Question

The C.I. on \[{{Rs}}.16,000\] at \[15\% \] p.a. for \[2\dfrac{1}{3}\] years is:
A) \[Rs.6,418\]
B) \[Rs.5,000\]
C) \[Rs.22,218\]
D) \[Rs.6,218\]

Answer 1

Hint:
Here we will first find the value of amount by substituting the given values of principal, time and rate of interest in the formula of amount. Then we will find the compound interest which will be equal to the difference between the amount and the principal.

Formula used:
We will use the following formulas:
1) \[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}\], where \[A\] is the amount, \[P\] is the principal interest, \[R\] is the rate of interest and \[T\] is the time period.
2) \[SI = \dfrac{{P \times R \times T}}{{100}}\] where \[SI\] is the simple interest, \[P\] is the principal interest, \[R\] is the rate of interest and \[T\] is the time period.

Complete step by step solution:
It is given that:
Principal amount \[ = Rs.16,000\]
Rate of interest \[ = 15\% \]
Time period \[ = 2\dfrac{1}{3}{\rm{years}}\]
Now, we will calculate the amount for the first 2 year.
We know the formula to calculate the amount is given by \[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}\].
Now, we will substitute the value of principal amount and rate of interest here.
\[ \Rightarrow A = 16000{\left( {1 + \dfrac{{15}}{{100}}} \right)^2}\]
By taking LCM inside the bracket, we get
\[ \Rightarrow A = 16000 \times {\left( {\dfrac{{115}}{{100}}} \right)^2}\]
On further simplification, we get
\[ \Rightarrow A = Rs.21,160\]
Now, we will find the compound interest.
We know that the compound interest is equal to the difference between the amount and the principal.
Therefore,
\[CI = 21,160 - 16,000 = 5,160\]
The principal interest at the end of the 2nd year will be equal to sum of the principal and the compound interest.
Therefore,
Principal at the 2nd year \[ = Rs.5,160 + Rs.16000 = Rs.21,160\]
So we will find the simple interest for next \[\dfrac{1}{3}year\].
We know the formula of simple interest is \[SI = \dfrac{{P \times R \times T}}{{100}}\].
Now, we will substitute the value of principal amount, time and rate of interest in the formula of simple interest here
\[SI = \dfrac{{21160 \times 15 \times \dfrac{1}{3}}}{{100}}\]
On further simplification, we get
\[ \Rightarrow SI = Rs.1058\]
Therefore, the required compound will be equal to the sum of compound interest for 2nd year and the simple interest for the next half year.
Therefore, required compound interest \[ = 5160 + 1058 = {\rm{Rs}}.6,218\]

Hence, the correct option is option D.

Note:
Here we have obtained the compound interest. Simple interest and compound interest are the terms used in banking and financial sector. Compound interest is also known as compounding interest and is defined as the interest which is calculated on the initial principal and which also includes all of the accumulated interest from the previous periods on a loan or deposit. Here, we have calculated simple interest on the principal for 4 months. Simple interest is the interest which is calculated on principal only.