Question

Calculate the amount if Rs 18,000 is invested at \[15\% \] p.a. compounded annually for 3 years.

Answer 1

Hint: Here, we need to find the amount for the given principal. We will use the formula for amount when a principal is compounded for a period of time. Then, we will simplify the expression to find the required amount. Amount is the money obtained after adding the principal amount to the interest incurred during a particular period of time.

Formula used:
The amount \[A\] of an investment after \[t\] years is given by \[A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}\], where \[P\] is the amount invested, \[n\] is the number of compounding periods in a year and \[r\] is the interest rate compounded annually.

Complete step-by-step answer:
As the sum is compounded annually, the number of compounding periods in a year is 1.
Substituting \[n = 1\], \[t = 3\], \[P = 18,000\] and \[R = 15\% \] in the formula \[A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}\], we get
\[A = 18000{\left( {1 + \dfrac{{15\% }}{1}} \right)^{1 \times 3}}\]
Simplifying the expression, we get
\[ \Rightarrow A = 18000{\left( {1 + 15\% } \right)^3}\]
Rewriting the percentage as a fraction, we get
\[ \Rightarrow A = 18000{\left( {1 + \dfrac{{15}}{{100}}} \right)^3}\]
Taking the L.C.M. in the parentheses, we get
\[ \Rightarrow A = 18000{\left( {\dfrac{{100 + 15}}{{100}}} \right)^3}\]
Adding the terms in the numerator, we get
\[ \Rightarrow A = 18000{\left( {\dfrac{{115}}{{100}}} \right)^3}\]
Both 115 and 100 are divisible by 5.
Simplifying the fraction in the parentheses, we get
\[ \Rightarrow A = 18000{\left( {\dfrac{{23}}{{20}}} \right)^3}\]
The expression \[{\left( {\dfrac{{23}}{{20}}} \right)^3}\] can be written as the product of \[\dfrac{{23}}{{20}}\], \[\dfrac{{23}}{{20}}\], and \[\dfrac{{23}}{{20}}\].
Therefore, we can rewrite the equation as
\[ \Rightarrow A = 18000 \times \dfrac{{23}}{{20}} \times \dfrac{{23}}{{20}} \times \dfrac{{23}}{{20}}\]
Simplifying the expression by cancelling the common factors, we get
\[ \Rightarrow A = 9 \times \dfrac{{23}}{1} \times \dfrac{{23}}{2} \times \dfrac{{23}}{2}\]
Multiplying the terms in the expression, we get
\[ \Rightarrow A = \dfrac{{109503}}{4}\]
Writing the amount in decimal form, we get
\[ \Rightarrow A = 27375.75\]
Therefore, we get the required amount as \[{\rm{Rs}}.27375.75\] in decimal form.


Note: We can verify our answer by calculating simple interest on the amounts at the end of each year.
The simple interest is given by \[S.I. = \dfrac{{P \times R \times T}}{{100}}\], where \[P\] is the principal amount, \[R\] is the rate of interest, and \[T\] is the time period.
Therefore, we get
Simple interest on Rs. 18,000 for the first year \[ = \dfrac{{18000 \times 15 \times 1}}{{100}} = 2700\]
The amount is the sum of the principal amount and the interest. It is given by the formula \[A = P + S.I.\], where \[P\] is the principal amount and \[S.I.\] is the simple interest.
Therefore, we get
Amount at the end of first year \[ = 18000 + 2700 = 20700\]
The simple interest for the second year will be calculated on Rs. 20700.
Simple interest on Rs. 20,700 for the second year \[ = \dfrac{{20700 \times 15 \times 1}}{{100}} = 3105\]
Amount at the end of second year \[ = 20700 + 3105 = 23805\]
The simple interest for the third year will be calculated on Rs. 23,805.
Simple interest on Rs. 23,805 for the third year \[ = \dfrac{{23805 \times 15 \times 1}}{{100}} = 3570.75\]
Amount at the end of third year \[ = 23805 + 3570.75 = 27375.75\]
Therefore, we have verified using simple interest that the amount is \[{\rm{Rs}}.27375.75\] in decimal form.