Final answer:
To find the amounts invested at different interest rates, we formulate a system of equations: one representing the total amount invested and another representing the total annual interest earned. We then solve these algebraically, taking into account simple interest calculations. Compound interest was mentioned for additional context but is not needed for solving the system.
Step-by-step explanation:
To find the amount invested in each fund, we will establish a system of equations based on the total investment and the interest earned from each investment. Let the amounts invested in the first and second funds be $x and $y, respectively. The total invested is $13,000, so we have the equation $x + $y = $13,000.
Let's assume the annual interest rates for the first and second funds are r1 and r2, respectively, and let the total interest earned be $I. The interest from each fund would be $x × r1 and $y × r2. Adding these interests gives the total annual interest, which is $I. Thus, the second equation is $x × r1 + $y × r2 = $I.
With two equations, we can solve for $x and $y using common algebraic methods such as substitution or elimination. An example to better understand simple interest is, if you deposit $100 at a simple interest rate of 5% held for three years, the total interest earned will be: $100 × 0.05 × 3 = $15.
Compound interest, however, takes into account the interest earned on both the principal and the accumulated interest over time. While this was not part of the original question, it's useful to know that compound interest can substantially increase the total amount earned on an investment over a longer period.