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Pragya borrowed \[5500\] for \[3\] years and \[7000\] for 2 years at the same rate of interest. The total interest she had to pay was \[1500\] . Find the rate of interest.

Last updated date: 21st Jul 2024
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Hint: In this question we are going to find the rate of interest. For solving this problem, let us know about Simple Interest. If the interest on a sum borrowed for a certain period is reckoned uniformly, it is called Simple Interest (S.I). Here we use terms “principal amount, rate, time and interest”.
Principal (P): The money borrowed or lent out for a certain period.
Interest (S.I): Extra money paid for using another's money.
Rate (R): Rate is the rate of interest at which the principal amount is given to someone for a certain time. Rate is in percentage (\[\% \]) and is to be written as \[\dfrac{R}{{100}}\] .
Time (T): Time is the duration for which the principal amount is given to someone. Time will be in years.
Finally, the formula for Simple Interest is, \[S.I = \dfrac{{P \times R \times T}}{{100}}\].

Complete step-by-step answer:
In the given question Pragya had borrowed two principal amounts for two different times. So first we find the rate of interest (R) for the two principal’s separately by using the S.I formula.
Let as assume Principal 1 as \[{\text{(}}{P_1}{\text{)}} = 5500\] and Time 1 \[{\text{(}}{{\text{T}}_1}{\text{)}} = 3\] years.
Let as assume Principal 2 \[{\text{(}}{P_2}{\text{)}} = 7000\] and Time 2 \[{\text{(}}{{\text{T}}_{\text{2}}}{\text{)}} = 2\] years.
Here we use the subscript \[1\] and \[2\] to differentiate the two principals and the time corresponding to it.
We use the formula for finding the simple Interest is \[S.I = \dfrac{{P \times R \times T}}{{100}}\] -----------(1)
Total interest she pays (I) \[ = 1500\].
We have to find out the total interest, then we can write formula as
\[S.{I_1} = \dfrac{{{P_1} \times R \times {T_1}}}{{100}}\] ------------(2)
\[S.{I_2} = \dfrac{{{P_2} \times R \times {T_2}}}{{100}}\] ----------(3)
Here we do not use subscript for rate because we know that Pragya borrows two principal for the same rate of interest.
By substituting the given values in the equation,
From the equation (2), we can find simple interest 1,
\[ \Rightarrow S.{I_1} = \dfrac{{5500 \times R \times 3}}{{100}}\]
 \[ = 55 \times R \times 3\]
\[S.{I_1} = 165R\]
From the equation (3), we can find simple interest 2,
\[ \Rightarrow S.{I_2} = \dfrac{{7000 \times R \times 2}}{{100}}\]
 \[ = 70 \times R \times 2\]
\[S.{I_2} = 140R\].
We already know that the total interest paid by Pragya was \[1500\], i.e., \[S.I = 1500\].
Therefore, The total interest, \[S.I = S.{I_1} \] +\[ S.{ I_2} \]
By substituting values,
\[1500 = 165R + 140R\]
\[1500 = 305R\]
Now, we need to find out the rate of interest (in percentage),
\[R = \dfrac{{1500}}{{305}} = 4.91803279\]
Here the values after decimal were going on. Usually we only consider two digits after decimal. By rounding off the decimal values we will get \[4.92\]. Also that the rate will be in \[\% \] .
Hence, \[R = 4.92\% \].
So, the correct answer is “ \[R = 4.92\% \]”.

Note: Read the question carefully, then only we can get an idea about how to solve it.
Simple interest (S.I) is a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time. In simple interest, the principal amount is always the same.
Remind the formula, \[S.I = \dfrac{{P \times R \times T}}{{100}}\], by using this we can find the formulas of \[P{\text{,}}R{\text{,}}T\].
Simply we can say that, \[R = \dfrac{{100 \times S.I}}{{P \times T}}\]. But in this question we cannot use this formula because there are two principal amounts of two different times.