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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Important Questions from Compound Interest
At what rate percent per annum will Rs. 7200 amount to Rs. 7938 in one year, if interest is compounded half yearly?
What is the compound interest (in Rs.) on a sum of Rs. 8192 for \(1 \frac{1}{4}\) years at 15% per annum, if interest is compounded 5-monthly ?
What is the difference (in Rs.) between the interests on Rs. 50,000 for one year at 8% per annum compounded half yearly and yearly?
A sum of money becomes Rs. 11,880 after 4 years and Rs. 17,820 after 6 years on compound interest, if the interest is compounded annually. What is the half of the sum (in Rs.)?
A sum invested at compound interest amounts to Rs. 7,800 in 3 years and Rs. 11,232 in 5 years. What is the rate per cent?
