Question

The \[CI \geqslant 50,000\] invested for one year, when interest is compounded half-yearly at 10% p.a is
A. Rs. 5000
B. Rs. 5125
C. Rs. 6000
D. Rs. 5500

Answer 1

Hint: Here in this question we have to determine the compound interest. The compound interest is a summation of the amount and the principal amount. In the question they have mentioned the compound interest, time and rate. Hence we determine the value of amount first and then compound interest.

Complete step-by-step solution:
Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words it is defined as interest on interest.
To determine the compound interest we have a formula and it is given by C.I = A – P
By the given data we have
The Principal amount is given by \[P = 50,000\]
The time is given by \[t = 1\] year = 2 half years
The rate of interest is given by \[r = \dfrac{{10}}{2} = 5\% \] per half year.
By using these data we determine the value of the amount. The formula for the amount is given by
\[A = P{\left[ {1 + \dfrac{r}{{100}}} \right]^n}\]
Substituting these values in the formula, and here the value of n will be 2.
Hence we have
\[ \Rightarrow A = 50,000{\left[ {1 + \dfrac{5}{{100}}} \right]^2}\]
Taking LCM for the terms present in the bracket
\[ \Rightarrow A = 50,000{\left[ {\dfrac{{105}}{{100}}} \right]^2}\]
Divide the numbers 105 and 100 by 5 we get
\[ \Rightarrow A = 50,000{\left[ {\dfrac{{21}}{{20}}} \right]^2}\]
Expanding the square
\[ \Rightarrow A = 50,000\left[ {\dfrac{{21}}{{20}}} \right]\left[ {\dfrac{{21}}{{20}}} \right]\]
On simplifying we get
\[ \Rightarrow A = 55125\]
hence we have determined the amount
Therefore the compound interest is determined by
\[C.I = A - P\]
By substituting the values we get
\[ \Rightarrow C.I = 55125 - 50000\]
On simplifying we get
\[ \Rightarrow C.I = 5125\]

Hence the correct answer is option ‘B’.

Note: The compound interest is interest calculated on the amount that includes principal and accumulated interest of the previous period whereas simple interest is interest on the invested amount for the entire period. This is the difference between the simple interest and compound interest. To find the value of amount where principal amount, rate of interest and time is known we use the standard formula \[A = P{\left[ {1 + \dfrac{r}{{100}}} \right]^n}\] to determine the value of A. We can also determine the compound interest by subtracting the initial principal amount from the amount.