mr. and mrs. garcia have a total of 100000$ to be invested in stocks, bonds, and a money market account. the stocks have a rate of return of 12% per year, while the bonds and the money market account pays 8% per year and 4% per year, respectively. the garcias 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 10000$ from their investments ?
amount invested in stocks ---- s
amount invested in bonds ---- b
amount invested in money market = .2s + .1b
s + b + .2s + .1b = 100,000
12s + 11b = 1,000,000 **
.12s + .08b + .04(.2s+.1b) = 10,000
.128s + .084b = 10,000
128s + 84b = 10,000,000
32s + 21b = 2,500,000 ***
solve ** and *** using your favourite method, it comes out nicely
To solve this problem, we can set up an equation based on the given information.
Let's denote the amount invested in stocks as S, the amount invested in bonds as B, and the amount invested in the money market account as M.
From the given information, we can determine certain relationships between these investments:
1. The total amount invested: S + B + M = $100,000
2. The annual income from stocks: 0.12S
3. The annual income from bonds: 0.08B
4. The annual income from the money market account: 0.04M
Based on the stipulation given, we know 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:
M = 0.2S + 0.1B
We are also told that the Garcias require an annual income of $10,000 from their investments. Therefore, the total annual income from all their investments should equal $10,000:
0.12S + 0.08B + 0.04M = $10,000
Now we can solve this system of equations to find the allocation of their resources.
First, we can substitute M in terms of S and B from the stipulation equation into the total investment equation:
S + B + (0.2S + 0.1B) = $100,000
Simplifying this equation:
1.2S + 0.2B = $100,000
Next, we substitute M in terms of S and B into the equation for total annual income:
0.12S + 0.08B + 0.04(0.2S + 0.1B) = $10,000
Simplifying this equation:
0.12S + 0.08B + 0.008S + 0.004B = $10,000
Combining like terms:
0.128S + 0.084B = $10,000
Now we have a system of two equations with two unknowns:
1.2S + 0.2B = $100,000 (Equation 1)
0.128S + 0.084B = $10,000 (Equation 2)
We can solve this system using various methods such as substitution or elimination. For simplicity, we'll use the substitution method.
We'll solve Equation 1 for S:
1.2S = $100,000 - 0.2B
S = ($100,000 - 0.2B) / 1.2
Substituting this value of S into Equation 2:
0.128[($100,000 - 0.2B) / 1.2] + 0.084B = $10,000
Simplifying and rearranging terms:
0.128($100,000 - 0.2B) + 0.084B = $12,000
12,800 - 25B + 8.4B = 12,000
-16.6B = -800
Dividing both sides by -16.6:
B = $48,192.77 (rounded to the nearest cent)
Substituting this back into Equation 1 to solve for S:
1.2S + 0.2($48,192.77) = $100,000
1.2S + $9,638.55 = $100,000
1.2S = $90,361.45
S = $75,301.21 (rounded to the nearest cent)
Finally, we can find the value of M using the stipulation equation:
M = 0.2S + 0.1B
M = 0.2($75,301.21) + 0.1($48,192.77)
M = $15,060.24 + $4,819.28
M = $19,879.52 (rounded to the nearest cent)
Therefore, the Garcias should allocate approximately:
$75,301.21 in stocks, $48,192.77 in bonds, and $19,879.52 in the money market account to meet their requirements of an annual income of $10,000.
To find how the Garcias should allocate their resources, we need to set up a system of equations using the given information.
Let's assign the following variables:
- Amount invested in stocks: S
- Amount invested in bonds: B
- Amount invested in the money market account: M
According to the stipulation, 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. Mathematically, this can be written as:
M = 0.20S + 0.10B Equation 1
The total amount invested should equal $100,000:
S + B + M = $100,000 Equation 2
The Garcias also require an annual income of $10,000 from their investments. We can calculate the income from each type of investment using their respective rates of return:
Income from stocks = 0.12S
Income from bonds = 0.08B
Income from money market account = 0.04M
The total annual income should equal $10,000:
0.12S + 0.08B + 0.04M = $10,000 Equation 3
Now we have a system of three equations (Equations 1, 2, and 3) that we can solve to find the values of S, B, and M.
1. Substitute Equation 1 into Equation 2 to eliminate M:
S + B + (0.20S + 0.10B) = $100,000
1.20S + 0.10B = $100,000
2. Solve Equation 3 for M:
0.04M = $10,000 - 0.12S - 0.08B
M = ($10,000 - 0.12S - 0.08B) / 0.04
3. Substitute the value of M from Step 2 into Equation 1:
($10,000 - 0.12S - 0.08B) / 0.04 = 0.20S + 0.10B
$10,000 - 0.12S - 0.08B = 0.20S + 0.10B
4. Rearrange the equation to isolate S:
0.20S + 0.10B - $10,000 = 0.12S + 0.08B
0.08S + 0.02B = $10,000
5. Substitute the value of M from Step 2 into Equation 2:
S + B + ($10,000 - 0.12S - 0.08B) / 0.04 = $100,000
0.04S + 0.04B + ($10,000 - 0.12S - 0.08B) = $100,000
0.04S + 0.04B - 0.12S - 0.08B = $100,000 - $10,000
6. Simplify and combine like terms in equations from Steps 4 and 5:
0.08S - 0.06S + 0.04B - 0.08B = $90,000
0.02S - 0.04B = $90,000
We now have two equations with two variables. We can solve this system of equations to find the values of S and B.