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 years
B. 20 years
C. 22 years
D. 24 years

Answer

To solve the problem, we need to find the time (in years) when the total amount of debt for both A and B will be equal.

### Step-by-Step Solution:

1. **Identify the given values:**
- A borrows Rs 800 at an interest rate of 12% per annum.
- B borrows Rs 910 at an interest rate of 10% per annum.

2. **Write the formula for the total amount (A) for A and B:**
- The formula for the total amount \( A \) after \( t \) years with simple interest is:
\[ A = P + \text{SI} = P + \left( \frac{P \times r \times t}{100} \right) \]
- For A:
\[ A_A = 800 + \left( \frac{800 \times 12 \times t}{100} \right) \]
- For B:
\[ A_B = 910 + \left( \frac{910 \times 10 \times t}{100} \right) \]

3. **Set the amounts equal to each other:**
\[ 800 + \left( \frac{800 \times 12 \times t}{100} \right) = 910 + \left( \frac{910 \times 10 \times t}{100} \right) \]

4. **Simplify the equation:**
- Calculate the simple interest for both A and B:
\[ A_A = 800 + \left( \frac{9600t}{100} \right) = 800 + 96t \]
\[ A_B = 910 + \left( \frac{9100t}{100} \right) = 910 + 91t \]
- Set the two equations equal:
\[ 800 + 96t = 910 + 91t \]

5. **Rearrange the equation:**
- Move all terms involving \( t \) to one side and constant terms to the other side:
\[ 96t - 91t = 910 - 800 \]
\[ 5t = 110 \]

6. **Solve for \( t \):**
\[ t = \frac{110}{5} = 22 \]

### Final Answer:
The time in which their amounts of debt will be equal is **22 years**.

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