Could you please answer this?
Investment. Part of $25,000 is invested at 10 percent, another part is invested at 12 percent, and a third part is invested at 16 percent. The total yearly income from these three investments is $3,200. Furthermore, the income from the 16 percent investment yields the same amount as the sum of the incomes from the other two investments. How much is invested at each rate?
Let A, B, C be the amounts invested at 10, 12, and 16 percent.
A*.10+B*.12+C*.16=3200
C*.16=A*.10+B*.12
A+B+C=25,000
Can you solve it now?
To solve this problem, we need to assign variables to the unknowns and use algebraic equations to represent the given information.
Let's start by assigning variables:
Let x be the amount invested at 10 percent.
Let y be the amount invested at 12 percent.
Let z be the amount invested at 16 percent.
From the problem, we know:
x + y + z = $25,000 (equation 1)
0.10x + 0.12y + 0.16z = $3,200 (equation 2)
0.16z = 0.10x + 0.12y (equation 3)
Now, we can solve these equations simultaneously to find the values of x, y, and z.
First, we need to solve equation 3 to relate the income from the 16 percent investment to the incomes from the other two investments. We can simplify equation 3 by multiplying it by 100 to get rid of the decimal points:
16z = 10x + 12y (equation 4)
Now we have a system of three equations with three variables:
x + y + z = $25,000 (equation 1)
0.10x + 0.12y + 0.16z = $3,200 (equation 2)
16z = 10x + 12y (equation 4)
There are several methods to solve this system of equations, such as substitution or elimination. Let's use the substitution method for this example:
From equation 4, we can express z in terms of x and y:
z = (10/16)x + (12/16)y = (5/8)x + (3/4)y (equation 5)
Now, substitute equations 5 and 1 into equation 2:
0.10x + 0.12y + 0.16[(5/8)x + (3/4)y] = $3,200
Simplify the equation by multiplying through by 100:
10x + 12y + 16[(5/8)x + (3/4)y] = $320,000
Distribute the 5/8 and 3/4:
10x + 12y + (5/8)x + (3/4)y = $320,000
Combine like terms:
(85/8)x + (15/4)y = $320,000
Multiply through by 8 to eliminate the fractions:
85x + 30y = $2,560,000 (equation 6)
Now, we have a system of two linear equations:
(85/8)x + (15/4)y = $320,000 (equation 6)
x + y = $25,000 (equation 1)
You can then solve this system of equations using any method you prefer, such as substitution or elimination, to find the values of x and y. Once you have x and y, you can substitute them back into equation 1 to find z, and therefore, determine how much is invested at each interest rate.