Question

Q) Nita has deposited Rs.25,000 in a deposit scheme which gives 8% p.a.(per annum) compounded half-yearly.How much amount will she get back at the end of 1 year 6 months to the nearest rupee? (with step-by-step explanation)​

Answer 1

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

Given that,

Nita has deposited Rs.25,000 in a deposit scheme which gives 8% per annum compounded half-yearly.

So, we have

Principal, P = Rs 25000

Rate of interest, r = 8 % per annum compounded half yearly.

Time, n = 1 year 6 months = 3/2 years

We know,

Amount on a certain sum of money of Rs P invested at the rate of r % per annum compounded half yearly 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

$\:\rm \: Amount \: = \: 25000 {\bigg[1 + \dfrac{8}{200} \bigg]}^{3} \: \:$

$\rm \: \: = \: 25000 {\bigg[1 + \dfrac{1}{25} \bigg]}^{3} \: \:$

$\rm \: \: = \: 25000 {\bigg[\dfrac{25 + 1}{25} \bigg]}^{3} \: \:$

$\rm \: \: = \: 25000 {\bigg[\dfrac{26}{25} \bigg]}^{3} \: \:$

$\rm \: =  \: 28121.60$

$\rm \: =  \: 28122 \: \: \: \: \: \: \{to \: nearest \: rupee \}$

Hence,

Amount received after one year and six months = Rs 28122.

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Additional Information :-

1. Amount 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 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 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, the conversion period is $\dfrac{1}{2}$ year.

Thus, in 1 year 6 months, there will be 3 conversion periods. Original principal =₹ 25,000

Interest for the first half-year

$\red{ \tt=₹ \dfrac{25000 \times \cancel 8^{ {}^{ {}^{4} } } \times \dfrac{1} {\cancel{2}}}{100} =₹250 \times 4=₹ 1,000 }$

∴ Amount at the end of first half-year,

i.e., the principal for the second-year

=₹ 25,000+₹ 1,000=₹ 26,000

Interest for the second half-year

$\pink{ \tt=₹ \: \: \dfrac{26000 \times \cancel 8 ^{ {}^{ {}^{4} } } \times \dfrac{1} {\cancel{2}}}{100} = ₹ \: 260 \times 4=₹ \: \: 1,040 .}$

∴ Amount at the end of second half-year, i.e., The principal for the third half-year

=₹ 26,000+₹ 1,040=₹ 27,040

Interest for the third half-year

$\color{maroon} \tt=₹ \: \: \: \dfrac{27040 \times \cancel 8 {}^{ {}^{ {}^ {\cancel{4}} } } \times \dfrac{1} {\cancel{2}}} {\cancel{100_{_{_{25}}}}} = ₹ \: \: \dfrac{27040}{25} =₹ \: \: 1,081.60$

∴ Amount at the end of third half-year =₹

$\color{darkcyan} \begin{array}{l} \tt 27,040+ ₹1,081.60 \\ \tt =₹ \: \: 28,121.60 \\ \tt=₹ \: \: 28,122 \: \: \: \: \: \: \: \text{(to the nearest rupee)} \end{array}$

Thus, Nita will get back ₹28,122 at the end of 1 year 6 months.