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# A sum of Rs. 10,000 is borrowed at 8% per annum compounded annually. If the amount is to be paid in three equal installments, what will be the annual installment?

Last updated date: 20th Sep 2024
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Hint: In his type of question use the concept that the amount is borrowed it has to pay in three equal installments so this is equal to (first installment / (1 + rate of interest)) + (second installment / (1 + rate of interest)$^2$) + (third installment / (1 + rate of interest)$^3$), so use this concept to reach the solution of the question.

Given data:
Sum of money which is borrowed = Rs. 10,000
Rate of interest at which it is borrowed = 8% per annum compounded annually.
Now it is given that the amount is paid in three equal installments.
Now we have to calculate the annual installment such that the amount is paid in three equal annual installments.
Let the annual installment be Rs. X.
So the amount borrowed = (first installment / (1 + rate of interest)) + (second installment / (1 + rate of interest)$^2$) + (third installment / (1 + rate of interest)$^3$).
As all the installments are the same, the first installment = second installment = third installment = X.
Now substitute the values we have,
$\Rightarrow 10000 = \dfrac{X}{{\left( {1 + \dfrac{8}{{100}}} \right)}} + \dfrac{X}{{{{\left( {1 + \dfrac{8}{{100}}} \right)}^2}}} + \dfrac{X}{{{{\left( {1 + \dfrac{8}{{100}}} \right)}^3}}}$
Now simplify this equation we have,
$\Rightarrow 10000 = \dfrac{X}{{\left( {\dfrac{{108}}{{100}}} \right)}} + \dfrac{X}{{{{\left( {\dfrac{{108}}{{100}}} \right)}^2}}} + \dfrac{X}{{{{\left( {\dfrac{{108}}{{100}}} \right)}^3}}}$
$\Rightarrow 10000 = \dfrac{{25X}}{{27}} + \dfrac{{{{\left( {25} \right)}^2}X}}{{{{\left( {27} \right)}^2}}} + \dfrac{{{{\left( {25} \right)}^3}X}}{{{{\left( {27} \right)}^3}}}$
$\Rightarrow 10000 = X\left( {\dfrac{{25}}{{27}} + \dfrac{{{{\left( {25} \right)}^2}}}{{{{\left( {27} \right)}^2}}} + \dfrac{{{{\left( {25} \right)}^3}}}{{{{\left( {27} \right)}^3}}}} \right)$
$\Rightarrow 10000 = X\left( {\dfrac{{25{{\left( {27} \right)}^2} + {{\left( {25} \right)}^2}27 + {{\left( {25} \right)}^3}}}{{{{\left( {27} \right)}^3}}}} \right)$
$\Rightarrow 10000 = X\left( {\dfrac{{18225 + 16875 + 15625}}{{19683}}} \right)$
$\Rightarrow 10000 = X\left( {\dfrac{{50725}}{{19683}}} \right)$
$\Rightarrow X = \left( {\dfrac{{19683}}{{50725}}} \right)10000 = 3880.335$ Rs.
So this is the required installment he has to pay in each year such that the amount is paid in three equal annual installments.

Note: Whenever we face such types of questions if he has to pay a single amount in after three years then the amount he has to pay is equal to $P{\left( {1 + \dfrac{r}{{100}}} \right)^n}$, where P is the amount which he borrowed, r is the rate of interest and n is the period of time.