Question

A man borrowed Rs. 8000 at \[60\% \] per annum simple interest and immediately lent the whole sum at \[60\% \] per annum compound interest. What does he gain at the end of 2 years?
A.Rs. 60
B.Rs. 80
C.Rs. 1000
D.Rs. 120

Answer 1

Hint: First, we will use the formula of simple interest, \[{\text{S.I.}} = \dfrac{{{\text{P}} \times {\text{T}} \times {\text{R}}}}{{100}}\], where \[{\text{P}}\] is principal starting amount of money, \[{\text{R}}\] is the interest rate per year and \[{\text{T}}\] is the time the money is invested in years and the amount when interest is compounded is \[A = {\text{P}}{\left( {1 + \dfrac{{\text{R}}}{{100}}} \right)^n}\], where \[{\text{R}}\] is the interest rate per year. Apply these formulae, and then use the given conditions to find the required value.

Complete step-by-step answer:
Let us assume that \[{\text{P}}\] represents the principle and R represents the rate of interest.
We are given that the man borrowed Rs. 8000 at \[60\% \] per annum simple interest and immediately lent the whole sum at \[60\% \] per annum compound interest.
We know that the formula to calculate the amount when interest is compounded is \[A = {\text{P}}{\left( {1 + \dfrac{{\text{R}}}{{100}}} \right)^T}\], where \[{\text{R}}\] is the interest rate per year and T is the time in the years.
\[
   \Rightarrow A = 8000{\left( {1 + \dfrac{{10}}{{100}}} \right)^2} \\
   \Rightarrow A = 8000{\left( {\dfrac{{100 + 10}}{{100}}} \right)^2} \\
   \Rightarrow A = 8000{\left( {\dfrac{{110}}{{100}}} \right)^2} \\
   \Rightarrow A = 8000{\left( {\dfrac{{11}}{{10}}} \right)^2} \\
   \Rightarrow A = 8000\left( {\dfrac{{121}}{{100}}} \right) \\
   \Rightarrow A = 80\left( {121} \right) \\
   \Rightarrow A = 9680 \\
 \]
Substituting the value of P and A in the formula of compound interest, \[{\text{C.I.}} = {\text{A}} - {\text{P}}\], where A is the amount when interest is compounded, we get
\[
   \Rightarrow {\text{C.I.}} = 9680 - 8000 \\
   \Rightarrow {\text{C.I.}} = 1680 \\
 \]

We know that the formula of simple interest, \[{\text{S.I.}} = \dfrac{{{\text{P}} \times {\text{T}} \times {\text{R}}}}{{100}}\], where \[{\text{P}}\] is principal starting amount of money, \[{\text{R}}\] is the interest rate per year and \[{\text{T}}\] is the time the money is invested in years.
First, we have to find the value of \[{\text{T}}\] for the above formula of simple interest.
\[{\text{T}} = 2\]
We will now substitute the above value, P and R to compute the simple interest using the above formula.
\[
   \Rightarrow {\text{S.I.}} = \dfrac{{8000 \times 10 \times 2}}{{100}} \\
   \Rightarrow {\text{S.I.}} = 1600 \\
 \]
Since C.I. is greater than S.I., so we have gained.
Finding the difference of C.I. and S.I. to find the amount of gain, we get
\[
   \Rightarrow Gain = 1680 - 1600 \\
   \Rightarrow Gain = 80 \\
 \]
Thus, the option B is correct.

Note: In solving these types of questions, you should be familiar with the formulae of simple interest and compound interest. Students should note here that the sum of compound interest and simple interest is the principal amount. It is also important to understand in applying both the simple interest and compound interest formula accordingly. One should remember that simple interest is computed only on the principle, but the compound interest is calculated on both the accumulated interest and the principal or else the answer will be wrong.