Question

Anuj and Rajesh each lent the same sum 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 interest received.

Answer 1

Hint: Here first we will assume the sum of the money i.e. principal value to be x and then we will find the interests of Anuj and Rajesh separately using the formulas of simple interest and compound interest respectively and then form a linear equation according to the given condition in the question and then solve for the value of x.

Complete step-by-step answer:
Let the sum of money i.e. principal value to be x.
Then it is given that Anuj lent the sum of money for 2 years at 8% simple interest.
Now we know that, the formula of simple interest is given by:-
\[S.I. = \dfrac{{P \times R \times T}}{{100}}\] where P is the principal value, R is the rate of interest and T is the time period.
Hence, applying this formula for the given situation we get:-
\[{\text{Anuj's interest}} = \dfrac{{x \times 8 \times 2}}{{100}}\]
Simplifying it we get:-
\[{\text{Anuj's interest}} = \dfrac{{16x}}{{100}}\]………………. (1)
Now it is given that Rajesh lent the sum of money for 2 years at 8% compound interest.
Now we know the formula of compound interest is given by:-
\[A = p{\left( {1 + \dfrac{r}{{100}}} \right)^t}\] where P is the principal value, R is the rate of interest and t tis the time period.
\[C.I. = A - P\]
Hence, applying this formula for the given situation we get:-
\[{\text{A}} = x{\left( {1 + \dfrac{8}{{100}}} \right)^2}\]
Taking the LCM we get:-
\[{\text{A}} = x{\left( {\dfrac{{100 + 8}}{{100}}} \right)^2}\]
\[ \Rightarrow {\text{A}} = x{\left( {\dfrac{{108}}{{100}}} \right)^2}\]
Simplifying it further we get:-
\[{\text{A}} = x{\left( {\dfrac{{27}}{{25}}} \right)^2}\]
\[ \Rightarrow {\text{A}} = \dfrac{{729x}}{{625}}\]
\[{\text{Rajesh's interest = }}\dfrac{{729x}}{{625}} - x\]
\[{\text{Rajesh's interest = }}\dfrac{{104x}}{{625}}\]…………………….. (2)
Now it is given that,
Rajesh received Rs. 64 more than Anuj
Hence, \[{\text{Rajesh's interest}} = {\text{Anuj's interest}} + 64\]
Putting the values from equation 1 and 2 we get:-
\[\dfrac{{104x}}{{625}} = \dfrac{{16x}}{{100}} + 64\]
Simplifying it we get:-
\[ \Rightarrow \dfrac{{104x}}{{625}} = \dfrac{{4x}}{{25}} + 64\]
\[ \Rightarrow \dfrac{{104x}}{{625}} - \dfrac{{4x}}{{25}} = 64\]
Taking the LCM we get:-
\[\dfrac{{104x - 100x}}{{625}} = 64\]
Simplifying it further we get:-
\[\dfrac{{4x}}{{625}} = 64\]
\[ \Rightarrow 4x = 64 \times 625\]
Solving for x we get:-
\[x = \dfrac{{64 \times 625}}{4}\]
\[x = 10000\]
Therefore, the principal value was Rs. 10000 which was lent by Anuj and Rajesh.
Now Putting the value of x in equation 1 we get:-
\[{\text{Anuj's interest}} = \dfrac{{16\left( {10000} \right)}}{{100}}\]
Simplifying it we get:-
\[{\text{Anuj's interest}} = 1600\]Rs.
Now putting the value of x in equation 2 we get:-
\[{\text{Rajesh's interest = }}\dfrac{{104\left( {10000} \right)}}{{625}}\]
Simplifying it we get:-
\[{\text{Rajesh's interest = }}1664\]Rs.

Hence, Anuj’s Interest is Rs. 1600 and Rajesh’s interest is Rs. 1664.

Note: Students should note that, the difference between simple interest and compound interest is that simple interest is based on principal amount whereas compound interest is based on the principal amount and the interest compounded for a particular period of time.