Mr. and Mrs. Garcia have a total of $ 100,000 to be invested in stocks, bonds, and a money market account.The stocks have a rate of return of 12%/ year, while the bonds and the money market account pay 8% and 4%/ year, respec-tively. They have stipulated that the amount invested in the money market account should be equal to the sum of 20% of the amount invested in stocks and 10% of the amount invested in bonds. How should the Garcias allocate their resources if they require an annual income of $ 10,000 from their investments?

huy

Let's set up an equation for the total annual income from the investments.

Let x be the amount invested in stocks.
The annual income from stocks would be 12% of x, or 0.12x.

The amount invested in bonds would be (100,000 - x - 0.2x - 0.1(100,000 - x)), or (100,000 - x - 0.2x - 10,000 + 0.1x).
Simplifying this expression, we get (90,000 - 0.1x).
The annual income from bonds would be 8% of this amount, or 0.08(90,000 - 0.1x).

The amount invested in the money market account would be (0.2x + 0.1(90,000 - 0.1x)), or (0.2x + 9,000 - 0.01x).
Simplifying this expression, we get (9,000 + 0.19x).
The annual income from the money market account would be 4% of this amount, or 0.04(9,000 + 0.19x).

To find the total annual income, we add up the annual income from each investment:
0.12x + 0.08(90,000 - 0.1x) + 0.04(9,000 + 0.19x) = 10,000

Now, we can solve this equation to find the value of x.

To determine how the Garcias should allocate their resources, we need to find the amounts invested in stocks, bonds, and the money market account.

Let's start by assigning variables to the amounts invested in each category:
- Let S be the amount invested in stocks.
- Let B be the amount invested in bonds.
- Let M be the amount invested in the money market account.

According to the problem, we know the following:
1. The total amount invested is $100,000, so we have the equation S + B + M = $100,000.
2. The amount invested in the money market account should be equal to 20% of the amount invested in stocks plus 10% of the amount invested in bonds. Mathematically, this can be represented as M = 0.2S + 0.1B.

Now, let's determine the annual income from each investment:
- The stocks have a rate of return of 12% per year, so the annual income from stocks is 12% of the amount invested, which is 0.12S.
- The bonds have a rate of return of 8% per year, so the annual income from bonds is 8% of the amount invested, which is 0.08B.
- The money market account has a rate of return of 4% per year, so the annual income from the money market account is 4% of the amount invested, which is 0.04M.

The Garcias require an annual income of $10,000 from their investments, so we have the equation 0.12S + 0.08B + 0.04M = $10,000.

Now, we can solve the system of equations:

1. S + B + M = $100,000
2. M = 0.2S + 0.1B
3. 0.12S + 0.08B + 0.04M = $10,000

We can use substitution or elimination to solve for the variables. Let's use substitution:

From equation 2, we have M = 0.2S + 0.1B.

Substituting this into equation 3, we get:
0.12S + 0.08B + 0.04(0.2S + 0.1B) = $10,000

Simplifying the equation:
0.12S + 0.08B + 0.008S + 0.004B = $10,000
0.128S + 0.084B = $10,000

Now, we can rewrite equation 1 as:
S + B = $100,000 - M

Substituting the value of M from equation 2, we get:
S + B = $100,000 - (0.2S + 0.1B)
S + B = $100,000 - 0.2S - 0.1B
1.2S + 1.1B = $100,000

So, we have two equations:
1.2S + 1.1B = $100,000
0.128S + 0.084B = $10,000

Now, you can solve this system of equations using any method of your choice, such as substitution or elimination, to find the amount invested in stocks (S), bonds (B), and the money market account (M).