Hello there. To solve this question, we'll have to remember some properties about investments.
Knowing Brent borrowed $2000 from his brother and will pay him the same amount of interest that he would have received if the money had been invested at 2.75% compounded quarterly, we need to find how much money will Brent repay his brother.
First, we have to remember the formula to find the amount a principal value P will become after t years, invested at an interesting rate r, compounded quarterly.
It is given by:
![A=P\cdot\mleft(1+(r)/(4)\mright)^(4t)](https://img.qammunity.org/2023/formulas/mathematics/college/11ofnqvht8vfjckfdvqljkjvlhdk8qx12m.png)
Plugging in P = 2000, r = 2.75% (converted in decimals, divide by 100) and t = 3, we get:
![A=2000\cdot\mleft(1+(0.0275)/(4)\mright)^(4\cdot3)=2000\cdot\mleft(1+0.006875\mright)^(12)](https://img.qammunity.org/2023/formulas/mathematics/college/4vtb5revhg7hvscqspchds814iex08w92v.png)
Add the values inside parenthesis and calculate the power (with the help of a calculator)
![A=2000\cdot1.006875^(12)\approx2000\cdot1.0857=2171.38](https://img.qammunity.org/2023/formulas/mathematics/college/9uxym1tqsfqw3r02gt096swdqgredq4jwh.png)
This is the amount Brent had to repay his brother.