Question

What will be the compound interest on \[Rs.25000\] for \[3\] years at \[6\% \] per annum, compounded annually?

Answer 1

Hint: Here we are asked to find the compound interest from the given data. Since we are provided with all the values like principal, year, and rate of interest we will directly substitute these values in the formula of compound interest to find the answer.
Formula: formulas that we need to know before solving this problem:
Compound interest: \[P\left[ {{{\left( {1 + \dfrac{r}{{100}}} \right)}^n} - 1} \right]\] where \[P - \]the principal amount, \[r - \]rate of interest, and \[n - \]the number of years.
\[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\]

Complete step-by-step solution:
It is given that a principal amount of Rs.\[25000\] is compounded annually for \[3\] years at \[6\% \] per annum.
We aim to find the compound interest for the given data.
We know that the compound interest of a principal amount \[P\] for \[n\] years at \[r\% \] per annum is given by \[CI = P\left[ {{{\left( {1 + \dfrac{r}{{100}}} \right)}^n} - 1} \right]\]
From the question, we already have all the values in the formula on substituting them in that we get
\[CI = 25000\left[ {{{\left( {1 + \dfrac{6}{{100}}} \right)}^3} - 1} \right]\]
Let us simplify this to find the required value. First, let us simplify the inner part of the above expression.
\[CI = 25000\left[ {{{\left( {\dfrac{{100 + 6}}{{100}}} \right)}^3} - 1} \right]\]
      \[ = 25000\left[ {{{\left( {\dfrac{{106}}{{100}}} \right)}^3} - 1} \right]\]
Let us simplify it further. Simplifying the inner fraction term, we get
     \[ = 25000\left[ {{{\left( {\dfrac{{53}}{{50}}} \right)}^3} - 1} \right]\]
     \[ = 25000\left[ {\left( {\dfrac{{{{53}^3}}}{{{{50}^3}}}} \right) - 1} \right]\]
     \[ = 25000\left[ {\dfrac{{{{53}^3} - {{50}^3}}}{{{{50}^3}}}} \right]\]
Now we can see that the numerator of the fraction term is in the form \[{a^3} - {b^3}\] so let us use the formula \[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\] to simplify it.
     \[ = 25000\left[ {\dfrac{{\left( {53 - 50} \right)\left( {{{53}^3} + \left( {53} \right)\left( {50} \right) + {{50}^2}} \right)}}{{{{50}^3}}}} \right]\]
On further simplification we get
     \[ = 25000\left[ {\dfrac{{3 \times \left( {2809 + 2650 + 2500} \right)}}{{50 \times 50 \times 50}}} \right]\]
     \[ = 25000\left[ {\dfrac{{3 \times 7959}}{{50 \times 50 \times 50}}} \right]\]
     \[ = \left[ {\dfrac{{3 \times 7959}}{5}} \right]\]
     \[ = \dfrac{{23877}}{5}\]
\[CI = 4775.4\]
Thus, we have found the compound interest of the given data that is \[Rs.4775.4\].

Note: Here all the values that we needed to calculate the compound interest are given so it was easier for us to find the interest. During the calculation, we can also use any standard formulas that are available to make our calculation simpler. Here we have used the formula of \[{a^3} - {b^3}\] which helped us to carry over the calculation further.