Question

Vikram borrowed Rs . 2000 from a bank at \[10\% \] per annum simple interest. He lent it to his friend Venkat at the same rate but compounded annually. Find his gain after \[2\dfrac{1}{2}\] years .
A. \[280\]
B. \[38.117\]
C. \[370\]
D. \[347.13\]

Answer 1

Hint: Here we need to use the expression for amount , principal , rate of interest and time then we can get the value of simple interest and compound interest by the expression of compound interest where we are taking the difference of amount and principal.
Formula used for simple interest
Simple interest \[\left( {S.I} \right) = \dfrac{{P \times R \times T}}{{100}}\]
Also formula used for compound interest
Amount \[\left( A \right) = P{\left[ {1 + \dfrac{R}{{100}}} \right]^n}\]
Where \[A = \]Amount
\[P = \]principal
\[R = \]Rate of interest
And \[n = \]Time

Complete step-by-step solution:
We have principal \[\left( P \right) = 2000\]
Time\[\left( n \right)\]or\[\left( T \right) = 2\dfrac{1}{2}\]years
\[ = \dfrac{5}{2}\]years
And Rate of interest \[\left( R \right) = 10\% \] per annum
Now Vikram has to pay a bank so , we have calculated simple interest then
We can use formula of simple interest
Simple interest \[\left( {S.I} \right) = \dfrac{{P \times R \times T}}{{100}}\]
Substituting the value of \[P,R\]and \[T\], we get
\[S.I = \dfrac{{2000 \times 10 \times \dfrac{5}{2}}}{{100}}\]
\[ \Rightarrow S.I = \dfrac{{2000 \times 10 \times 5}}{{100 \times 2}}\]
\[ \Rightarrow \]\[S.I = 20 \times 5 \times 5\]
\[ \Rightarrow S.I = 500\]
So ,vikram has to pay \[2000 + 500 = Rs.2500\] to the bank after \[\dfrac{5}{2}\] years .
Again vikram lent \[Rs.2000\]to venkat at same rate compounded annually
So , we have calculated compound interest then
We can use formula for the amount here and we have the formula
\[\left( A \right) = P{\left[ {1 + \dfrac{R}{{100}}} \right]^n}\]
Substituting the value of \[P,R\]and \[n\],we get
\[A = 2000{\left[ {1 + \dfrac{{10}}{{100}}} \right]^{\dfrac{5}{2}}}\]
Solving the bracket first , we get
\[A = 2000{\left[ {\dfrac{{100 + 10}}{{100}}} \right]^{2 + \dfrac{1}{2}}}\]
\[ \Rightarrow A = 2000{\left[ {\dfrac{{110}}{{100}}} \right]^{2 + \dfrac{1}{2}}}\]
\[ \Rightarrow A = 2000{\left( {\dfrac{{11}}{{10}}} \right)^2}{\left( {\dfrac{{11}}{{10}}} \right)^{\dfrac{1}{2}}}\]
As the power is two and half so , we can write
\[ \Rightarrow A = 2000 \times \dfrac{{11}}{{10}} \times \dfrac{{11}}{{10}} \times \sqrt {\dfrac{{11}}{{10}}} \]
\[ \Rightarrow A = 20 \times 11 \times 11 \times \sqrt {1.1} \]
\[ \Rightarrow A = 2420 \times 1.04880884817\]
\[ \Rightarrow A = 2538.117\]
So , Amount will be
\[A = 2538.117\]
Now to find compound interest we need to subtract principal from amount so ,
Compound interest \[ = \]Amount \[ - \]principal
\[C.I = A - P\]
Substituting the value of principal and amount ,we get
\[C.I = 2538.117 - 2000\]
\[\therefore C.I = 538.117\]
So ,vikram gain \[ = 538.117 - 500\]
\[ = 38.117\]
Option B is the correct answer.

Note: Compound interest is an interest rate that is calculated on the principle and that interest accumulated over the previous period and compound in interest is different from the simple interest where interest is not added to the principle while calculating the interest during the next period.