In what time will Rs.64000 amount to Rs.68921 at 5% interest per annum, interest being compounded half annually?

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1 \frac{1}{2} years

\frac{2}{3} years

2 year

None

answer is A.

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Detailed Solution

P = Rs.64000
A = Rs.68921
r = 5%
Now, the rate of interest is compounded half annually.
So we will have to decrease the rate of interest and increase the time period i.e. rate of interest will become 52% per half annum or we can say r =

\frac{5}{2}

% per half annum.
Also, our time period will become twice. So, we can say t = 2T, where T is in years.

A = P {(1 + \frac{r}{100})}^{T}

68921 = 64000 {(1 + \frac{\frac{5}{2}}{100})}^{2 T}

\frac{68921}{64000} = {(1 + \frac{5}{200})}^{2 T}

\Rightarrow \frac{68921}{64000} = {(\frac{200 + 5}{200})}^{2 T}

\Rightarrow \frac{68921}{64000} = {(\frac{205}{200})}^{2 T}

\Rightarrow \frac{68921}{64000} = {(\frac{41}{40})}^{2 T}

Now, we know that 68921 can be written as 41×41×41 and 64000 can be written as 40×40×40 so we get:

\Rightarrow {(\frac{41}{40})}^{3} = {(\frac{41}{40})}^{2 T}

⇒3=2T

\Rightarrow T = \frac{3}{2}

Or we can write

\frac{3}{2} as 1 \frac{1}{2}

years.
Hence the required time period is

1 \frac{1}{2}

years.
Hence option 1 is the correct answer.

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