Mon Feb 3 webassignnet Need Help? Read it Wakehlt SUBMIT ANSWER [-/11.12 Points] DETAILS MY NOTES TANFIN12 2.3.041. ASK YOUR TEACHER 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 $$ 8 \% / $$ year and $$ 4 \% / $$ year, respectively. The Garcias have stipulated that the amount invested in stocks should be equal to the sum of the amount invested in bonds and 3 times the amount invested in the money market account. How should the Garcias allocate their resources if they require an annual income of $$ \$ 10,000 $$ from their investments? Give two speciffic options. (Let $$ x_{1}, y_{1} $$, and $$ z_{1} $$ refer to one option for investing money in stocks, bonds, and the money market account respectively. Let $$ x_{2}, y_{2} $$, and $$ z_{2} $$ refer to a second option for investing money in stocks, bonds, and the money market account respectively.) $$ \left\{\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right)\right\}=\{\square $$ Need Help? Read It Watch it SUBMIT ANSWER

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Sure, I can help you work through this investment problem step by step. Let's break down the information and set up the necessary equations to find the two possible investment options for Mr. and Mrs. Garcia.

### **Given:**
- **Total Investment:** \$100,000
- **Investment Types and Rates of Return:**
  - **Stocks:** 12% per year
  - **Bonds:** 8% per year
  - **Money Market Account:** 4% per year
- **Constraint:** The amount invested in stocks should be equal to the sum of the amount invested in bonds and three times the amount invested in the money market account.
- **Desired Annual Income:** \$10,000

### **Let:**
- $$ x $$ = Amount invested in **Stocks**
- $$ y $$ = Amount invested in **Bonds**
- $$ z $$ = Amount invested in **Money Market Account**

### **Step 1: Set Up the Equations**

Based on the information provided, we can establish the following equations:

1. **Total Investment:**
   $$
   x + y + z = 100,000 \quad \text{(Equation 1)}
   $$

2. **Investment Constraint:**
   $$
   x = y + 3z \quad \text{(Equation 2)}
   $$

3. **Desired Annual Income:**
   $$
   0.12x + 0.08y + 0.04z = 10,000 \quad \text{(Equation 3)}
   $$

### **Step 2: Substitute Equation 2 into Equations 1 and 3**

From **Equation 2**:
$$
x = y + 3z
$$

**Substitute $$ x $$ in Equation 1:**
$$
(y + 3z) + y + z = 100,000 \
2y + 4z = 100,000 \quad \text{(Simplified Equation 1)}
$$

**Substitute $$ x $$ in Equation 3:**
$$
0.12(y + 3z) + 0.08y + 0.04z = 10,000 \
0.12y + 0.36z + 0.08y + 0.04z = 10,000 \
0.20y + 0.40z = 10,000 \
y + 2z = 50,000 \quad \text{(Simplified Equation 3)}
$$

### **Step 3: Solve the System of Equations**

Now, we have two simplified equations:
1. $$ 2y + 4z = 100,000 $$
2. $$ y + 2z = 50,000 $$

Notice that **Equation 2** is simply **Equation 1** divided by 2, which means there are infinitely many solutions along the line defined by $$ y + 2z = 50,000 $$.

To find specific investment options, we'll assign values to $$ z $$ within logical bounds (i.e., $$ z \geq 0 $$ and $$ y \geq 0 $$).

### **Option 1: $$ z = \$0 $$**
- **Calculate $$ y $$:**
  $$
  y + 2(0) = 50,000 \
  y = 50,000
  $$
- **Calculate $$ x $$:**
  $$
  x = y + 3z = 50,000 + 3(0) = 50,000
  $$
  
**Investment Allocation:**
$$
(x_1, y_1, z_1) = (\$50,000, \$50,000, \$0)
$$

### **Option 2: $$ z = \$10,000 $$**
- **Calculate $$ y $$:**
  $$
  y + 2(10,000) = 50,000 \
  y = 50,000 - 20,000 = 30,000
  $$
- **Calculate $$ x $$:**
  $$
  x = y + 3z = 30,000 + 3(10,000) = 60,000
  $$
  
**Investment Allocation:**
$$
(x_2, y_2, z_2) = (\$60,000, \$30,000, \$10,000)
$$

### **Summary of Investment Options:**
$$
\left\{\left(\$50,000, \$50,000, \$0\right), \left(\$60,000, \$30,000, \$10,000\right)\right\}
$$

These two options satisfy all the given constraints:
- The total investment sums to \$100,000.
- The investment in stocks equals the sum of bonds and three times the money market account.
- The annual income from the investments is \$10,000.

You can choose different values for $$ z $$ (between \$0 and \$25,000) to generate other valid investment options if needed.
The Garcias can allocate their \$100,000 investment in the following two ways to achieve an annual income of \$10,000: 1. **Option 1:** - **Stocks:** \$50,000 - **Bonds:** \$50,000 - **Money Market Account:** \$0 2. **Option 2:** - **Stocks:** \$60,000 - **Bonds:** \$30,000 - **Money Market Account:** \$10,000 These allocations satisfy all the given constraints.

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