A sum of money amounts to Rs. 1500 after 3 years and Rs. 2000 after 5

1.	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» $\large\underline{\sf{Given- }}$ $\sf{\rm :\longmapsto\: Amount,\: \: a_1 \:=Rs \: 1500}$ $\sf{\rm :\longmapsto\: Time,\: \: t_1 \:=3\: years}$ $\sf{\rm :\longmapsto\: Amount,\: \: a_2 \:=Rs \: 2000}$ $\sf{\rm :\longmapsto\: Time,\: \: t_2 \:=5\: years}$ 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 $\boxed{ \rm{ Amount = P\bigg(1+\dfrac{r}{400}\bigg)^{4n}}}$

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»

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

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

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

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

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

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

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