Question

Shruti lent some money to Pallavi at $5\%$ p.a. simple interest. Pallavi lent the whole amount to Niki on the same day at $8\dfrac{1}{2}\%$ p.a. In this transaction after a year Pallavi earned a profit of Rs. $350$. Find the sum of money lent by Shruti to Pallavi.

Answer 1

Hint: To solve this question, firstly, we will assume the sum of money. Then we will use the formula of simple interest rate for Pallavi. After that we will use the formula of simple interest again and will do the same procedure for calculating interest for Niki. Then, we will use the given condition and simplify the equation to get the required answer.

Complete step by step solution:
Here, we will consider that the sum of money that Shruti lent to Pallavi is $P$.
Then, the interest that shruti will get will be calculated by using the formula as:
$\Rightarrow S.I.=\dfrac{P\times r\times t}{100}$
Now, we will substitute the corresponding values according to Shruti as:
\[\Rightarrow S.{{I}_{shruti}}=\dfrac{P\times 5\times 1}{100}\]
Here, we will get the interest as:
\[\Rightarrow S.{{I}_{shruti}}=\dfrac{5P}{100}\]
Now, we will use the formula for Pallavi as:
$\Rightarrow S.I.=\dfrac{P\times r\times t}{100}$
And will substitute the corresponding values according to Pallavei. Since, the given interest is in mixed ratio, we will convert it into improper fraction as:
$\Rightarrow 8\dfrac{1}{2}\%=\dfrac{17}{2}\%$
Now, from the formula, we will have:
$\Rightarrow S.{{I}_{Pallavi}}=\dfrac{P\times \dfrac{17}{2}\times 1}{100}$
Here, we will do required calculation as:
$\begin{align}
  & \Rightarrow S.{{I}_{Pallavi}}=\dfrac{17P}{2\times 100} \\
 & \Rightarrow S.{{I}_{Pallavi}}=\dfrac{17P}{200} \\
\end{align}$
Now, according to question, Pallavi earned $350$. So, we can write it as:
$\Rightarrow $ Pallavi’s earning $=S.{{I}_{Pallavi}}-S.{{I}_{Shruti}}$
Here, we will substitute the obtained values as:
$\Rightarrow 350=\dfrac{17P}{200}-\dfrac{5P}{100}$
Now, we will use the subtraction of fraction as:
$\Rightarrow 350=\dfrac{17P-10P}{200}$
Here, after doing subtraction, we will have the above step as:
$\Rightarrow 350=\dfrac{7P}{200}$
Now, we will do the multiplication of $\dfrac{200}{7}$ both sides as:
$\Rightarrow 350\times \dfrac{200}{7}=\dfrac{7P}{200}\times \dfrac{200}{7}$
Here, we will cancel out the equal like terms as:
$\Rightarrow 50\times 200=P$
After multiplication, we will get the sum of money as:
$\Rightarrow 10000=P$
Hence, the option $B$ is the correct option.

Note: Simple interest is the interest on the principal amount only but compound interest is the interest on both the principal amount and the earned interest that accumulates on it every year or within a time period.
Formula for simple interest:
$\Rightarrow S.I.=\dfrac{P\times r\times t}{100}$
Formula of Compound interest:
$\Rightarrow C.I.=P\left[ {{\left( 1+\dfrac{r}{100} \right)}^{t}}-1 \right]$
Where, $P$ is principal amount, $r$ is rate of interest and $t$ is time period.