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Question

Here is the problem Please show work not just answer ( I have more problems like these and I don't know how to set up equation or anything. Thanks

Your grandmother needs your help. She has $50,000 to invest. Part of this money is to be invested in non insured bonds paying 15% annual interest. The rest of this money is to be invested in a government-insured certificate of deposit paying 7% annual interest. She told you that she requires $6000 per year in extra income from both of these investments. How much money should be placed in each investment?

Answer 1

The answer above is NOT correct, although the methof is substantially correct.  6000 - 7500 = -1500

The correct answer, however, involves an inequality.  I assume grandma would not mind getting more than $6000, as long as she gets at least $6000.

Set up two equations/inequalities:

let x = amount to invest @ 15%

let y = amount to invest @ 7%

x + y = 50000  ( the amount she has to invest)

so, you can solve for x or y (your choice)

x = 50000 - y  (my choice)

then, .15x + .07y >or= 6000

In other words, the interest from x and the interest from y must be greater than or equal to $6000.  Grandma will be able to get by...

substitute for x: .15(50000 - y) + .07y >or= 6000

distribute and collect like terms=>

7500 - .08y >or= 6000

-.08y >or= -1500

solve for y:  remember that multiplication (or division) by a negative reverses the direction of the inequality

y < or = 18750

x > or =  31250

If grandma invests at least 31250 in 15% bonds and no more than 18750 in 7% bonds, she is cool!

Answer 2

We need to find out how much she should split her $50,000 in amount X and Y. X being the amount of money invested into non insured bonds. Y being money invested into government insured cert.

So X + Y = 50,000

We know that grandma will get 15% of X each year, and 15% of X can be expressed as 0.15X. Similarly we have 0.07Y for 7% of Y for each year. We also know that grandma wants to make a total of $6000 each year just from interest alone, meaning the total of what she will get from her interest.

So 0.15X + 0.07Y = 6000

Now we have two equations and two variables. Solve for X and Y.

X + Y = 50000

We can solve for X by moving Y to the other side of the equation. In order to do this you will need to minus Y from both sides of the equation.

X + Y - Y = 50000 - Y

obviously Y - Y cancels out to 0 so you get:

X = 50000 - Y

Now we have an equation showing us what X is equal to. In order to figure out what X is obviously we need to find out what Y is. We can do this by substituting this equation into:

0.15X + 0.07Y = 6000

You will get:

0.15(50000 - Y) + 0.07Y = 6000

In order to solve this equation you will need to simplify it by opening up the brackets, which we will look at it by itself.

0.15(50000 - Y)

In order to open up the brackets you will need to times whatever the number is outside of the bracket with each individual number and variable within it.

0.15 x 50000 - 0.15Y

So 7500 - 0.15Y

put this back to 0.15(50000 - Y) + 0.07Y = 6000 you get:

7500 - 0.15Y + 0.07Y = 6000

Always combine the same variables within the same equation before continuing to solve anything else.

So -0.15Y + 0.07Y = -0.08Y

You get -0.08Y by solving

-0.15 + 0.07

(Note the minus in front of 0.15 making it a negative number. I hope you understand the concepts of negative and positive numbers. If you don't it is very important for you to ask someone to explain it to you. Ask a friend or teacher. By just getting this simple part wrong can give you a totally different answer. It is a very common mistake when solving equations when people forget to pay attention to it)

So now we have:

7500 - 0.08Y = 6000

Solve for Y by keeping Y on one side of the equation and moving everything else to the other side of the equation.

7500 - 0.08Y - 7500 = 6000 - 7500

-0.08Y = -500 (if you forgot how I did this check above how i solved for X)

To get Y by itself you divide both side by whatever the number is in front of Y which is -0.08

-0.08Y divided by -0.08 results in Y (since -0.08/-0.08 = 1)

-500 divided by -0.08 = 6250

So Y = 6250

Substitute this back into X = 50000 - Y

X = 50000 - 6250

X = 43750

So grandma should invest $43750 into non insured and $6250 into government insured.

I hope this helps you with your other questions and you will be able to solve them by yourself too. The hardest part is setting up the equation, if you can set that up correctly obviously even if you solve for X and Y correctly you will still get the wrong answer. I hope you aren't just going to copy this answer but to also try and understand how i got it for your tests and exams.