Find the compound interest on Rs. 25,000 for two years at 10% per annum if the interest is compounded (a) yearly (b) half-yearly (c) quarterly

Let's calculate the compound interest for the given principal amount of Rs. 25,000 over 2 years at an interest rate of 10% per annum, compounded at different intervals.

The compound interest formula is:

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

Where:
- \(A\) = final amount (principal + interest)
- \(P\) = principal amount = Rs. 25,000
- \(r\) = annual interest rate = 10% or 0.10
- \(n\) = number of times interest is compounded per year
- \(t\) = time period in years = 2 years

The compound interest is given by:

\[
\text{Compound Interest} = A - P
\]

### (a) Compounded Yearly
- \(n = 1\) (compounded once per year)

\[
A = 25,000 \left(1 + \frac{0.10}{1}\right)^{1 \times 2} = 25,000 \times (1.10)^2
\]

Now, let's calculate:

\[
A = 25,000 \times 1.21 = 30,250
\]

Compound Interest = \(30,250 - 25,000 = Rs. 5,250\)

### (b) Compounded Half-Yearly
- \(n = 2\) (compounded twice per year)

\[
A = 25,000 \left(1 + \frac{0.10}{2}\right)^{2 \times 2} = 25,000 \times \left(1 + 0.05\right)^4 = 25,000 \times (1.05)^4
\]

Now, let's calculate:

\[
A = 25,000 \times 1.21550625 = 30,387.66
\]

Compound Interest = \(30,387.66 - 25,000 = Rs. 5,387.66\)

### (c) Compounded Quarterly
- \(n = 4\) (compounded four times per year)

\[
A = 25,000 \left(1 + \frac{0.10}{4}\right)^{4 \times 2} = 25,000 \times \left(1 + 0.025\right)^8 = 25,000 \times (1.025)^8
\]

Now, let's calculate:

\[
A = 25,000 \times 1.21840281 = 30,460.07
\]

Compound Interest = \(30,460.07 - 25,000 = Rs. 5,460.07\)

### Summary of Results:
- **Yearly Compound Interest**: Rs. 5,250
- **Half-Yearly Compound Interest**: Rs. 5,387.66
- **Quarterly Compound Interest**: Rs. 5,460.07

Would you like more details on any of these steps?

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Here are 5 related questions:
1. How does the frequency of compounding affect the compound interest?
2. What would the compound interest be for 3 years instead of 2 years?
3. How would the formula change if the interest rate was 15% instead of 10%?
4. Can you calculate compound interest for monthly compounding?
5. How is compound interest different from simple interest?

**Tip:** Compound interest grows faster with more frequent compounding periods (e.g., quarterly or monthly) compared to yearly compounding.

Calculate the compound interest on Rs. 25,000 over 2 years at 10% per annum for different compounding intervals: yearly, half-yearly, and quarterly. This detailed solution uses the compound interest formula and demonstrates how the compounding frequency affects the interest. Suitable for Grades 9-12.

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