Question

The CI on a certain sum for 2 years is Rs. 41 and SI is Rs. 40. Then, the rate percent per annum is:
A. 4%
B. 5%
C. 6%
D. 8%

Answer 1

Hint: We will first let the principle as \[P\] and rate of interest as \[R\% \]. Then we will find the equations from the data given using the formula for Compound interest, \[CI = P{\left( {1 + \dfrac{R}{{100}}} \right)^2} - P\] and the formula for simple interest, \[SI = \dfrac{{PRT}}{{100}}\] by substituting the values. Now, further using the substitution method we will solve the equations to determine the value of rate percent per annum.

Complete step by step Answer:

We will first let the principle be \[P\] Rs. And the rate of interest as \[R\% \]. We have already given the time of 2 years in the question.
The aim is to find the rate of percent per annum.
Now, we will first use the formula for compound interest, \[CI = P{\left( {1 + \dfrac{R}{{100}}} \right)^2} - P\] where \[P\] represents the principle amount, \[R\] represents the rate of interest and substitute the values we have in this and find an equation,
Thus, we get,
\[ \Rightarrow CI = P\left[ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^2} - 1} \right]\]
We can use the identity \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\] and simplify the expression,
\[
   \Rightarrow 41 = P\left[ {1 + \dfrac{{2R}}{{100}} + \dfrac{{{R^2}}}{{10000}} - 1} \right] \\
   \Rightarrow 41 = \dfrac{{2PR}}{{100}} + \dfrac{{P{R^2}}}{{10000}} \\
   \Rightarrow 41 = \dfrac{{2PR}}{{100}} + \dfrac{{PR\left( R \right)}}{{10000}} \\
 \]
Hence, we get the first equation as \[ \Rightarrow 41 = \dfrac{{2PR}}{{100}} + \dfrac{{PR\left( R \right)}}{{10000}}\]----(1)
Now, we will use the formula of simple interest, \[SI = \dfrac{{PRT}}{{100}}\] where \[P\] represents the principle amount, \[R\] represents the rate of interest, \[T\] represents the time period and substitute the values to find another equation,
Thus, we get,
\[
   \Rightarrow 40 = \dfrac{{PR\left( 2 \right)}}{{100}} \\
   \Rightarrow 40 = \dfrac{{PR}}{{50}} \\
   \Rightarrow PR = 2000 \\
 \]
Hence, we get the second equation as \[PR = 2000\]-----(2)
Now, we will substitute the value from (2) equation into (1) equation,
\[
   \Rightarrow 41 = \dfrac{{2\left( {2000} \right)}}{{100}} + \dfrac{{2000\left( R \right)}}{{10000}} \\
   \Rightarrow 41 = 40 + \dfrac{{1\left( R \right)}}{5} \\
   \Rightarrow 41 - 40 = \dfrac{{1\left( R \right)}}{5} \\
   \Rightarrow R = 1\left( 5 \right) \\
   \Rightarrow R = 5\% \\
 \]
Hence, we get the rate of percent per annum as 5%.
Thus, option B is correct.

Note: We have used the formula for compound interest and simple interest so, we need to remember these formulas for such questions, obtain the values from the one equation and substitute it in another equation to determine the required value. Also, we have used the basic identity, \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\] to simplify the calculations. We have to form two equations using the basic formulas to find the rate of interest.