Determine the rate per cent per annum if Rs 25,000 amounts to 26,010 in 6 months, interest being com- pounded quarterly.

Answer

To determine the rate per cent per annum when Rs 25,000 amounts to Rs 26,010 in 6 months with interest compounded quarterly, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Principal (P) = Rs 25,000
- Amount (A) = Rs 26,010
- Time (T) = 6 months = 0.5 years

2. **Convert Time for Quarterly Compounding:**
- Since the interest is compounded quarterly, we need to convert the time into quarters.
- Number of quarters in 6 months = 6 months / 3 months per quarter = 2 quarters.
- Therefore, T = 2 quarters.

3. **Set Up the Compound Interest Formula:**
- The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100n}\right)^{nt} \]
- Here, \( n \) is the number of times interest is compounded per year (quarterly means \( n = 4 \)).
- We can rewrite the formula for our case:
\[ 26,010 = 25,000 \left(1 + \frac{R}{400}\right)^{2} \]

4. **Divide Both Sides by the Principal:**
\[ \frac{26,010}{25,000} = \left(1 + \frac{R}{400}\right)^{2} \]
- Calculate the left side:
\[ 1.0404 = \left(1 + \frac{R}{400}\right)^{2} \]

5. **Take the Square Root of Both Sides:**
\[ \sqrt{1.0404} = 1 + \frac{R}{400} \]
- Calculate the square root:
\[ 1.02 = 1 + \frac{R}{400} \]

6. **Isolate R:**
\[ 1.02 - 1 = \frac{R}{400} \]
\[ 0.02 = \frac{R}{400} \]

7. **Multiply Both Sides by 400:**
\[ R = 0.02 \times 400 \]
\[ R = 8 \]

8. **Conclusion:**
- The rate of interest per annum is **8%**.