Question

The Fiedler family has up to $130,000 to invest. They decide that they want to have at least $40,000 invested in stable bonds yielding 5.5% and that no more than $60,000 should be invested in more volatile bonds yielding 11%. How much should they invest in each type of bond to maximize income if the amount in the stable bond should not exceed the amount in the more volatile bond? What is the maximum income?

Answer 1

This is an optimization problem. I don't know how to actually solve the entire problem, but this deals with derivatives. To find the maximum interest it can get, you would have to get the function and find the derivative and equate it to 0.

Answer 2

I don't really get what you said there but thanks anyways

Answer 3

If you look at the mimimum and maximum given vs the total you have, it should be pretty easy to find.

Answer 4

Is there a lot of work to this question ?

Answer 5

Not really. By logic I saw it in seconds.

Answer 6

the min is 40,000 and the max is 60,000 ?

Answer 7

Ah, but min what and max what? They are diffeent.

Also, the goal is to get the income as high as possible. Well, which investment gives you more income? And you must have at minimum a certain ammount in one investment, if you put that minimum in, how much is left over?

Answer 8

Idk I thought that was right , im kinda really confused now

Answer 9

Well, you have 130k to invest. If you just do the min and the max, will you have invested 130k?

Answer 10

No?

Answer 11

"at least $40,000 invested in stable bonds yielding 5.5%" So it is a min, but a min in stable bonds. So 40K S bonds min.

"no more than $60,000 should be invested in more volatile bonds yielding 11%." That is the max alright, but it is in volatile bonds. So no more than 60K V Bonds.

So if 40K Sb + 60K Vb = 120K total but you need to invest 130K, where can you put the extra 10K? And, would it make the most income?

Answer 12

From the the volatile bonds?

Answer 13

Yes, the most income is form V bonds. But Vbonds have a max! (Which is why this is simple.) If you can't do more than the maximum... where must you put the money?

Answer 14

In the stable bonds?

Answer 15

Yep. You have no other choice. You maximize the Vbonds to maximize the profits, but when you reach their limit, the rest MUST go into Sbonds. No other choice. And when you look at the numbers you get, you see if they break any other rules in the question (they wont) and you are done.

Answer 16

Okay so for "How much should they invest in each type of bond to maximize income if the amount in the stable bond should not exceed the amount in the more volatile bond? " I would just put the max and the min? And the answer for the second question would be how much was put in the stable bonds? Which is 10k ?

Answer 17

Or not that's wrong hold on let me think this through

Answer 18

Kk. You are close... You will have more than the minimum in Sb.

Answer 19

Wait isn't 40k plus 60k just 100k ?

Answer 20

Oops... Yah. Hmmm... Made a mental messup myself there!

Answer 21

So you won't be able to invest all of it.

Answer 22

"if the amount in the stable bond should not exceed the amount in the more volatile bond" limits you.

Answer 23

so the maximum income would be 120k? And both bonds would be at they're max ?

Answer 24

The max invested is 120k, not income. The income you have to figure.

Answer 25

Oh ..wait okay im lost again

Answer 26

the max income would be what can't be invested ?

Answer 27

Well, you know that 60k is the max volitile. If stable can not exceed volitile you can do 60k stable. So yours first part is right there.

"How much should they invest in each type of bond to maximize income if the amount in the stable bond should not exceed the amount in the more volatile bond?"

If you know you should invest as much as you can of V, but you are limited on S, then the 60K each is it.


The second question is, " What is the maximum income?" Well, you need to find the interest earned.

Answer 28

and how do I find that ?

Answer 29

I think they ar doing simple interest, so it would be the percentage. Know how to get a percentage of somthing?

Answer 30

Nope? Or im not sure if you can, give me an example or like show me the steps ?

Answer 31

Well, lets start with the simple example. 10 cents is 10% of a dollar.

A dollar s 100 cents. 10% in decimal is .10 or just .1.
100 * .1 = 10.

OR, you can use the definition of "per cent" which means "out of 100" to make a ratio (fraction) and use that to multiply by:

\(100 \times \dfrac{10}{100}=10\)

OK. That is the simple example. Now for some numbers that are not quite as round.

Answer 32

Do you want it in decimal or in ratio form?

Answer 33

um this questions involves decimals right ? So...I guess decimal?:)

Answer 34

Some classes go heavy on the ratio... hehe. OK, so decimal.

\(7.65\%\) means \(\dfrac{7.65}{100}\) So as a decimal, you just shift over two to the left.

\(7.65\%=.0765\)

OK. Now, lets say I got a \(7.65\%\) return on 57300. So I just multiply!

\(.0765 \times 57300\)
Do you know how to calculate that? There is the calculator, and other methods.

Answer 35

4383.45 right ?

Answer 36

Yep! So you just want to do the same sort of thing with yours. For each 60k, multiply it by the % as a decimal. That will get you two numbers. Add them.

Answer 37

for every 60k its 11% right ?

Answer 38

Well, one 60k is 11% and the other is 5.5% since you have 60k in each investment.

Answer 39

\($60,000\times 5.5\%+$60,000\times 11\%=?\)

And if you are good with algebra, there is a little bit of a shortcut. But it is not a huge deal.

Answer 40

6963? Idk if I did that right

Answer 41

Hmmm. You have an error somewhere.

What is 11% of 60k?

Answer 42

6600?

Answer 43

Yes.

OK, now the 5.5% of 60k?

Answer 44

3300:p so both added would be 9900?

Answer 45

Yep! That is it. Not sure where you went wrong last time, but this time you did it perfect. =)

Answer 46

Sorry bout the early confuson on the 40+60=120. LOL. Dunno where my brain was at that point... probably thinking ahead to how you could only invest 120. Always pays to slow down and really pay attention. Even for me!

Answer 47

And what Boblovesmath suggested could be done. You could use calculus to do this. However, because it is a simple problem without lots of changing things, that is overkill.

Answer 48

Thanks you! So the income is 9900? :) Thanks so much for explaining to me , im a slow thinker lol

Answer 49

Well, speed comes with practice. If you were doing this a few hours a day for a few months you would be super fast. But everyone in school hopps from topic to topic so they don't always get to develop speed.

Answer 50

Yeah thank you:) , if I post another question can you help me on that one ?