Q63. - Algebra 2

Question

Suppose you deposit a principal amount of P dollars in a bank account that pays compound interest. If the annual interest rate is r (expressed as a decimal) and the bank makes interest payments n times every year, the amount of money A you would have after t years is given by $A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$

Write an equation giving the amount of money you would have after t years if you deposit $1000 into an account paying 4% annual interest compounded quarterly (four times per year).

Step-by-Step Solution

Answer

The equation giving the amount of money after t years if$1000 is deposited into an account paying 4% annual interest compounded quarterly (four times per year) is $A\left(t\right)=1000{\left(1.01\right)}^{4t}\mathrm{\pi }$ .

Step 1. Given Information.

Given, if a principal amount of P dollars is deposited in a bank account that pays compound interest and the annual interest rate is r (expressed as a decimal) and the bank makes interest payments n times every year, the amount of money A after t years is given by 

$A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$

An equation giving the amount of money after t years if $1000 is deposited into an account paying 4% annual interest compounded quarterly (four times per year) is to be determined.

Step 2. Explanation .

From the given information, the principal amount is $1000.

Hence $P=1000$

The rate of interest per annum in decimal form is given as:

$\begin{array}{l}r=4\%\\ r=\frac{4}{100}\\ r=0.04\end{array}$

The number of compounding per year is four times per year i.e., $n=4$

Plugging the values in the equation:

$\begin{array}{l}A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}\\ A\left(t\right)=1000{\left(1+\frac{0.04}{4}\right)}^{4t}\\ A\left(t\right)=1000{\left(1+0.01\right)}^{4t}\\ A\left(t\right)=1000{\left(1.01\right)}^{4t}\end{array}$

So, the equation is $A\left(t\right)=1000{\left(1.01\right)}^{4t}$

Step 3. Conclusion .

Hence the equation giving the amount of money after t years if $1000 is deposited into an account paying 4% annual interest compounded quarterly (four times per year) is $A\left(t\right)=1000{\left(1.01\right)}^{4t}$