Question

at what time will rupees 1600 amount to rupees 1681 at 5% per annum if the interest is being compounded half yearly​

Answer 1

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

Given that,

• A sum of Rs 1600 amounts to Rs 1681 at the rate of 5 % per annum compounded half - yearly.

So, we have

Sum of money invested, p = Rs 1600

Amount = Rs 1681

Rate of interest, r = 5 % per annum compounded half yearly.

Let suppose that time period be 'n' 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

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: Amount = p{\bigg[1 + \dfrac{r}{200} \bigg]}^{2n} \: }}}$

So, on substituting the values, we get

$\rm :\longmapsto\:1681 = 1600 {\bigg[1 + \dfrac{5}{200} \bigg]}^{2n}$

$\rm :\longmapsto\:1681 = 1600 {\bigg[1 + \dfrac{1}{40} \bigg]}^{2n}$

$\rm :\longmapsto\:1681 = 1600 {\bigg[\dfrac{40 + 1}{40} \bigg]}^{2n}$

$\rm :\longmapsto\: \dfrac{1681}{1600} ={\bigg[\dfrac{40 + 1}{40} \bigg]}^{2n}$

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

$\rm \implies\:2n = 2$

$\bf\implies \:n \: = \: 1 \: year$

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1. Amount on a certain sum of money of Rs p invested at the rate of r % per annum compounded yearly for n years is

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: Amount = p{\bigg[1 + \dfrac{r}{100} \bigg]}^{n} \: }}}$

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

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: Amount = p{\bigg[1 + \dfrac{r}{400} \bigg]}^{4n} \: }}}$

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

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: Amount = p{\bigg[1 + \dfrac{r}{1200} \bigg]}^{12n} \: }}}$