1. 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?

Question

1. 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?

Answer 1

Suppose x is invested at 10%, y is invested at 12% and z is invested at 16%. Here is what you know:

x + y + z = 25,000
0.10 x + 0.12 y + 0.16 z = 3200
0.16 z = 0.10 x + 0.12 y

You can immediatley write
2*(0.16 z) = 0.32z = 3200
So z = $10,000

Therefere x + y = $15,000.
You also know that
0.10x + 0.12 y = 1600

Now finish it. You have two equations in two unknowns left.

Answer 2

To solve this problem, we can use a system of equations.

Let's assume that the amount invested at 10 percent is x dollars, the amount invested at 12 percent is y dollars, and the amount invested at 16 percent is z dollars.

From the given information, we can set up the following equations:

Equation 1: x + y + z = $25,000 (since the total investment is $25,000)
Equation 2: 0.10x + 0.12y + 0.16z = $3,200 (since the total income is $3,200)

We also know that the income from the 16 percent investment is equal to the sum of the incomes from the other two investments. Mathematically, we can express this as:

Equation 3: 0.16z = 0.10x + 0.12y

Now, we can solve this system of equations to find the values of x, y, and z.

One possible approach is to use substitution:

1. Solve Equation 3 for x: x = (0.16z - 0.12y) / 0.10
2. Substitute this value of x into Equation 1: (0.16z - 0.12y) / 0.10 + y + z = $25,000
3. Simplify and rearrange the equation to isolate y: 1.6z - 1.2y + 10y + 10z = $250,000
Combine like terms: 12y + 11z = $250,000
4. Solve Equation 2: 0.10(0.16z - 0.12y) / 0.10 + 0.12y + 0.16z = $3,200
5. Simplify and rearrange the equation to isolate y: 0.16z - 0.12y + 0.12y + 0.16z = $3,200
Combine like terms: 0.32z = $3,200
Divide both sides by 0.32: z = $10,000
6. Substitute z = $10,000 into Equation 1: 12y + 11($10,000) = $250,000
Simplify: 12y + $110,000 = $250,000
Subtract $110,000 from both sides: 12y = $140,000
Divide both sides by 12: y = $11,666.67
7. Substitute y = $11,666.67 and z = $10,000 into Equation 1: x + $11,666.67 + $10,000 = $25,000
Simplify: x + $21,666.67 = $25,000
Subtract $21,666.67 from both sides: x = $3,333.33

Therefore, $3,333.33 is invested at 10 percent, $11,666.67 is invested at 12 percent, and $10,000 is invested at 16 percent.

Note: Since the investments are specified in dollars, the solution is given to the nearest cent.