Question

Find the compound interest on \[{\rm{Rs}}5000\]for 3 years at \[5\% \]. Find the simple interest also for \[{\rm{Rs}}5000\] for 3 years at \[5\% \]p.a. and calculate the difference in the interests.

Answer 1

Hint:
Here, we will find simple interest and compound interest by substituting the given values in the respective formulas. Then we will subtract the interest which is less from the interest which is more to find the required difference.

Formula Used:
We will use the following formulas:
1) \[S.I = \dfrac{{P \cdot R \cdot T}}{{100}}\], where \[S.I\] is the Simple Interest, \[P\] is the Principal, \[R\] is the rate of interest per annum and \[T\] is the time period.
2) \[C.I = P{\left( {1 + \dfrac{R}{{100}}} \right)^n} - P\], where \[C.I\]is the Compound Interest, \[P\] is the Principal, \[R\] is the rate of interest per annum and \[n\] is the time period.

Complete step by step solution:
According to the question,
The given sum of money or the Principal, \[P = {\rm{Rs}}5000\]
Given time period, \[T = 3{\rm{years}}\]
Rate of interest per annum, \[R = 5\% \]
Hence, substituting the given values in this formula of simple interest, \[S.I = \dfrac{{P \cdot R \cdot T}}{{100}}\], we get,
\[S.I = \dfrac{{\left( {5000} \right) \cdot \left( 5 \right) \cdot \left( 3 \right)}}{{100}}\]
Simplifying the expression, we get
\[ \Rightarrow S.I = 15 \times 50 = {\rm{Rs750}}\]
Now we will find the compound interest.
Hence, substituting the given values in this formula of compound interest \[C.I = P{\left( {1 + \dfrac{R}{{100}}} \right)^n} - P\], we get,
\[C.I = 5000{\left( {1 + \dfrac{5}{{100}}} \right)^3} - 5000\]
Dividing the terms inside the bracket, we get
\[ \Rightarrow C.I = 5000{\left( {1 + \dfrac{1}{{20}}} \right)^3} - 5000\]
Taking the LCM inside the bracket, we get
\[ \Rightarrow C.I = 5000{\left( {\dfrac{{21}}{{20}}} \right)^3} - 5000\]
Applying the exponent on the terms, we get
\[ \Rightarrow C.I = 5\left( {\dfrac{{9261}}{8}} \right) - 5000\]
Here, taking 5 common, we get,
\[ \Rightarrow C.I = 5\left[ {\left( {\dfrac{{9261}}{8}} \right) - 1000} \right]\]
Taking the LCM inside the bracket, we get
\[ \Rightarrow C.I = 5\left( {\dfrac{{9261 - 8000}}{8}} \right) = \dfrac{5}{8} \times 1261\]
Multiplying the terms, we get
\[ \Rightarrow C.I = \dfrac{{6305}}{8} = {\rm{Rs}}788.125\]
Now, we are required to calculate the difference in the interests.
Hence, we will subtract the simple interest from the compound interest to find the required difference.
\[C.I - S.I = 788.125 - 750 = {\rm{Rs}}38.125\]

Therefore, the required difference in the interests is \[{\rm{Rs}}38.125\]
Hence, this is the required answer.

Note:
In this question, we have used the formula of Simple Interest as well as compound interest. Simple Interest is the interest earned on the Principal or the amount of loan. The second type of interest is Compound Interest. Compound Interest is calculated both on the Principal as well as on the accumulated interest of the previous year. Hence, this is also known as ‘interest on interest’.