Question

Magesh invested ₹ \[{\text{5000}}\] at \[12\% \] p.a for one year. If the interest is compounded half yearly, find the amount at the end of the year.

Answer 1

Hint: First we have to find the rate of interest for half yearly, then use the compound interest formula to find the total interest for one year if the interest is compounded half yearly.

Complete step by step solution:
First understand some basic definitions and formulas.
Simple interest \[ = \dfrac{{P \times T \times R}}{{100}}\] where \[P\] is Principal, T is time and \[R\] is the rate of interest. In case of compound interest, the interest will be added to the initial principal after every compounding period. Hence, compound interest keeps on increasing after every compounding period. Hence, Compound interest if given by
 \[CI = P{\left[ {1 + \dfrac{R}{{100}}} \right] ^N} - P\] .
If the interest is compounded half yearly, In this case, after every 6 months, interest well be added to the principal.
 \[CI = P{\left[ {1 + \dfrac{{\left( {\dfrac{R}{2}} \right)}}{{100}}} \right] ^N} - P\] ----(1), Where \[R\] is the rate of interest for one year.
Given \[P = 5000\] , \[N = 2\] , \[R = 12\% \] \[ \Rightarrow \dfrac{R}{2} = 6\% \]
 \[CI = 5000 \times {\left[ {1 + \dfrac{6}{{100}}} \right] ^2} - 5000 = \] \[618\]
Hence total amount end of the year \[ = \] Principal+ Compound interest \[ = 5000 + 618 = 5618\]
Hence the amount at the end of the year is ₹ \[5618\] .
So, the correct answer is “₹ \[5618\] ”.

Note: Note that the unit of rate of interest and time should be the same. So, if the rate of interest is ‘per year’, then time should also be in ‘year’. Similarly, if the rate of interest is ‘per six months’, then time should also be ‘per six months’.