Question

Anuj and Rajesh each lent the same amount of money for 2 years at 8 % simple interest and compound interest, respectively. Rajesh received Rs. 64 more than Anuj. Find the money lent by each and the interest received.

Answer 1

Hint: Assume the money lent by each as P, which is also the principal amount. Calculate the interest received by Anuj by applying the formula of simple interest, given by : $S.I=\dfrac{P\times R\times T}{100}$, where S.I is the simple interest, P is the principal amount, R is the rate and T is the time in years. Now, calculate the compound interest received by Rajesh by using the formula : $C.I=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}-P$, where C.I is the compound interest, P is the principal amount, r is the rate, t is the time in years and n is the number of times interest is paid or received in one year. Finally subtract the interest of Anuj from the interest of Rajesh and equate it with 64 to get the answer or value of P. Substitute this value of P in the S.I and C.I formula to find the interest received by Anuj and Rajesh respectively.

Complete step by step answer:
Let us assume the same sum of money lent by Anuj and Rajesh as P. This is also called the principal amount. Now, let us find the interest received by Anuj and Rajesh.
1. For Anuj :
We have been given that Anuj has lent the money for 2 years at 8 % simple interest. So, we have,
Principal amount = P
Rate = R = 8 %
Time = T = 2 years
Now, we know that simple interest is given by :
$S.I=\dfrac{P\times R\times T}{100}$, where S.I is the simple interest.
Therefore, substituting the values of P, R, and T we get,
$\begin{align}
  & \Rightarrow S.I=\dfrac{P\times 8\times 2}{100} \\
 & \Rightarrow S.I=\dfrac{4P}{25}\ldots \ldots \ldots \left( i \right) \\
\end{align}$
2. For Rajesh :
We have been given that Rajesh has lent the money for 2 years at 8 % compound interest. So, we have,
Principal amount = P
Rate = r = 8 % = $\dfrac{8}{100}$
Time = t = 2 years
Now, we know that compound interest is given by :
$C.I=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}-P$
Here, C.I is the compound interest and n is the number of times interest is paid or received in one year. Since, here no information is given about n, so we will assume its value as 1. Therefore, substituting the values of P, r, n and t, we get,
$\begin{align}
  & \Rightarrow C.I=P{{\left( 1+\dfrac{8}{100\times 1} \right)}^{2\times 1}}-P \\
 & \Rightarrow C.I=P{{\left( 1+\dfrac{2}{25} \right)}^{2}}-P \\
 & \Rightarrow C.I=P\left[ {{\left( 1+\dfrac{2}{25} \right)}^{2}}-1 \right]\ldots \ldots \ldots \left( ii \right) \\
\end{align}$
Now, we have been provided with the information that Rajesh received Rs. 64 more than Anuj. So, the difference of their interests must be 64.
$\begin{align}
  & \Rightarrow C.I-S.I=64 \\
 & \Rightarrow P\left[ {{\left( 1+\dfrac{2}{25} \right)}^{2}}-1 \right]-\dfrac{4P}{25}=64 \\
 & \Rightarrow P\left[ {{\left( \dfrac{27}{25} \right)}^{2}}-1 \right]-\dfrac{4P}{25}=64 \\
 & \Rightarrow P\left[ \dfrac{{{\left( 27 \right)}^{2}}-{{\left( 25 \right)}^{2}}}{{{\left( 25 \right)}^{2}}} \right]-\dfrac{4P}{25}=64 \\
\end{align}$
Using the identity ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$, we get.
$\begin{align}
  & \Rightarrow P\left[ \dfrac{\left( 27+25 \right)\left( 27-25 \right)}{{{\left( 25 \right)}^{2}}} \right]-\dfrac{4P}{25}=64 \\
 & \Rightarrow P\left[ \dfrac{52\times 2}{{{\left( 25 \right)}^{2}}} \right]-\dfrac{4P}{25}=64 \\
 & \Rightarrow \dfrac{P}{{{\left( 25 \right)}^{2}}}\left[ \dfrac{104-100}{1} \right]=64 \\
 & \Rightarrow \dfrac{P}{{{\left( 25 \right)}^{2}}}\times 4=64 \\
 & \Rightarrow P=\dfrac{{{\left( 25 \right)}^{2}}\times 64}{4} \\
 & \Rightarrow P={{\left( 25 \right)}^{2}}\times 16 \\
 & \Rightarrow P=10000 \\
\end{align}$
Hence, the amount of money lent by each is Rs. 10000.
Now, substituting P = 10000 in equation (i), we get,
$\begin{align}
  & \Rightarrow S.I=\dfrac{4\times 10000}{25} \\
 & \Rightarrow S.I=1600 \\
\end{align}$
Since, compound interest is Rs. 64 more than simple interest, therefore, we have,
$\begin{align}
  & \Rightarrow C.I=1600+64 \\
 & \Rightarrow C.I=1664 \\
\end{align}$

Hence, the interest received by Anuj and Rajesh are Rs. 1600 and Rs. 1664 respectively.

Note: One may note that we can also find the value of C.I by substituting P = 10000 in equation (ii), but here the calculation will be more. That is why we have used the information given in the question and simply added 64 in the value of S.I. Now, to solve the above question, we must remember the formulas of S.I and C.I, otherwise it will be very difficult to find the required values.