Question : At what time will Rs. 64000 amount to Rs. 68921 at 5% per annum, with interest being compounded half-yearly?
Option 1: $3$ years
Option 2: $2\frac{1}{2}$ years
Option 3: $2$ years
Option 4: $1\frac{1}{2}$ years
Correct Answer: $1\frac{1}{2}$ years
Solution :
Sum, $P$ = Rs. 64000
Rate, $R$ = 5%
Total amount = Rs. 68921
When compounded half-yearly,
Total amount = $P(1+\frac{\frac{R}{2}}{100})^{2n} $ where $P$ is principal, $R$ is the rate of interest per annum compounded annually and $n$ is time in years.
⇒ 68921 = $64000(1+\frac{5}{200})^{2n}$
⇒ $\frac{68921}{64000}$ = $(1.025)^{2n}$
⇒ $(1.025)^{3}$ = $(1.025)^{2n}$
⇒ $ 2n = 3$
⇒ $n$ = $\frac{3}{2}$ = $1\frac{1}{2}$ years
Hence, the correct answer is $1\frac{1}{2}$ years.
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