest is compounded per year.
• t is the number of years the money is invested or borrowed for (Time = 1 year and 6 months = 1.5 years).

Adapting the Formula for Half-Yearly Compounding

In this case, the interest is compounded half-yearly. This means:

• n = 2 (since interest is compounded twice a year).
• The interest rate per compounding period is $ \frac{r}{n} = \frac{r}{2} $.
• The total number of compounding periods is $ nt = 2 \times 1.5 = 3 $.

The formula becomes:

$$ A = P \left(1 + \frac{r/2}{100}\right)^{3} $$

Or simplified:

$$ A = P \left(1 + \frac{r}{200}\right)^{3} $$

Calculating the Interest Rate

We are given:

• P = ₹15,625
• A = ₹19,683
• Number of periods = 3

Substitute these values into the formula:

$$ 19,683 = 15,625 \left(1 + \frac{r}{200}\right)^{3} $$

Step 1: Isolate the rate term

Divide both sides by the principal amount (₹15,625):

$$ \frac{19,683}{15,625} = \left(1 + \frac{r}{200}\right)^{3} $$

Step 2: Find the cube root

To solve for the term inside the parenthesis, we need to find the cube root of both sides:

$$ \sqrt[3]{\frac{19,683}{15,625}} = 1 + \frac{r}{200} $$

We need to find the cube roots of 19,683 and 15,625.

Let's check potential cubes:

• $ 25^3 = 25 \times 25 \times 25 = 625 \times 25 = 15,625 $
• $ 27^3 = 27 \times 27 \times 27 = 729 \times 27 = 19,683 $

So, the cube root is:

$$ \frac{27}{25} = 1 + \frac{r}{200} $$

Step 3: Solve for 'r'

Now, isolate the rate ($r$):

$$ \frac{r}{200} = \frac{27}{25} - 1 $$

Find a common denominator:

$$ \frac{r}{200} = \frac{27}{25} - \frac{25}{25} $$

$$ \frac{r}{200} = \frac{2}{25} $$

Multiply both sides by 200:

$$ r = \frac{2}{25} \times 200 $$

$$ r = 2 \times \frac{200}{25} $$

$$ r = 2 \times 8 $$

$$ r = 16 $$

Conclusion

The annual rate of interest ($r$) is 16%.