In how many years will a sum of Rs. 800 at 10% per annum compounded semi-annually becomes Rs. 926.10?

Answer

To solve the problem of how many years it will take for a sum of Rs. 800 at 10% per annum compounded semi-annually to become Rs. 926.10, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 800
- Amount (A) = Rs. 926.10
- Annual Interest Rate (R) = 10%
- Compounding frequency = Semi-annually

### Step 2: Calculate the semi-annual interest rate
Since the interest is compounded semi-annually, we need to divide the annual interest rate by 2:
- Semi-annual rate (r) = R / 2 = 10% / 2 = 5%

### Step 3: Write the compound interest formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- A = Amount after time t
- P = Principal
- r = Rate of interest per period
- t = Number of compounding periods

### Step 4: Substitute the known values into the formula
We need to find t, so we substitute the known values into the formula:
\[ 926.10 = 800 \left(1 + \frac{5}{100}\right)^t \]

### Step 5: Simplify the equation
First, simplify the term inside the parentheses:
\[ 926.10 = 800 \left(1 + 0.05\right)^t \]
\[ 926.10 = 800 \left(1.05\right)^t \]

### Step 6: Divide both sides by 800
\[ \frac{926.10}{800} = (1.05)^t \]
\[ 1.163875 = (1.05)^t \]

### Step 7: Take the logarithm of both sides
To solve for t, we can take the logarithm of both sides:
\[ \log(1.163875) = t \cdot \log(1.05) \]

### Step 8: Solve for t
Now, we can isolate t:
\[ t = \frac{\log(1.163875)}{\log(1.05)} \]

### Step 9: Calculate the values
Using a calculator:
- \(\log(1.163875) \approx 0.0645\)
- \(\log(1.05) \approx 0.0212\)

Now, substituting these values:
\[ t \approx \frac{0.0645}{0.0212} \approx 3.04 \text{ half-years} \]

### Step 10: Convert half-years to years
Since t is in half-years, we convert it to years:
\[ t \approx 3.04 \text{ half-years} = \frac{3.04}{2} \approx 1.52 \text{ years} \]

### Final Answer
Thus, it will take approximately **1.52 years** for Rs. 800 to grow to Rs. 926.10 at 10% per annum compounded semi-annually.

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