Question

The C.I on Rs. 10000 at 20% per annum at the end of 1 year 6 month, if the interest is calculated half-yearly will be
(A) 4340
(B) 5320
(C) 3310
(D) 4590

Answer 1

Hint: It is given that the principal is Rs. 10000. The rate of interest is 20% per annum. The rate of interest for 6 months is half of the rate of interest per annum. Since the rate is imposed per half-year so, the number of times that the interest will be imposed for 1 year 6 months is \[\dfrac{18}{6}=3\] . Now, use the formula, \[\text{Amount=Principal}{{\left( 1+\dfrac{\text{rate}}{100} \right)}^{n}}\] and calculate the amount. Now, put the value of the amount and the principal in the formula, \[\text{Interest=Amount-Principal}\] and calculate the value of interest.

Complete step by step solution:
According to the question, it is given that the principal is Rs. 10000 and the rate of compound interest is 20% per annum. We have to calculate the compound interest at the end of 1 year 6 months if the interest is calculated half-yearly.
The principal = Rs. 10000 ……………………….(1)
We have the rate of interest per annum that is for 12 months.
Since the interest is calculated half-yearly, the rate of interest will be half of the rate of interest per annum.
The rate of interest for 12 months = 20%.
Using the unitary method,
The rate of interest for 1-month = \[\dfrac{20\%}{12}\] .
The rate of interest for 6 months = \[\dfrac{20\%}{12}\times 6=\dfrac{20\%}{2}=10\%\] ……………………….(2)
We have to calculate the compound interest for 1 year 6 months if the compound interest is compounded half-yearly. It means that the interest is imposed once in 6 months.
We have 18 months for 1 year 6 months=
The number of times that the interest will be imposed in 1 year 6 months = \[\dfrac{18}{6}=3\] ………………….(3)
Let us calculate the amount.
We know the formula, \[\text{Amount=Principal}{{\left( 1+\dfrac{\text{rate}}{100} \right)}^{n}}\] . Here, n is the number of times that the interest is imposed …………………………(4)
Putting the values of the principal from equation (1), the rate of interest from equation (2), and the number of times that the interest will be imposed from equation (3), in the formula of equation (4), we get
\[\begin{align}
  & \text{Amount=10000}{{\left( 1+\dfrac{10}{100} \right)}^{3}} \\
 & \Rightarrow \text{Amount}=10000{{\left( 1+\dfrac{1}{10} \right)}^{3}} \\
 & \Rightarrow \text{Amount}=10000{{\left( \dfrac{10+1}{10} \right)}^{3}} \\
 & \Rightarrow \text{Amount}=10000{{\left( \dfrac{11}{10} \right)}^{3}} \\
 & \Rightarrow \text{Amount}=10000\times \dfrac{11}{10}\times \dfrac{11}{10}\times \dfrac{11}{10} \\
 & \Rightarrow \text{Amount}=10\times 11\times 11\times 11 \\
 & \Rightarrow \text{Amount=13310} \\
\end{align}\]
The amount after 1 year 6 months = 13310 …………………………(5)
We know the formula, \[\text{Interest=Amount-Principal}\] ………………………………..(6)
Now, putting the values of principal from equation (1), and amount from equation (5), in equation (5), we get
\[\begin{align}
  & \Rightarrow \text{Interest=13310-10000} \\
 & \Rightarrow \text{Interest=3310} \\
\end{align}\]
Therefore, the interest for 1 year 6 months is equal to Rs. 3310.
Hence, the correct option is (C).

Note: In this question, one might make a silly mistake. Like,
The rate of interest per annum is 20% ………………………(1)
We have to calculate the interest for 1 year and 6 months.
The number of times the interest is going to be imposed = \[\dfrac{1.5}{1}\] = 1.5 ……………….(2)
We have, the principal = Rs. 10000 ……………………….(3)
We have the formula, \[\text{Amount=Principal}{{\left( 1+\dfrac{\text{rate}}{100} \right)}^{n}}\] .
Now, using the formula and equation (1), equation (2), and equation (3), we get
\[\text{Amount=10000}{{\left( 1+\dfrac{20}{100} \right)}^{1.5}}\]
This is wrong because the rate of interest is 20% for the whole year and here it is given that the interest is imposed half-yearly. So, the rate of interest for the half-year will be 10%.
Also, the number of times that the interest is going to be imposed is calculated under the condition that the interest is going to be imposed per annum. This is wrong because the interest is imposed per half-yearly. So, the number of times that the interest is going to be imposed is, \[\dfrac{18}{6}=3\] .