Question

What is the rate of interest p.c.p.a (percent compounded per annum)?
I. An amount doubles itself in 5 years on simple interest.
II. Difference between the compound interest and the simple interest earned on a certain amount in 2 years is Rs.400
III. Simple interest earned per annum is Rs.2000
A. I only
B. II and III only
C. All I, II and III
D. Any two of the three
E. I only or II and III only

Answer 1

Hint: In this question, we will consider the unknown quantities as variables and then we will use the formula for compound interest and simple interest to reach the solution of the given problem.

Complete step-by-step answer:
I. let \[P\] be the principal amount, \[I\] be the simple interest and \[R\] be the rate of interest per annum. Given in 5 years of time i.e., \[T = 5\] the final amount gets doubled the principal amount. So, the final amount is \[2P\].
We know that simple interest = final amount – principal amount = \[2P - P = P\]. So, \[I = P\].
We know that \[I = \dfrac{{P \times T \times R}}{{100}}\]
By substituting the above data, we have
\[
   \Rightarrow P = \dfrac{{P \times 5 \times R}}{{100}} \\
   \Rightarrow \dfrac{{100}}{5} = R \\
  \therefore R = 20 \\
\]
Thus, the rate of interest percent compounded per annum is 20.

II. Given the difference between the compound interest and the simple interest earned in two years on a certain amount is Rs.400. So, time period = 2 years.
We know that the compound interest is given by the formula \[C = \left[ {P{{\left( {1 + \dfrac{R}{{100}}} \right)}^T} - P} \right]\] where \[P\] is the principle amount, \[T\] is the time period and \[R\] is the rate on interest per annum.
According to the given condition,
\[
   \Rightarrow \left[ {P{{\left( {1 + \dfrac{R}{{100}}} \right)}^T} - P} \right] - \dfrac{{P \times T \times R}}{{100}} = 400 \\
   \Rightarrow \left[ {P{{\left( {1 + \dfrac{R}{{100}}} \right)}^2} - P} \right] - \dfrac{{P \times 2 \times R}}{{100}} = 400{\text{ }}\left[ {\because T = 2} \right] \\
\]
By simplifying the terms, we have
\[
   \Rightarrow \dfrac{{P{{\left( {100 + R} \right)}^2} - 10000P}}{{10000}} - \dfrac{{2PR}}{{100}} = 400 \\
   \Rightarrow P{\left( {100 + R} \right)^2} - 10000P - 200PR = 4000000 \\
   \Rightarrow 10000P + 200PR + P{R^2} - 10000P - 200\operatorname{P} R = 4000000 \\
  \therefore \operatorname{P} {R^2} = 4000000...........................................\left( 1 \right) \\
\]

III. Given simple interest earned per annum is Rs.2000
We know that the simple interest \[I\] is given by the formula \[I = \dfrac{{P \times T \times R}}{{100}}\] where \[P\] is the principle amount, \[T\] is the time period and \[R\] is the rate on interest per annum.
So, we have
\[
   \Rightarrow \dfrac{{P \times 1 \times R}}{{100}} = 2000{\text{ }}\left[ {\because T = 1} \right] \\
   \Rightarrow \dfrac{{PR}}{{100}} = 2000 \\
   \Rightarrow PR = 200000......................................\left( 2 \right) \\
\]
Dividing equation (1) with (2), we get
\[
   \Rightarrow \dfrac{{P{R^2}}}{{PR}} = \dfrac{{4000000}}{{200000}} \\
  \therefore R = 20 \\
\]
So, by using the II and III conditions we can have the rate of interest p.c.p.a or by using condition I only.
So, the correct answer is “Option E”.

Note: Simple interest \[I\] is given by the formula \[I = \dfrac{{P \times T \times R}}{{100}}\] where \[P\] is the principle amount, \[T\] is the time period and \[R\] is the rate on interest per annum. Compound interest is given by the formula \[C = \left[ {P{{\left( {1 + \dfrac{R}{{100}}} \right)}^T} - P} \right]\] where \[P\] is the principle amount, \[T\] is the time period and \[R\] is the rate on interest per annum.