Matrices Word Problem help needed

In summary, the retired couple should invest $150,000 in three areas: a high-risk stock with an expected rate of return of 15%, a low risk stock with an expected rate of return of 10%, and government bonds at 8% return. Twice as much money should be invested in the low-risk stock as the high-risk stock and the remainder should be used to buy bonds.
  • #1
DoctorB2B
15
0

Homework Statement


Any help would be greatly appreciated! I have to use matrices to figure these out.

1. A retired couple wishes to invest $150,000, diversifying the investment in three areas: a high-risk stock with an expected rate of return of 15%, a low risk stock with an expected rate of return of 10%, and government bonds at 8% return. To protect their investment, they wish wish to place twice as much money in the low-risk stock as the high risk stock and use the remainder to buy bonds. How should the money be allocated per investment to yield a total profit of $17,500 return on their investment?

2. Three solutions contain a certain acid. The first contains 10% acid, the second 30% , and the third 50%. A chemist wishes to use all three solutions to obtain a 40-liter mixture containing 35% acid. If the chemist wants to use twice as much of the 50% solution as of the 30% solution, how many liters of each solution should be used?

All I need help with is getting these bad boys started and I can go from there.

Homework Equations





The Attempt at a Solution

 
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  • #2
hmm, I'm not sure how to represent matrices in this window. I have a stab at helping out anyway. This is a general formula.

| percentage A | |amount of A |
| | | |
| percentage B | . |amount of B | = pA*aA + pB*aB + pC*aC
| | | |
| percentage C | |amount of C|


Remember, the amount of something multiplied by how much that something contains of a desired substance gives you how much of the desired substance you have in total.
 
  • #3
DoctorB2B said:

Homework Statement


Any help would be greatly appreciated! I have to use matrices to figure these out.

1. A retired couple wishes to invest $150,000, diversifying the investment in three areas: a high-risk stock with an expected rate of return of 15%, a low risk stock with an expected rate of return of 10%, and government bonds at 8% return. To protect their investment, they wish wish to place twice as much money in the low-risk stock as the high risk stock and use the remainder to buy bonds. How should the money be allocated per investment to yield a total profit of $17,500 return on their investment?
Let the amount invested at 15% be x, the amount invested at 10% be y, and the amount invested at 8% be z. Then, "A retired couple wishes to invest $150,000" so x+ y+ z= 150000. "To protect their investment, they wish wish to place twice as much money in the low-risk stock as the high risk stock and use the remainder to buy bonds" so y= 2x or 2x- y= 0. "How should the money be allocated per investment to yield a total profit of $17,500 return," so .15x+ .10y+ .08z= 17500. Do you know how to write the equations
x+ y+ z= 150000
2x- y = 0
.15x+ .10y+ .08z= 17500
as a matrix multiplication?

2. Three solutions contain a certain acid. The first contains 10% acid, the second 30% , and the third 50%. A chemist wishes to use all three solutions to obtain a 40-liter mixture containing 35% acid. If the chemist wants to use twice as much of the 50% solution as of the 30% solution, how many liters of each solution should be used?
Let x be the amount of the 10% acid solution, y the amount of the 30% solution, and z the mount of the 50% acid solution.

"A chemist wishes to use all three solutions to obtain a 40-liter mixture containing 35% acidA chemist wishes to use all three solutions to obtain a 40-liter mixture containing 35% acid" tells us two things: x+ y+ z= 40 and .10x+ .30y+ .50z= .35(40). "the chemist wants to use twice as much of the 50% solution as of the 30% solution" tells us z= 2y or 2y- z= 0.

Can you write
x+ y+ z= 40
.10+ .30y+ .50z= .35(40)= 14
2y- z= 0
as a matrix multipllication?

[quolte]All I need help with is getting these bad boys started and I can go from there.

Homework Equations





The Attempt at a Solution

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  • #4
Thank you very much ... you're the greatest!
 

1. What are matrices and why are they important in word problems?

Matrices are rectangular arrays of numbers or variables. They are important in word problems because they allow us to represent and solve systems of equations, which often occur in real-world scenarios.

2. How do I set up a matrix for a word problem?

To set up a matrix for a word problem, you first need to identify the variables and their corresponding equations in the problem. Then, you can arrange the coefficients of each variable in a row of the matrix, with the constant term in the last column. Repeat this process for each equation in the problem.

3. What is the difference between a matrix and a system of equations?

A matrix is a visual representation of a system of equations, while a system of equations is a set of equations that have a common solution. Matrices allow us to solve systems of equations more efficiently by using techniques such as row operations.

4. How do I solve a word problem using matrices?

To solve a word problem using matrices, you can use techniques such as Gaussian elimination or Cramer's rule. These methods involve manipulating the matrix to eliminate variables and solve for the remaining variables, ultimately leading to the solution of the problem.

5. Can matrices be used for all types of word problems?

While matrices are a useful tool for solving systems of equations, they may not be applicable to all types of word problems. Some problems may require other mathematical techniques or may not have a solution at all. It is important to carefully read and understand the problem before determining if matrices can be used for the solution.

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