An executive invests $24,000, some at 7% and some at 5% annual interest. If he receives an annual return of $1600, how much is invested at each rate?
Let amount invested at 7% be $x.
Amount invested at 5% = $(24000-x)
x*0.07+(24000-x)*0.05 = 1600
Solve for x.
To determine the amount invested at each rate, we can set up a system of equations.
Let's assume the executive invests x dollars at 7% and y dollars at 5%.
We know that the total amount invested is $24,000, so the first equation is:
x + y = 24,000 ----(Equation 1)
We also know that the annual return on the investment is $1600. The interest earned from the investment at 7% can be calculated as 0.07x, while the interest earned from the investment at 5% is 0.05y. Therefore, the second equation is:
0.07x + 0.05y = 1600 ----(Equation 2)
Now we have a system of two equations with two variables. By solving this system, we can find the values of x and y, which represent the amounts invested at each rate.
To solve this system of equations, we can use a method called substitution or elimination. Let's use the substitution method.
From Equation 1, we can solve for x:
x = 24,000 - y
Now, substitute this expression for x in Equation 2:
0.07(24,000 - y) + 0.05y = 1600
Simplify and solve for y:
1680 - 0.07y + 0.05y = 1600
0.02y = 1600 - 1680
0.02y = -80
y = -80 / 0.02
y = 4000
Now that we have found the value of y, substitute it back into Equation 1 to find x:
x + 4000 = 24,000
x = 24,000 - 4000
x = 20,000
Therefore, the executive invested $20,000 at 7% and $4,000 at 5%.