Sanju puts equal amount of money one at 10% per annum compound interest payable half yearly and the second at a certain rate per cent per annum compound interest payable yearly. If he gets equal amounts after 3 years, what is the value of the second rate per cent?

Options

A. `10(1)/(4)%`
B. `10%`
C. `9(1)/(2)%`
D. `8(1)/(4)%`

Answer

To solve the problem, we need to find the second rate of interest at which Sanju invests his money, given that he receives equal amounts from two different investments after 3 years.

### Step-by-Step Solution:

1. **Define the Principal Amount**:
Let the principal amount invested by Sanju be \( P \).

2. **Calculate the Amount from the First Investment**:
Sanju invests part of his money at 10% per annum compound interest, compounded half-yearly.
- The effective rate for half-yearly compounding is \( \frac{10\%}{2} = 5\% \).
- The number of compounding periods in 3 years (half-yearly) is \( 3 \times 2 = 6 \).
- The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
- For the first investment:
\[ A_1 = P \left(1 + \frac{5}{100}\right)^6 \]
\[ A_1 = P \left(1.05\right)^6 \]

3. **Calculate the Amount from the Second Investment**:
Let the second rate of interest be \( R \) percent per annum compounded yearly.
- The amount after 3 years is given by:
\[ A_2 = P \left(1 + \frac{R}{100}\right)^3 \]

4. **Set the Two Amounts Equal**:
Since the amounts from both investments are equal after 3 years:
\[ P \left(1.05\right)^6 = P \left(1 + \frac{R}{100}\right)^3 \]
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \left(1.05\right)^6 = \left(1 + \frac{R}{100}\right)^3 \]

5. **Calculate \( (1.05)^6 \)**:
- Using a calculator:
\[ (1.05)^6 \approx 1.340095 \]

6. **Set Up the Equation**:
Now we have:
\[ 1.340095 = \left(1 + \frac{R}{100}\right)^3 \]

7. **Take the Cube Root**:
To solve for \( R \), take the cube root of both sides:
\[ 1 + \frac{R}{100} = (1.340095)^{\frac{1}{3}} \]
- Using a calculator:
\[ (1.340095)^{\frac{1}{3}} \approx 1.100 \]

8. **Solve for \( R \)**:
\[ 1 + \frac{R}{100} \approx 1.100 \]
\[ \frac{R}{100} \approx 0.100 \]
\[ R \approx 10 \]

### Conclusion:
The value of the second rate per cent is approximately **10%**.