Question

If \[P=Rs.1,000,R=5\%,p.a,n=4;\]. What is the amount and CI?
\[\begin{align}
  & A)Rs.1,215.50\And 215.50 \\
 & B)Rs.1,215\And 125 \\
 & C)Rs.2,115\And 115 \\
 & D)\text{None of these} \\
\end{align}\]

Answer 1

Hint: We should know that if P is the initial amount, R is the interest, then the interest for one year is equal to I then the value of I is equal to \[\dfrac{P\times R\times 1}{100}\]. Now the amount after the first year is equal to the sum of P and \[\dfrac{P\times R\times 1}{100}\]. We know that compound interest after n years is equal to the difference between the amount obtained after n years and initial amount. By using this concept, we can find the value of amount and C.I.

Complete step by step answer:
Before solving the problem, we should know that if P is the initial amount, R is the interest, then the interest for one year is equal to I then the value of I is equal to \[\dfrac{P\times R\times 1}{100}\]. Now the amount after the first year is equal to the sum of P and \[\dfrac{P\times R\times 1}{100}\].
From the question, it was given that \[P=Rs.1,000,R=5\%,p.a,n=4;\]. Now we should calculate the value of the amount after 4 years and the value of C.I where C.I represents compound interest.
Now, we know that if P is the initial amount, R is the interest, then the interest for one year is equal to I then the value of I is equal to \[\dfrac{P\times R\times 1}{100}\]. Now the amount after the first year is equal to the sum of P and \[\dfrac{P\times R\times 1}{100}\].
So, now we will apply the above formula for this first year.
Now we get
Interest for the first year \[=Rs.\dfrac{1000\times 5\times 1}{100}=Rs.50\]
Amount after first year \[=Rs.1000+Rs.50=Rs.1050\]
So, now we will again apply this for the second year.
Now we will get
Interest for the second year \[=Rs.\dfrac{1050\times 5\times 1}{100}=Rs.52.5\]
Amount after second year \[=Rs.1050+Rs.52.5=Rs.1102.5\]
So, now we will again apply this for the third year.
Now we will get
Interest for the third year \[=Rs.\dfrac{1102.5\times 5\times 1}{100}=Rs.55.125\]
Amount after third year \[=Rs.1102.5+Rs.55.125=Rs.1157.625\]
So, now we will again apply this for the fourth year.
Now we will get
Interest for the fourth year \[=Rs.\dfrac{1157.625\times 5\times 1}{100}=Rs.57.88125\]
Amount after fourth year \[=Rs.1157.625+Rs.57.88125=Rs.1215.50\]
So, it is clear that the total amount after 4 years is equal to \[Rs.1215.50\].
We know that compound interest after n years is equal to the difference between the amount obtained after n years and initial amount.
So, let us assume the compound interest is equal to C.I.
\[\begin{align}
  & \Rightarrow C.I=Rs.1215.50-Rs.1000 \\
 & \Rightarrow C.I=Rs.215.50 \\
\end{align}\]
So, it is clear that the compound interest is equal to \[Rs.215.50\].

So, the correct answer is “Option A”.

Note: This problem can be solved in an alternative manner. We know that if P is the initial amount, R is the interest, then the interest for n year is equal to I then the value of I is equals \[A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}\]. From the question, it was given that \[P=Rs.1,000,R=5\%,p.a,n=4;\]. Now we should calculate the value of the amount after 4 years and the value of C.I where C.I represents compound interest.
\[\begin{align}
  & \Rightarrow A=\left( 1000 \right){{\left( 1+\dfrac{5}{100} \right)}^{4}} \\
 & \Rightarrow A=\left( 1000 \right){{\left( \dfrac{105}{100} \right)}^{4}} \\
 & \Rightarrow A=1215.50 \\
\end{align}\]
So, it is clear that the total amount after 4 years is equal to \[Rs.1215.50\].
We know that compound interest after n years is equal to the difference between the amount obtained after n years and initial amount.
So, let us assume the compound interest is equal to C.I.
\[\begin{align}
  & \Rightarrow C.I=Rs.1215.50-Rs.1000 \\
 & \Rightarrow C.I=Rs.215.50 \\
\end{align}\]
So, it is clear that the compound interest is equal to \[Rs.215.50\].
Hence, option A is correct.