A borrows Rs. 800 at the rate of 12% per annum simple interest and B borrows Rs. 910 at the rate of 10% per annum, simple interest. In how many years will their amounts of debt be equal ?

Options

A. 18
B. 20
C. 22
D. 24

Answer

To solve the problem, we need to find out in how many years the debts of A and B will be equal. We will use the formula for calculating the amount of debt using simple interest.

### Step-by-Step Solution:

1. **Identify the Principal and Interest Rates:**
- A borrows Rs. 800 at a rate of 12% per annum.
- B borrows Rs. 910 at a rate of 10% per annum.

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

3. **Calculate the Amount for A after T years:**
For A:
\[ A_A = 800 + \left( \frac{800 \times 12 \times T}{100} \right) \]
Simplifying this:
\[ A_A = 800 + \left( \frac{9600T}{100} \right) = 800 + 96T \]
So,
\[ A_A = 800 + 96T \]

4. **Calculate the Amount for B after T years:**
For B:
\[ A_B = 910 + \left( \frac{910 \times 10 \times T}{100} \right) \]
Simplifying this:
\[ A_B = 910 + \left( \frac{9100T}{100} \right) = 910 + 91T \]
So,
\[ A_B = 910 + 91T \]

5. **Set the Amounts Equal to Each Other:**
To find when their debts will be equal, we set \( A_A \) equal to \( A_B \):
\[ 800 + 96T = 910 + 91T \]

6. **Solve for T:**
Rearranging the equation:
\[ 96T - 91T = 910 - 800 \]
This simplifies to:
\[ 5T = 110 \]
Dividing both sides by 5:
\[ T = \frac{110}{5} = 22 \]

7. **Conclusion:**
The amount of debt will be equal after **22 years**.

### Final Answer:
The answer is **22 years**.