Akbar lends twice the interest received from' Birbal to Chanakya at the half of the interest at which he lent to Birbal. If Akbar lent ₹ P@ `r%` per annum for 1 year to Birbal -then the interest received by Akbar from Chanakya is:

Options

A. (Pr^2)/100`
B. `((Pr)/10)^2`
C. `P(r/10)^2`
D. `P(r/100)^2`

Answer

To solve the problem step by step, let's break down the information provided:

1. **Understanding the Lending to Birbal**:
- Akbar lends an amount \( P \) at an interest rate of \( r\% \) per annum for 1 year to Birbal.
- The interest received by Akbar from Birbal can be calculated using the formula for simple interest:
\[ \text{Interest from Birbal} = \frac{P \times r \times 1}{100} = \frac{Pr}{100} \]

2. **Lending to Chanakya**:
- According to the problem, Akbar lends twice the interest received from Birbal to Chanakya. Therefore, the amount lent to Chanakya is:
\[ \text{Amount lent to Chanakya} = 2 \times \text{Interest from Birbal} = 2 \times \frac{Pr}{100} = \frac{2Pr}{100} \]

3. **Interest Rate for Chanakya**:
- The interest rate at which Akbar lends to Chanakya is half of the interest rate at which he lent to Birbal. Thus, the interest rate for Chanakya is:
\[ \text{Rate for Chanakya} = \frac{r}{2} \]

4. **Calculating Interest from Chanakya**:
- Now, we can calculate the interest received from Chanakya using the same simple interest formula:
\[ \text{Interest from Chanakya} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} \]
- Substituting the values:
\[ \text{Interest from Chanakya} = \frac{\left(\frac{2Pr}{100}\right) \times \left(\frac{r}{2}\right) \times 1}{100} \]
- Simplifying this:
\[ = \frac{2Pr \times \frac{r}{2}}{100 \times 100} = \frac{Pr^2}{10000} \]

5. **Final Result**:
- Therefore, the interest received by Akbar from Chanakya is:
\[ \text{Interest from Chanakya} = \frac{Pr^2}{10000} \]