Question

A sum of money amounts to Rs. 1500 after 3 years and Rs. 2000 after 5 years at the same rate of SI. Find the rate of interest per annum ?​

Answer 1

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

$\sf{\rm :\longmapsto\: Amount,\: \: a_1 \:=Rs \: 1500}$

$\sf{\rm :\longmapsto\: Time,\: \: t_1 \:=3\: years}$

$\sf{\rm :\longmapsto\: Amount,\: \: a_2 \:=Rs \: 2000}$

$\sf{\rm :\longmapsto\: Time,\: \: t_2 \:=5\: years}$

Let assume that sum of money invested be Rs P at the rate of r % per annum.

We know,

Simple interest (SI) on a certain sum of money of Rs P invested at the rate of r % per annum for n years is always same for every year and is given by

$\boxed{ \rm{ SI = \frac{P \times r \times n}{100}}}$

and

$\boxed{ \rm{ Amount = P + SI}}$

Now,

According to statement it is given that

Amount on a certain sum of money is Rs 1500 after 3 years.

Let assume that simple interest for 1 year on Rs P at the rate of r % per annum is SI.

So,

$\rm :\longmapsto\:Amount = 1500$

$\rm :\longmapsto\:P + 3SI = 1500 - - - (1)$

and

Further it is given that Amount on certain sum of money is Rs 2000 after 5 years.

So,

$\rm :\longmapsto\:Amount = 2000$

$\rm :\longmapsto\:P + 5SI = 2000 - - - (2)$

On Subtracting equation (1) from equation (2), we get

$\rm :\longmapsto\:2SI = 500$

$\bf\implies \:SI = Rs \: 250$

On substituting the value of SI in equation (1) we get

$\rm :\longmapsto\:P + 3 \times 250 = 1500$

$\rm :\longmapsto\:P + 750 = 1500$

$\rm :\longmapsto\:P= 1500 - 750$

$\bf :\longmapsto\:P=Rs \: 750$

Now, we have

Simple Interest for 1 year = Rs 250

Sum of money invested, P = Rs 750

Time, n = 1 year

We know,

$\boxed{ \rm{ SI = \frac{P \times r \times n}{100}}}$

On substituting the values, we get

$\rm :\longmapsto\:250 = \dfrac{750 \times 1 \times r}{100}$

$\bf\implies \:r = \dfrac{100}{3} = 33\dfrac{1}{3} \%$

Additional Information :-

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

$\boxed{ \rm{ Amount = P\bigg(1+\dfrac{r}{100}\bigg)^{n}}}$

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

$\boxed{ \rm{ Amount = P\bigg(1+\dfrac{r}{200}\bigg)^{2n}}}$

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

$\boxed{ \rm{ Amount = P\bigg(1+\dfrac{r}{400}\bigg)^{4n}}}$

Answer 2

According to the question;

Amount ($a_{1}$) = Rs. 1500 ; Time ($t_{1}$) = 3 years

Amount ($a_{2}$) = Rs. 2000; Time ($t_{2}$) = 5 years

let the princple amount be p and rate of intrest = r%

S.I = $\frac{p* r * n }{100}$

amount = S.I + p

now, for 1 year, on Rs. p at r% is S.I

Amount ($a_{1}$) = Rs. 1500

p + 3 S.I = 1500   (1 equation)

Amount on a certain sum of money is Rs. 2000 after 5 year

Amount = Rs. 2000

p + 5 S.I = 2000  ( 2 equation)

on subtracting equation 1 & 2 we get;

2 S.I = 500

S.I = 250

Put S.I in 1 equation

p + 3 (250) = 1500

p = 1500 - 750

p = 750

now we get;

Simple intrest for 1 years = Rs. 250

Sum of money invested p = Rs. 750

Time, n = 1 years

S.I = $\frac{p* r * n }{100}$

$250 = \frac{750* r * 1}{100}$

r = 33.33%

A sum of money amounts to Rs. 1500 after 3 years and Rs. 2000 after 5 years at the same rate of SI, the rate of interest per annum is 33.33%​

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