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A certain sum of money amounts to $Rs.6300/ - $ in $2$ years $Rs.7875/ - $ in $3$ years and $9$ months at simple interest. Find the rate of interest per annum.

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Last updated date: 13th Jun 2024
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Answer
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Hint:In this question we are given that certain sum of money amounts to a particular amount in some given time at simple interest.Use the formula of simple interest for amount $Rs.6300/-$ and consider it as equation 1 .Then apply simple interest formula again for amount $Rs.7875/ - $ and consider it as equation 2.Now divide both equations to get the value of rate of interest per annum.

Formula used:$Amount = P\left( {1 + \dfrac{{T \times R}}{{100}}} \right)$ where $P$ is principal amount, $R\% $ is rate of interest, $T$ is the time period.

Complete step-by-step answer:
According to the question we need to find the rate of interest i.e. simple interest.
So, we will use the formula for simple interest,
$Amount = P\left( {1 + \dfrac{{T \times R}}{{100}}} \right)\,\,\,\,\,\,\,\, \to (1)$ where $P$ is principal amount, $R\% $ is rate of interest, $T$ is the time period. We are given that the sum of money amounts to $Rs.6300/ - $in $2$ years. So let $Rs.\,P$ be the principal amount and $R\% $ be the rate of interest per annum. Now using (1),
$6300 = P\left( {1 + \dfrac{{2 \times R}}{{100}}} \right)\,\,\,\,\,\,\,\,\,\,\, \to (2)$
Also we are given that a sum of money amounts to $Rs.7875/ - $in $3$ years and $9$ months.
Now the time period is not in years,
Time period $ = $ $3$years and $9$ months
$ = 3\dfrac{9}{{12}} = \dfrac{{15}}{4}$years
Now using (1) we get,
$7375 = P\left( {1 + \dfrac{{15 \times R}}{{400}}} \right)\,\,\,\,\,\,\,\,\,\, \to (3)$
Now dividing (2) by (3) we get,
$
  \dfrac{{6300}}{{7375}} = \dfrac{{P\left( {1 + \dfrac{{2 \times R}}{{100}}} \right)\,}}{{P\left( {1 + \dfrac{{15 \times R}}{{400}}} \right)\,}} \\
  \dfrac{4}{5} = \dfrac{{(100 + 2R)400}}{{(400 + 15R)100}} \\
  \dfrac{4}{5} = \dfrac{{(100 + 2R)4}}{{(400 + 15R)}} \\
  400 + 15R = 500 + 10R \\
  15R - 10R = 500 - 100 \\
  5R = 400 \\
  R = 20\% \\
 $
So we get the rate of interest per annum equal to $20\% $.

Note:In this question we take time period in years and generally we make this type of mistakes by not converting time period in years.