In how many years will a sum of ₹16,000 at 10% per annum compounded semi-annually become ₹18,522?
1.5 years
To determine in how many years a sum of ₹16,000 will become ₹18,522 at 10% per annum compounded semi-annually, we use the compound interest formula:
\(A = P \left(1 + \frac{r}{n}\right)^{nt}\)
where:
- \(A\) is the amount after time \(t\).
- \(P\) is the principal amount (₹16,000).
- \(r\) is the annual interest rate (10% or 0.10).
- \(n\) is the number of times the interest is compounded per year (2 times for semi-annual).
- \(t\) is the time in years.
Given \(A = 18522, P = 16000, r = 0.10,\) and \(n = 2\), we need to find \(t\).
Plug in the values into the formula:
\(18522 = 16000 \left(1 + \frac{0.10}{2}\right)^{2t}\)
Calculate \(\left(1 + \frac{0.10}{2}\right)\):
\(\left(1 + \frac{0.10}{2}\right) = 1.05\)
Rewrite the equation:
\(18522 = 16000 \times (1.05)^{2t}\)
Divide both sides by 16000:
\(\frac{18522}{16000} = (1.05)^{2t}\)
Simplify the fraction:
\(1.157625 = (1.05)^{2t}\)
To solve for \(t\), take the logarithm of both sides:
\(\log(1.157625) = 2t \cdot \log(1.05)\)
Calculate the logarithms (using a calculator):
- \(\log(1.157625) \approx 0.063198\)
- \(\log(1.05) \approx 0.021189\)
Substitute these values into the equation:
\(0.063198 = 2t \cdot 0.021189\)
Solve for \(t\):
\(2t = \frac{0.063198}{0.021189}\)
\(2t \approx 2.982\)
\(t \approx \frac{2.982}{2} = 1.491\)
Round to the nearest half year, we find \(t = 1.5\) years.
Thus, the correct answer is 1.5 years.
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