Calculating Compound Interest Rate Per Annum

The question asks us to find the annual rate of compound interest at which a given principal amount grows to a specific amount over a certain period, with interest compounded half-yearly.

Let's identify the key information provided:

• Principal Amount (P) = Rs. 20000
• Amount (A) = Rs. 23152.50
• Time Period (t) = 1 year 6 months = 1.5 years
• Interest is compounded half-yearly.

When interest is compounded half-yearly, the number of compounding periods per year (n) is 2. The total number of compounding periods over the given time will be $n \times t$. The rate of interest applied per compounding period will be the annual rate divided by n, i.e., $\frac{r}{n}$, where r is the annual interest rate.

The formula for compound interest is:

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

Now, we substitute the given values into the formula:

$$ 23152.50 = 20000 \left(1 + \frac{r}{2}\right)^{2 \times 1.5} $$

$$ 23152.50 = 20000 \left(1 + \frac{r}{2}\right)^{3} $$

To find the rate (r), we need to isolate the term containing r. First, divide both sides of the equation by the principal amount, 20000:

$$ \frac{23152.50}{20000} = \left(1 + \frac{r}{2}\right)^{3} $$

Calculate the value on the left side:

$$ 1.157625 = \left(1 + \frac{r}{2}\right)^{3} $$

Now, take the cube root of both sides to eliminate the exponent of 3:

$$ \sqrt[3]{1.157625} = 1 + \frac{r}{2} $$

The cube root of 1.157625 is 1.05:

$$ 1.05 = 1 + \frac{r}{2} $$

Next, subtract 1 from both sides:

$$ 1.05 - 1 = \frac{r}{2} $$

$$ 0.05 = \frac{r}{2} $$

Finally, multiply both sides by 2 to find the value of r:

$$ r = 0.05 \times 2 $$

$$ r = 0.10 $$

The value of r is 0.10. Since r is the annual rate expressed as a decimal, we convert it to a percentage by multiplying by 100:

$$ \text{Annual Rate} = 0.10 \times 100\% = 10\% $$

So, the compound interest rate per annum is 10%.

Let's summarize the calculation steps:

The calculated annual compound interest rate is 10%.

Revision Table: Compound Interest Concepts

Additional Information: Compound Interest Compounding Periods

The compounding frequency significantly impacts the total amount earned or paid. More frequent compounding leads to higher interest over the same period and rate because interest starts earning interest sooner.

Here are common compounding frequencies and their 'n' values:

• Annually: n = 1
• Half-yearly (Semi-annually): n = 2
• Quarterly: n = 4
• Monthly: n = 12
• Daily: n = 365 (usually)

In this problem, 'half-yearly' compounding was key, meaning we used n=2 and adjusted both the rate per period ($\frac{r}{2}$) and the total number of periods ($nt = 1.5 \times 2 = 3$). Understanding the compounding period is crucial for solving compound interest problems correctly.