To solve this problem, we need to find the time it takes for Ashok to repay a compound interest of Rs 4,488 on a loan of Rs 55,000 with a compound interest rate of 4 paisa per rupee per year.
### Step-by-Step Solution:
1. Convert the Interest Rate:
The interest rate given is 4 paisa per rupee per year. We convert this rate into rupees. Since there are 100 paisa in 1 rupee, the annual interest rate is:
[tex]\[
\text{Rate per year} = \frac{4}{100} = 0.04
\][/tex]
2. Set Up the Compound Interest Formula:
The formula for compound interest is:
[tex]\[
A = P(1 + r/n)^{nt}
\][/tex]
Where:
- [tex]\( A \)[/tex] is the total amount after interest
- [tex]\( P \)[/tex] is the principal amount (initial loan)
- [tex]\( r \)[/tex] is the annual interest rate (which we've determined to be 0.04)
- [tex]\( n \)[/tex] is the number of times interest applied per time period (since it's yearly, [tex]\( n = 1 \)[/tex])
- [tex]\( t \)[/tex] is the time (in years)
Given [tex]\( n = 1 \)[/tex], the formula simplifies to:
[tex]\[
A = P(1 + r)^t
\][/tex]
3. Calculate the Total Amount [tex]\( A \)[/tex]:
The total amount after interest is the principal amount plus the compound interest:
[tex]\[
A = P + \text{Compound Interest} = 55000 + 4488 = 59488
\][/tex]
4. Rearrange the Formula to Solve for Time [tex]\( t \)[/tex]:
[tex]\[
\frac{A}{P} = (1 + r)^t
\][/tex]
Taking the natural logarithm (log) of both sides:
[tex]\[
\log\left(\frac{A}{P}\right) = t \cdot \log(1 + r)
\][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{\log\left(\frac{A}{P}\right)}{\log(1 + r)}
\][/tex]
5. Substitute the Known Values to Find [tex]\( t \)[/tex]:
[tex]\[
\frac{A}{P} = \frac{59488}{55000}
\][/tex]
Hence,
[tex]\[
t = \frac{\log\left(\frac{59488}{55000}\right)}{\log(1 + 0.04)}
\][/tex]
Using logarithm properties:
[tex]\[
t = \frac{\log(1.0816)}{\log(1.04)}
\][/tex]
From this calculation, we find that:
[tex]\[
t \approx 2 \text{ years}
\][/tex]
Therefore, Ashok pays off the compound interest of Rs 4,488 in approximately 2 years.
