Suppose you have $\$ 2350$ in your savings account at the end of a certain period of time. You invested $\$ 2000$ at a $2.25 \%$ simple annual interest rate. How long, in years, was your money invested? State your result to the nearest hundredth of a year.
Solution
Step 1 ：Let's denote the principal amount (initial investment) as P, the interest as I, the annual interest rate (in decimal form) as R, and the time the money is invested for (in years) as T.
Step 2 ：The formula for simple interest is \(I = PRT\). We can rearrange this formula to solve for T: \(T = I / (PR)\).
Step 3 ：We know that the total amount in the savings account at the end of a certain period of time is $2350, and the initial investment was $2000. So, the interest I is $2350 - $2000 = $350.
Step 4 ：Substitute P = $2000, I = $350, and R = 2.25% = 0.0225 into the formula to find T.
Step 5 ：Calculate T: \(T = 350 / (2000 * 0.0225) = 7.777777777777778\).
Step 6 ：Round the result to the nearest hundredth of a year: \(T = 7.78\) years.
Step 7 ：Final Answer: The money was invested for approximately \(\boxed{7.78}\) years.
