If A borrowed Rs. P at x% and B borrowed Rs Q `(gt P)` at y% per annum at simple interset at the same time, then the amount of their debts will be equal after

Options

A. `100 ((Q - P)/(Px - Qy))` years
B. `100 ((Px - Qy)/(Q-P)` years
C. `100 ((Px - Qy)/(P-Q))` years
D. `100 ((P-Q)/(Px - Qy))` years

Answer

To solve the problem, we need to find the time (T) after which the amounts owed by A and B will be equal. We will use the formula for calculating the total amount due under simple interest.

### Step-by-Step Solution:

1. **Understand the Problem**:
- A borrows Rs. P at an interest rate of x% per annum.
- B borrows Rs. Q (where Q > P) at an interest rate of y% per annum.
- We need to find the time (T) when both their debts will be equal.

2. **Formula for Simple Interest**:
- The formula for the amount (A) after time T at simple interest is:
\[ A = P + \text{SI} = P + \left(\frac{P \times r \times T}{100}\right) \]
- Where P is the principal, r is the rate of interest, and T is the time in years.

3. **Calculate Amount for A**:
- For A, the amount after time T will be:
\[ A_A = P + \left(\frac{P \times x \times T}{100}\right) \]
- Simplifying this:
\[ A_A = P + \frac{PxT}{100} \]

4. **Calculate Amount for B**:
- For B, the amount after time T will be:
\[ A_B = Q + \left(\frac{Q \times y \times T}{100}\right) \]
- Simplifying this:
\[ A_B = Q + \frac{QyT}{100} \]

5. **Set the Amounts Equal**:
- We need to set the amounts equal to each other:
\[ A_A = A_B \]
- Substituting the expressions for A_A and A_B:
\[ P + \frac{PxT}{100} = Q + \frac{QyT}{100} \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ \frac{PxT}{100} - \frac{QyT}{100} = Q - P \]
- Factoring out T/100:
\[ \frac{T}{100}(Px - Qy) = Q - P \]

7. **Solving for T**:
- Now, isolate T:
\[ T = \frac{(Q - P) \times 100}{(Px - Qy)} \]

### Final Answer:
The time (T) after which the amounts of A and B will be equal is:
\[ T = \frac{(Q - P) \times 100}{(Px - Qy)} \]