Question

Ashima took a loan of Rs.1,00,000 at \[12%~\] p.a., compounded half yearly. She paid Rs.1,12,360. If \[\left( 1.06 \right)2~\] is equal to 1.1236, then the period for which she took the loan is
A) 2 years
B) 1 years
C) 6 months
D) 11.2 years

Answer 1

Hint:
Here we have to find the time period for which she took the loan. Here principal amount, rate of interest and amount paid is given. We will assume the period to be any variable and then we will use the formula for amount when compounded half-yearly. From there, we will get the value of the required time period.

Complete step by step solution:
It is given that:-
Principal amount $\left( P \right)$ $=Rs1,00,000$
Rate of interest $\left( R \right)$ $=12%$
Amount paid $=Rs1,12,360$
Let $k$ be the time period.
We know the formula of amount when compounded half- yearly.
$A=P{{\left( 1+\dfrac{R}{200} \right)}^{2n}}$
Here $A$ is the amount paid, $P$ is the principal amount, $R$ is the rate of interest and $n$ is the time period.
Substituting the value of amount paid, principal amount, rate of interest and time period, we get
$\Rightarrow 112360=100000{{\left( 1+\dfrac{12}{200} \right)}^{2k}}$
Simplifying all the terms, we get
$\Rightarrow \dfrac{112360}{100000}={{\left( \dfrac{212}{200} \right)}^{2k}}$
Simplifying the fractions of both sides, we get
$\Rightarrow {{\left( \dfrac{53}{50} \right)}^{2}}={{\left( \dfrac{53}{50} \right)}^{2k}}$
On comparing both sides, we get
$\Rightarrow 2=2k$
Dividing 2 on both sides, we get
$\begin{align}
& \Rightarrow \dfrac{2}{2}=\dfrac{2k}{2} \\
& \Rightarrow k=1 \\
\end{align}$
Thus, the time period for which she took the loan is 1 year.

Hence, the correct option is option B.

Note:
Always keep in mind that if two exponentials are equal and they have the same base then their exponents would also be equal. We can find the value of the variable exponents of the exponentials using this property. Always remember that if the rate of interest is annual and the interest is compounded half-yearly then the number of years is doubled and the rate of annual interest gets halved. In such cases, we use the following formula for amount when the interest is compounded half-yearly;
$A=P{{\left( 1+\dfrac{R}{200} \right)}^{2n}}$
Here $A$ is the amount paid, $P$ is the principal amount, $R$ is the rate of interest and $n$ is the time period.