A person borrows 400 Taka at 10% simple interest and after 1 year borrows 500 Taka at 12% simple interest. After how many years of taking the first loan will the interest on both loans be equal?  

To find out after how many years the interest on both loans will be equal, let's calculate the interest for each loan.

### 1. Loan 1: 
- Principal \( P_1 = 400 \, \text{Taka} \)
- Interest rate \( r_1 = 10\% \)
- Time \( t = t \, \text{years} \) (since we need to find the time)

The formula for simple interest is:
\[
\text{Interest} = P \times r \times t
\]
For the first loan:
\[
I_1 = 400 \times \frac{10}{100} \times t = 40t
\]

### 2. Loan 2:
- Principal \( P_2 = 500 \, \text{Taka} \)
- Interest rate \( r_2 = 12\% \)
- Time \( t_2 = t - 1 \, \text{years} \) (because this loan is taken 1 year after the first loan)

For the second loan:
\[
I_2 = 500 \times \frac{12}{100} \times (t - 1) = 60(t - 1)
\]

### 3. Setting the interests equal:
We need to find when the interest from both loans will be the same, so set \( I_1 = I_2 \):
\[
40t = 60(t - 1)
\]

Now solve for \( t \):

\[
40t = 60t - 60
\]
\[
60 = 20t
\]
\[
t = 3 \, \text{years}
\]

### Conclusion:
The interest on both loans will be equal after 3 years of taking the first loan.

Would you like more details on this, or do you have any further questions?

Here are 5 related questions to deepen your understanding:
1. What is the difference between simple and compound interest?
2. How does changing the interest rate affect the time when the interests are equal?
3. What if the second loan was taken 2 years after the first? How would that change the result?
4. How can you calculate the total interest paid after a certain time for both loans combined?
5. How would the result change if the interest rates were swapped between the loans?

**Tip:** When dealing with simple interest, always ensure the time for each loan is considered correctly, especially if loans are taken at different times.

A person borrows 400 Taka at 10% simple interest and after 1 year borrows 500 Taka at 12% simple interest. After how many years of taking the first loan will the interest on both loans be equal?

Find out how many years it takes for the interest on two simple interest loans to be equal. This problem involves loans of 400 Taka at 10% and 500 Taka at 12%, with the second loan taken 1 year later. Detailed step-by-step solution provided. Suitable for students in Grades 8-10.

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