Question

Find the rate percent per annum if Rs.2,00,000 amount to Rs. 2,31,525 in $1\dfrac{1}{2}$ year interest being compounded half yearly.
(A) 5%
(B) 10%
(C) 15%
(D) 10.5%

Answer 1

Hint: we start solving this question by first considering the formula for amount in compound interest, $A=P{{\left( 1+\dfrac{R}{100} \right)}^{n}}$. Then we find the value of n as it is the number of times interest is paid. Then we substitute the obtained value and the values of P and A from the given information in the question and then simplify the obtained equation and find the rate of interest per half-year. Then we multiply the obtained rate with 2 to find the rate of interest per annum.

Complete step by step answer:
We are given that Rs 2,00,000 amounts to Rs 2,31,525 in $1\dfrac{1}{2}$ year. We are also given that interest is being compounded half-yearly.
We need to find the rate of interest per annum.
Now let us consider the formula for an amount when interest is compounded.
$A=P{{\left( 1+\dfrac{R}{100} \right)}^{n}}$
where, A = Amount after the time period (or) Final Amount
P = Principle Value
R = Rate of Interest
n = Number of times interest is paid

Now comparing the given information with the formula above we get,
P = 2,00,000
A = 2,52,000
As the given time period is $1\dfrac{1}{2}$ year.
As the interest is compounded half yearly, in $1\dfrac{1}{2}$ year, interest is paid 3 times.
So, n = 3
So, by substituting these values in the above formula, we get,
\[\begin{align}
  & \Rightarrow 231525=200000\times {{\left( 1+\dfrac{R}{100} \right)}^{3}} \\
 & \Rightarrow {{\left( 1+\dfrac{R}{100} \right)}^{3}}=\dfrac{231525}{200000} \\
\end{align}\]
Simplifying it we get,
\[\Rightarrow {{\left( 1+\dfrac{R}{100} \right)}^{3}}=\dfrac{9261}{800}\]
Now we can write the value on the right-hand side as
\[\Rightarrow {{\left( 1+\dfrac{R}{100} \right)}^{3}}={{\left( \dfrac{21}{20} \right)}^{3}}\]
As both the sides are raised to the power 3, let us equate the bases. Then we get,
\[\begin{align}
  & \Rightarrow 1+\dfrac{R}{100}=\dfrac{21}{20} \\
 & \Rightarrow \dfrac{R}{100}=\dfrac{21}{20}-1 \\
 & \Rightarrow \dfrac{R}{100}=\dfrac{1}{20} \\
 & \Rightarrow R=\dfrac{1}{20}\times 100 \\
 & \Rightarrow R=5 \\
\end{align}\]
So, we get the rate as 5% half-yearly.
As we need the rate of interest per annum, let us multiply the obtained rate with 2. Then we get,
$\begin{align}
  & \Rightarrow 2\times 5\% \\
 & \Rightarrow 10\% \\
\end{align}$
Hence, we get the rate of interest per annum as 10%.
Hence answer is Option B.

Note:
There is a possibility of one making a mistake in this question by considering the formula for simple interest, $I=\dfrac{PRT}{100}$, and then using the formula $A=P+I$ and then find the value of the rate of interest. But here we need to remember that we are given that interest is compounded and use the formula accordingly. One might also make a mistake by not multiplying the value we got for R with 2 and mark the answer as Option A, which is 5%. But it is wrong as the value we got for R is the rate per half year and we need to find the rate per annum.