The rates of simple interest in two banks x and y are in the ratio of 10:8. Rajni wants to deposit his total savings in two banks in such a way that she receive equal half-yearly interest from both. She should deposit the savings in banks x and y in the ratio of

Options

A. 4:5
B. 3:5
C. 5:4
D. 2:1

Answer

To solve the problem, we need to determine the ratio in which Rajni should deposit her savings in banks X and Y so that she receives equal half-yearly interest from both banks.

### Step-by-Step Solution:

1. **Understand the Interest Rates**:
The rates of simple interest in banks X and Y are given in the ratio of 10:8.
Let's denote the rate of interest for bank X as \( R_x = 10x \) and for bank Y as \( R_y = 8y \).

2. **Set Up the Interest Formula**:
The formula for simple interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time period.

3. **Define the Principal Amounts**:
Let \( P_x \) be the amount deposited in bank X and \( P_y \) be the amount deposited in bank Y.

4. **Calculate the Simple Interest for Both Banks**:
- For bank X:
\[ SI_x = \frac{P_x \times R_x \times T}{100} = \frac{P_x \times 10x \times \frac{1}{2}}{100} = \frac{P_x \times 10x}{200} \]
- For bank Y:
\[ SI_y = \frac{P_y \times R_y \times T}{100} = \frac{P_y \times 8y \times \frac{1}{2}}{100} = \frac{P_y \times 8y}{200} \]

5. **Set the Interests Equal**:
Since Rajni wants to receive equal interest from both banks:
\[ SI_x = SI_y \]
Therefore,
\[ \frac{P_x \times 10x}{200} = \frac{P_y \times 8y}{200} \]

6. **Simplify the Equation**:
We can cancel out the common factor of 200:
\[ P_x \times 10x = P_y \times 8y \]

7. **Express the Ratio of Principal Amounts**:
Rearranging gives:
\[ \frac{P_x}{P_y} = \frac{8y}{10x} = \frac{4y}{5x} \]

8. **Final Ratio**:
Thus, the ratio in which Rajni should deposit her savings in banks X and Y is:
\[ P_x : P_y = 4y : 5x \]

### Conclusion:
Rajni should deposit her savings in banks X and Y in the ratio of \( 4y : 5x \).