How Can Matrices Solve Investment and Chemistry Problems?

[DoctorB2B]

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

 

[Clairefucious]

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.

 

[HallsofIvy]

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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[DoctorB2B]

Thank you very much ... you're the greatest!