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# 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.$8\%$

Last updated date: 13th Jun 2024
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Hint: We will use the formula of CI as: ${\text{CI}} = P{\left( {1 + \dfrac{R}{{100}}} \right)^2} - P$ where P is the principle and rate of interest is R%. After putting the value of CI, we will use the algebraic identity: ${\left( {a + b} \right)^2} = \left( {{a^2} + 2ab + {b^2}} \right)$ for simplifying it further. Then, for SI, we will use the formula: ${\text{SI}} = \dfrac{{PRT}}{{100}}$ , where T is the time period and from this equation, we will calculate the value of PR and then we will substitute the value of PR in the equation obtained from the formula of CI to calculate the value of R i.e. the rate percent per annum.

We are given that CI on a certain amount is Rs.41 and SI is Rs.40 on a time interval of 2 years.
We are required to calculate the value of the rate percent per annum.
Let the principle be P with rate of interest as R% for 2 years.
Then, using the formula of CI: ${\text{CI}} = P{\left( {1 + \dfrac{R}{{100}}} \right)^2} - P$, we can write after putting the value of CI as
$\Rightarrow {\text{CI}} = P{\left( {1 + \dfrac{R}{{100}}} \right)^2} - P \\ \Rightarrow 41 = P\left[ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^2} - 1} \right] \\$
Now, using the algebraic identity ${\left( {a + b} \right)^2} = \left( {{a^2} + 2ab + {b^2}} \right)$ to simplify it further, we get
$\Rightarrow 41 = P\left[ {1 + \dfrac{{2R}}{{100}} + \dfrac{{{R^2}}}{{10000}} - 1} \right] \\ \Rightarrow 41 = P\left[ {\dfrac{{2R}}{{100}} + \dfrac{{{R^2}}}{{10000}}} \right] \\ \Rightarrow 41 = \dfrac{{2PR}}{{100}} + \dfrac{{P{R^2}}}{{10000}} \\$
$\Rightarrow 41 = \dfrac{{2PR}}{{100}} + \dfrac{{PR\left( R \right)}}{{10000}}$ – equation (1)
Now, using the formula of SI given by: ${\text{SI}} = \dfrac{{PRT}}{{100}}$, where T is the time period = 2 years, we can write
$\Rightarrow {\text{SI}} = \dfrac{{PRT}}{{100}} \\ \Rightarrow 40 = \dfrac{{PR\left( 2 \right)}}{{100}} \\ \Rightarrow PR = \dfrac{{40\left( {100} \right)}}{2} = 2000 \\$
Using this value of PR in equation (1), we get
$\Rightarrow 41 = \dfrac{{2\left( {2000} \right)}}{{100}} + \dfrac{{R\left( {2000} \right)}}{{10000}}$
Simplifying this equation for the value of R, we get
$\Rightarrow 41 = \dfrac{{40000}}{{100}} + \dfrac{R}{5} \\ \Rightarrow 41 = 40 + \dfrac{R}{5} \\ \Rightarrow 41 - 40 = \dfrac{R}{5} \\ \Rightarrow 1\left( 5 \right) = R \\ \Rightarrow R = 5\% \\$
Hence, we get the value of the rate of interest as $5\%$.
Therefore, option (B) is correct.

Note: In this question, you may get confused while using the formulae of compound interest (CI) and simple interest (SI), since we have calculated the value of PR from SI and then substituted it in the equation obtained from the formula of CI. You may go wrong while solving for R from the equation obtained after putting PR = 2000. Such questions can be solved by direct implementation of the standard formulae related to the concepts given in the question.