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

Options

A. `1"" (1)/(2)`
B. `1"" (2)/(3)`
C. `2 "" (1)/(3)`
D. `2 "" (1)/(2)`

Answer

To solve the problem of how many years it will take for a sum of Rs. 800 at 10% per annum compound interest, 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
- Rate of interest (R) = 10% per annum
- Compounding frequency = Semi-annually

### Step 2: Adjust the rate and time for semi-annual compounding
Since the interest is compounded semi-annually, we need to adjust the rate and time:
- Semi-annual interest rate = R/2 = 10%/2 = 5% per 6 months.
- If T is the number of years, then the number of compounding periods (n) will be 2T (since there are 2 compounding periods in a year).

### Step 3: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^{nt} \]
Where:
- A = Amount after time t
- P = Principal
- r = Rate of interest per period
- n = Number of times interest is compounded per year
- t = Time in years

Substituting the values we have:
\[ 926.10 = 800 \left(1 + \frac{5}{100}\right)^{2T} \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 926.10 = 800 \left(1 + 0.05\right)^{2T} \]
\[ 926.10 = 800 \left(1.05\right)^{2T} \]

### Step 5: Divide both sides by 800
\[ \frac{926.10}{800} = (1.05)^{2T} \]
\[ 1.163875 = (1.05)^{2T} \]

### Step 6: Take logarithm on both sides
To solve for \(2T\), we can take the logarithm of both sides:
\[ \log(1.163875) = 2T \cdot \log(1.05) \]

### Step 7: Solve for \(2T\)
Now, calculate the logarithms:
- \( \log(1.163875) \approx 0.064 \)
- \( \log(1.05) \approx 0.0212 \)

So,
\[ 0.064 = 2T \cdot 0.0212 \]
\[ 2T = \frac{0.064}{0.0212} \approx 3.018 \]

### Step 8: Solve for \(T\)
Now, divide by 2 to find \(T\):
\[ T \approx \frac{3.018}{2} \approx 1.509 \text{ years} \]

### Step 9: Final answer
Thus, the time it will take for the sum to grow to Rs. 926.10 is approximately 1.5 years or \(1 \frac{1}{2}\) years.