Certainly! Let's solve this step-by-step.
1. Identify the given values:
- Principal (P): The initial amount of the loan is ₹8400.
- Final Amount (A): The amount to be paid to clear the debt is ₹10920.
- Rate of Interest (R): The simple interest rate is [tex]\(7 \frac{1}{2} \%\)[/tex] per annum, which can be written as 7.5%.
2. Convert the percentage to decimal form:
- [tex]\( R = 7.5 \% = \frac{7.5}{100} = 0.075 \)[/tex].
3. Recall the formula for the final amount in terms of simple interest:
[tex]\[
A = P + SI
\][/tex]
where [tex]\( SI \)[/tex] is the simple interest. The formula for simple interest is:
[tex]\[
SI = P \times R \times T
\][/tex]
Plugging in the simple interest formula into the final amount formula:
[tex]\[
A = P + (P \times R \times T)
\][/tex]
[tex]\[
A = P(1 + R \times T)
\][/tex]
4. Rearrange the formula to solve for time (T):
[tex]\[
A = P(1 + R \times T)
\][/tex]
[tex]\[
\frac{A}{P} = 1 + R \times T
\][/tex]
Simplify to isolate [tex]\(T\)[/tex]:
[tex]\[
\frac{A}{P} - 1 = R \times T
\][/tex]
[tex]\[
T = \frac{\frac{A}{P} - 1}{R}
\][/tex]
5. Substitute the values [tex]\(A = 10920\)[/tex], [tex]\(P = 8400\)[/tex], and [tex]\(R = 0.075\)[/tex] into the formula:
[tex]\[
T = \frac{\frac{10920}{8400} - 1}{0.075}
\][/tex]
6. Calculate the intermediate steps:
[tex]\[
\frac{10920}{8400} = 1.3
\][/tex]
[tex]\[
T = \frac{1.3 - 1}{0.075}
\][/tex]
[tex]\[
T = \frac{0.3}{0.075}
\][/tex]
7. Finally, compute the value of [tex]\(T\)[/tex]:
[tex]\[
T = \frac{0.3}{0.075} = 4
\][/tex]
So, the farmer will have to pay ₹10920 to clear the debt after 4 years.
