An executive invests $25,000, some at 8% and some at 4% annual interest. If he receives an annual return of $1600, how much is invested at each rate?
To solve this problem, we can use a system of equations. Let's assume the executive invests $x at 8% interest and $(25,000 - x) at 4% interest.
We can now set up the equations based on the given information:
Equation 1: The amount invested at 8% interest multiplied by the interest rate (0.08) should equal the interest earned at that rate: 0.08x = (Interest at 8%)
Equation 2: The amount invested at 4% interest multiplied by the interest rate (0.04) should equal the interest earned at that rate: 0.04(25,000 - x) = (Interest at 4%)
Since we know the annual return is $1600, we can create a third equation:
Equation 3: The sum of interests at both rates should equal the annual return: (Interest at 8%) + (Interest at 4%) = $1600
Now, let's solve this system of equations:
First, substitute Equation 1 and Equation 2 into Equation 3:
0.08x + 0.04(25,000 - x) = 1600
Expand the equation:
0.08x + 1000 - 0.04x = 1600
Combine like terms:
0.04x = 600
Divide by 0.04 on both sides:
x = 600 / 0.04
x = 15,000
So, $15,000 is invested at 8% interest.
Now, substitute the value of x into the initial equation:
25,000 - x = 25,000 - 15,000 = 10,000
Therefore, $10,000 is invested at 4% interest.
In summary, the executive invested $15,000 at 8% annual interest and $10,000 at 4% annual interest.