Question

In what time will Rs 64000 amount to Rs 68921 at 5% p.a. interest being compound semi – annually ?

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

Rs 64000 amount to Rs 68921 at 5% p.a. interest being compound semi – annually.

Let assume that the time period be n years.

Now, we have

Principal, P = Rs 64000

Rate of interest, r = 5 % per annum compounded semi - annually

Time period = n years

Amount = Rs 68921

We know that,

Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded semi annually for n years is given by

$\boxed{\sf{  \:\rm \: Amount \: = \: P {\bigg[1 + \dfrac{r}{200} \bigg]}^{2n} \: \: }}$

So, on substituting the values, we get

$\rm \: 68921 \: = \: 64000 {\bigg[1 + \dfrac{5}{200} \bigg]}^{2n} \: \:$

$\rm \: 68921 \: = \: 64000 {\bigg[1 + \dfrac{1}{40} \bigg]}^{2n} \: \:$

$\rm \: 68921 \: = \: 64000 {\bigg[ \dfrac{40 + 1}{40} \bigg]}^{2n} \: \:$

$\rm \: 68921 \: = \: 64000 {\bigg[ \dfrac{41}{40} \bigg]}^{2n} \: \:$

$\rm \: \frac{68921}{64000} \: = \: {\bigg[ \dfrac{41}{40} \bigg]}^{2n} \: \:$

$\rm \: {\bigg[\dfrac{41}{40} \bigg]}^{3} \: = \: {\bigg[ \dfrac{41}{40} \bigg]}^{2n} \: \:$

So, on comparing we get

$\rm \: 2n = 3$

$\rm\implies \:n \: = \: \frac{3}{2} \: years$

$\rm\implies \:n \: = \: 1 \: year \: and \: 6 \: months$

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Additional Information :-

1. Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded annually for n years is given by

$\boxed{\sf{  \:\rm \: Amount \: = \: P {\bigg[1 + \dfrac{r}{100} \bigg]}^{n} \: \: }}$

2. Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded quarterly for n years is given by

$\boxed{\sf{  \:\rm \: Amount \: = \: P {\bigg[1 + \dfrac{r}{400} \bigg]}^{4n} \: \: }}$

3. Amount received on a certain sum of money of Rs P invested at the rate of r % per annum compounded monthly for n years is given by

$\boxed{\sf{  \:\rm \: Amount \: = \: P {\bigg[1 + \dfrac{r}{1200} \bigg]}^{12n} \: \: }}$

Answer 2

Step-by-step explanation:

Here, Principal P= Rs 64000,

Amount A= Rs 68921 ,

rate R=5 % per annum.

Since the interest is compounded half-yearly.

$\tt\therefore \quad A=P\left(1+\dfrac{R}{200}\right)^{2 n}$

, where n is the number of years.

$\[ \begin{array}{l} \tt \Rightarrow 68921=64000\left(1+\dfrac{5}{200}\right)^{2 n} \\ \\ \tt\Rightarrow \dfrac{68921}{64000}=\left(\dfrac{41}{40}\right)^{2 n} \\ \\ \tt \Rightarrow\left(\dfrac{41}{40}\right)^{3}=\left(\dfrac{41}{40}\right)^{2 n} \\ \\ \tt\Rightarrow 2 n=3 \\ \\ \tt\Rightarrow n=\dfrac{3}{2} \text { years }=1 \dfrac{1}{2} \text { years } \end{array} \]$