Question

Determine the compound interest and compound amount on Rs.1000 at 6% compounded semiannually for 6 years. Given that ${{\left( 1+i \right)}^{n}}$=1.42576 for i = 3% and n = 12.
\[\begin{align}
  & A.\text{ }16000 \\
 & B.\text{ }630.50 \\
 & C.\text{ }425.76 \\
 & D.\text{ }1640 \\
\end{align}\]

Answer 1

Hint: In this question, we are given a principal amount, rate of interest compounded semiannually, and time period. As we are given rate as per annum but the rate is compounded semiannually, we will change it into rate as per half year and then change the time period into half years also as the rate is compounded semiannually. After that, we will use the formula of the compound amount and compound interest to find our required answer. The formula for compound amount is given by:
\[A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}\]
Where P is the principal amount, r is the numerical value of the rate of interest and n is the time period.
The formula for compound interest is given by:
Compound interest = Compound amount - Principal amount.

Complete step-by-step solution:
Here, we are given the principal amount as Rs.1000. Therefore, P = 1000
Now, we are given a rate of interest as 6% compounded semiannually but the rate of interest is per annum. So, we will change it according to half-year, that is, interest becomes half so that it can be compounded after every half year. Hence, the rate of interest becomes $\dfrac{6}{2}=3\%$.
Therefore, r = 3%
Now, as discussed earlier, interest is compounded semiannually, therefore, the time period should also be in half years. As we are given years to be six, so we will multiply it by 2 so that we get the required number of half years. Hence, time period becomes $2\times 6=12$.
Therefore, n = 12.
Now, we know the compound amount is given by $A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}$ where A is the compound amount, P is the principal amount, r is a rate of interest, and n is the time period.
Let us put all the values obtained earlier in the above formula, so we get:
\[A=1000{{\left( 1+\dfrac{3}{100} \right)}^{12}}\]
As $\dfrac{3}{100}$ can be written as 3% therefore, A becomes,
\[A=1000{{\left( 1+3\% \right)}^{12}}\cdots \cdots \cdots \cdots \cdots \left( 1 \right)\]
We are given in the question that, ${{\left( 1+i \right)}^{n}}$ for i = 3% and n = 12. Therefore,
${{\left( 1+3\% \right)}^{n}}=1.42576$
Putting this value in (1), we get:
\[\begin{align}
  & A=1000\times 1.42576 \\
 & \Rightarrow Rs.1425.76 \\
\end{align}\]
Hence, compound amount = Rs.1425.76
Now, as we know, Compound amount = Principal amount + Compound interest.
Therefore, compound interest can be calculated as
Compound interest = Compound amount - Principal amount.
As found earlier, compound amount = 1425.76 and principal amount = 1000.
Therefore, compound interest becomes,
Compound interest = 1425.76 - 1000 = Rs.425.76
Hence, compound amount = Rs.1425.76 and compound interest = Rs.425.76. Hence, option C is the correct answer.

Note: Students should not forget to change the rate of interest and time period according to which interest compounds. As we are given the value of ${{\left( 1+i \right)}^{n}}$ so, students should try to convert the formula into this form so as to simplify the calculation. Students should note that, in the formula for calculating the compound amount, r takes numeric value only and not the percentage.