SOLUTION: Betsy, a recent retiree, requires $6000 per year in extra income. She has $50,000 to invest and can invest in B-rated bonds paying 15% per year or in a certificate of deposit (CD)

Question 1199487: Betsy, a recent retiree, requires $6000 per year in extra income. She has $50,000 to invest and can invest in B-rated bonds paying 15% per year or in a certificate of deposit (CD) paying 7% per year. How much money should be invested in each to realize exactly $6000 in interest per year.

Let me see.

The equation I seek should show the following:

Bonds + CD = Total

Let x = amount of money that should be invested in each account

Bonds = 0.07x

CD = 0.15(50,000 - x)

Here is my equation:

0.07x + 0.15(50,000 - x) = 6000

Is this the correct set up?
Found 3 solutions by ikleyn, greenestamps, math_tutor2020:
Answer by ikleyn(52792)   (Show Source): You can put this solution on YOUR website!
.

The logic is broken in your solution.

This phrase is nonsense

"Let x = amount of money that should be invested in each account".

x   CAN NOT   be an amount invested in each account.

It only can be an amount invested in some ONE of the two accounts.

So, your solution must be re-written accordingly from the very beginning.

Also, on the way, percents are incorrectly connected with the amounts.

Be more attentive and try to keep right logic.

//////////////////

If you want to see many other similar solved problem (as your TEMPLATES), you can find them in the lesson
    - Using systems of equations to solve problems on investment
in this site.

You will find there different approaches (using one equation or a system of two equations in two unknowns), as well as
different methods of solution to the equations (Substitution, Elimination).

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic "Systems of two linear equations in two unknowns".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

Your equation is fine; when you solve it correctly you will have what you need to answer the question.

But you have not defined your variables and expressions correctly.

Your equation clearly shows that you are using x for the amount invested at 7% and (50000-x) for the amount invested at 15%. That is consistent with the given information.

But your opening statement (presumably hastily and carelessly written) says that x is the amount invested in each account. The given information certainly does not tell you that.

For beginning algebra students especially, it is necessary to start your setup with clear and correct definitions of the variables and expressions you are going to use. Even an experienced student, when working a long and involved problem, can get himself into trouble if he starts with sloppily written definitions which in fact are not correct and precise.

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Added after response from student....

Here is what you wrote in your post:

(1) The equation I seek should show the following:

(2) Bonds + CD = Total

(3) Let x = amount of money that should be invested in each account

(4) Bonds = 0.07x

(5) CD = 0.15(50,000 - x)

(6) Here is my equation:

(7) 0.07x + 0.15(50,000 - x) = 6000

(8) Is this the correct set up?

It appears to me that you need to take the time to write out CLEAR AND PRECISE definitions of the expressions you are going to use. In this case, you ended up with a (mostly) correct equation; but what you show before the equation is "shorthand" that doesn't tell what you REALLY mean.

Using what you show in your work as an outline, your actual work should look something like this:

(2) interest from bonds, plus interest from CD, equals total interest
[NOT just "bonds + CD = total"]

(3a) let x = amount invested in bonds
(3b) then 50,000-x = amount invested in CD

Your "Let x = amount of money that should be invested in each account" does NOT tell what you are really doing!

(4) interest from bonds (15% of x): 0.15x
[NOT just "bonds = ..."]
(5) interest from CD (7% of (50,000-x)): .07(50000-x)
[NOT just "CD = ..."]

(Note that, as the other tutors who responded noted (and I overlooked), your work has the interest rates from the two investments switched.)

(7) Now, using (4) and (5), you are ready to write the equation you said you were going to in (2), showing that the total interest should be $6000:

0.15x + 0.07(50,000-x) = 6000

Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

You have the bonds and CD interest rates mixed up.
Also, your phrasing to set up x is incorrect.

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This is one way to set up the equation.

x = amount invested in the bonds
50,000 - x = amount invested in the CD
The two amounts add to $50,000

The bonds account pays 15% return
0.15x = amount earned from the bonds

While the CD gives 7%
0.07(50,000-x) = amount earned from the CD

Add up those items to get the total amount earned
0.15x + 0.07(50,000-x) = 6000
it is set equal to 6000 since this is the desired amount Betsy wants to earn per year in extra income.

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An alternative equation.

y = amount invested in the CD
50,000 - y = amount invested in the bonds
I'm using another variable to avoid confusion.

The bonds account pays 15% return
0.15(50,000-y) = amount earned from the bonds

While the CD gives 7%
0.07y = amount earned from the CD

Add up those items to get the total amount earned
0.15(50,000-y) + 0.07y = 6000

This matches what you have, but you used x instead of y.

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