To determine the time it takes for ₹64,000 to grow to ₹88,360 at an annual interest rate of 17.5%, compounded yearly, we will use the compound interest formula:
[tex]\[ A = P(1 + r/n)^{nt} \][/tex]
Since the interest is compounded yearly, [tex]\( n = 1 \)[/tex]. Therefore, the formula simplifies to:
[tex]\[ A = P(1 + r)^t \][/tex]
Given:
- Principal amount [tex]\( P = ₹64,000 \)[/tex]
- Final amount [tex]\( A = ₹88,360 \)[/tex]
- Annual interest rate [tex]\( r = 17.5\% = \frac{17.5}{100} = 0.175 \)[/tex]
We need to solve for the time [tex]\( t \)[/tex]:
[tex]\[ 88,360 = 64,000(1 + 0.175)^t \][/tex]
To isolate [tex]\( t \)[/tex], we follow these steps:
1. Divide both sides by the principal [tex]\( P \)[/tex]:
[tex]\[ \frac{88,360}{64,000} = (1.175)^t \][/tex]
2. Simplify the left-hand side:
[tex]\[ 1.38 = (1.175)^t \][/tex]
3. Take the natural logarithm (log) of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ \log(1.38) = \log((1.175)^t) \][/tex]
4. Utilize the logarithm power rule, [tex]\(\log(a^b) = b \log(a)\)[/tex]:
[tex]\[ \log(1.38) = t \log(1.175) \][/tex]
5. Solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\log(1.38)}{\log(1.175)} \][/tex]
After evaluating this expression using logarithms:
[tex]\[ t \approx 2 \][/tex]
Therefore, it will take approximately 2 years for ₹64,000 to amount to ₹88,360 at an annual interest rate of 17.5%, compounded yearly.
