 QUESTION

# At what rate percent per annum will the simple interest on a sum of money be ​$\dfrac{2}{5}$ of the amount in 10 yrs.A. ${\text{4 % }}$B .$\dfrac{{17}}{3}{\text{ % }}$C. ${\text{6 % }}$D. $\dfrac{{20}}{3}{\text{ % }}$

HINT- Proceed the solution of this question using result which is Amount is equal to sum of principle and simple interest. So applying the given condition of question in this result, we can find the desired rate of interest.

Let the principal(P) be x, and given time (T) =10 years and Rate(R)= r % per annum
We know that simple interest ${\text{SI}} = \dfrac{{{\text{P}} \times {\text{R}} \times {\text{T}}}}{{100}}$
So on putting the values in above formula
$\Rightarrow {\text{SI}} = \dfrac{{{\text{P}} \times {\text{R}} \times {\text{T}}}}{{100}} = \dfrac{{{\text{x}} \times {\text{r}} \times 10}}{{100}}$
And we know that,
Amount = Principle + simple Interest
Amount = x + $\dfrac{{{\text{x}} \times {\text{r}} \times 10}}{{100}}$
In question, it is given that the simple interest on a sum of money be ​$\dfrac{2}{5}$ of the amount, therefore
$\Rightarrow \dfrac{{{\text{x}} \times {\text{r}} \times 10}}{{100}} = \dfrac{2}{5}\left( {{\text{x + }}\dfrac{{{\text{x}} \times {\text{r}} \times 10}}{{100}}} \right)$
$\Rightarrow \dfrac{{{\text{x}} \times {\text{r}}}}{{10}} = \dfrac{{2{\text{x}}}}{5} + \dfrac{{{\text{2}} \times {\text{x}} \times {\text{r}} \times 10}}{{5 \times 100}}$
On further solving
$\Rightarrow \dfrac{{{\text{x}} \times {\text{r}}}}{{10}} = \dfrac{{2{\text{x}}}}{5} + \dfrac{{{\text{2}} \times {\text{x}} \times {\text{r}}}}{{5 \times 10}}$
$\Rightarrow \dfrac{{{\text{x}} \times {\text{r}}}}{{10}} = \dfrac{{2{\text{x}}}}{5} + \dfrac{{{\text{x}} \times {\text{r}}}}{{25}}$
$\Rightarrow {\text{x}} \times {\text{r}}\left( {\dfrac{1}{{10}} - \dfrac{1}{{25}}} \right) = \dfrac{{2{\text{x}}}}{5}$
$\Rightarrow {\text{x}} \times {\text{r}}\left( {\dfrac{3}{{50}}} \right) = \dfrac{{2{\text{x}}}}{5}$
On cancelling x from both side
$\Rightarrow {\text{r}}\left( {\dfrac{3}{{50}}} \right) = \dfrac{2}{5}$
On doing cross multiplication
$\Rightarrow {\text{r}} = \dfrac{2}{5} \times \left( {\dfrac{{50}}{3}} \right) = \dfrac{{20}}{3}\%$
Hence rate percent per annum = $\dfrac{{20}}{3}\%$

Note- In such types of questions where we talk about simple interest, we should know that simple interest is calculated by multiplying the daily interest rate by the principal, by the number of days that pass by between payments. Simple interest is beneficial for those consumers who pay their loans on time or early each month. Auto loans and short-term personal loans are mainly based on simple interest loans