Question

At what rate of interest compounded annually will Rs. 12000 amount to Rs. 12730.80 in 2 years? (a) 3% (6) 6% (c) 9% (d) 1.5%​

Answer 1

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

Given that,

• Principal, P = Rs 12000

• Amount, A = Rs 12730.80

• Time, n = 2 years

Let assume that rate of interest be r % per annum compounded annually.

We know,

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

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

So, on substituting the values, we get

$\rm \: 12730.8 = 12000\bigg[1 + \dfrac{r}{100} \bigg]^{2}$

$\rm \: {\bigg[1 + \dfrac{r}{100} \bigg]}^{2} = \dfrac{127308}{120000}$

$\rm \: {\bigg[1 + \dfrac{r}{100} \bigg]}^{2} = 1.0609$

$\rm \: {\bigg[1 + \dfrac{r}{100} \bigg]}^{2} = {(1.03)}^{2}$

$\rm \: 1 + \dfrac{r}{100} = 1.03$

$\rm \: \dfrac{r}{100} = 1.03 - 1$

$\rm \: \dfrac{r}{100} = 0.03$

$\rm\implies \:\boxed{ \rm{ \:r \: = \: 3 \: \% \: \: }}$

$\rule{190pt}{2pt}$

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

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

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

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

3. 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

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

4. 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

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

Answer 2

Answer:

Given :-

• A sum of Rs 12000 of amount to Rs 12730.80 in 2 years.

To Find :-

• What is the rate of interest compounded annually.

Formula Used :-

$\clubsuit$ Amount formula when the interest is compounded annually :

$\bigstar \: \: \sf\boxed{\bold{\pink{A =\: P\bigg(1 + \dfrac{r}{100}\bigg)^n}}}\: \: \: \bigstar$

where,

• A = Amount
• P = Principal
• r = Rate of Interest
• n = Time Period

Solution :-

Given :

• Principal = Rs 12000
• Amount = Rs 12730.80
• Time Period = 2 years

According to the question by using the formula we get,

$\implies \bf A =\: P\bigg(1 + \dfrac{r}{100}\bigg)^n$

$\implies \sf 12730.80 =\: 12000\bigg(1 + \dfrac{r}{100}\bigg)^2$

$\implies \sf \dfrac{12730.80}{12000} =\: \bigg(1 + \dfrac{r}{100}\bigg)^2$

$\implies \sf 1.0609 =\: \bigg(1 + \dfrac{r}{100}\bigg)^2$

$\implies \sf \sqrt{1.0609} =\: \bigg(1 + \dfrac{r}{100}\bigg)$

$\implies \sf 1.03 =\: \bigg(1 + \dfrac{r}{100}\bigg)$

$\implies \sf 1.03 =\: 1 + \dfrac{r}{100}$

$\implies \sf 1.03 - 1 =\: \dfrac{r}{100}$

$\implies \sf 0.03 =\: \dfrac{r}{100}$

By doing cross multiplication we get,

$\implies \sf r =\: 0.03(100)$

$\implies \sf r =\: 0.03 \times 100$

$\implies \sf\bold{\red{r =\: 3\%}}$

$\sf\bold{\purple{\underline{\therefore\: The\: rate\: of\: interest\: is\: 3\%\: .}}}$

Hence, the correct options is option no (a) 3% .

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