SOLUTION: I don't understand how to set up this problem in order to solve it because the book examples do not break it up the same way. Dave invested half his money at 5%, one-third his mon

Question 160678: I don't understand how to set up this problem in order to solve it because the book examples do not break it up the same way.
Dave invested half his money at 5%, one-third his money at 4%, and the rest of his money at 3.5%. If his total annual investment income was $530, how much had he invested?
Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
first thing you have to do is find out what the rest is.
several ways to do it.
one way is as follows:
1/2 + 1/3 + x = 1
solve for x.
easiest way is to multiply both sides of the equation by 6 to remove the denominators.
equation becomes
3 + 2 + 6*x = 6
6*x = 6 - 3 - 2 = 1
x = 1/6
-----
to see if that's correct, add up 1/2 + 1/3 + 1/6 to see if it adds up to 1.
1/3 is the same as 2/6.
1/2 is the same as 3/6
so 2/6 + 3/6 + 1/6 = 6/6 = 1.
x = 1/6 is good.
-----
you know what the total income each year is.
it's $530.
-----
now you know that:
1/2 of the money is invested at 5%.
1/3 of the money is invested at 4%.
1/6 of the money is invested at 3.5%
-----
if 1/2 of the money is invested at 5%, then the amount of income from 1/2 of the money at the end of the year is 5% times (1/2 of the money).
this also means that 4% times (1/3 of the money) give you the amount of income from 1/3 of the money at the end of the year.
this also means that 3.5% times (1/6) of the money) gives you the amount of income from 1/6 of the money
-----
let y = total amount of money invested.
let $530 = total income from money invested.
equation becomes:
5%*y*1/2 + 4%*y*1/3 + 3.5%*y*1/6 = $530
-----
this is the same as (5%*y)/2 + (4%*y)/3 + (3.5%*y)/6 = $530.
-----
to get the interest rate you have to divide by 100%.
5% / 100% = .05
4% / 100% = .04
3.5% / 100% = .035
-----
you substitute interest rate for % interest and the equation becomes
-----
(.05*y)/2 + (.04*y)/3 + (.035*y)/6 = $530.
multiplying both sides of the equation by 6 to remove the denominators, and the equation becomes
3*.05*y + 2*.04*y + .035*y = 6*$530
which becomes
.15*y + .08*y + .035*y = $3180
which becomes
.265*y = $3180
which becomes
y = $3180/.265 which equals $12000.
-----
to prove that the answer is correct, you substitute $12000 for y and solve the original equation after you converted from % to rate.
that equation is .05*y*(1/2) + .04*y*(1/3) + .035*y*(1/6) = $530.
which becomes
.05*12000*(1/2) + .04*12000*(1/3) + .035*12000*(1/6) = $530.
which becomes
$300 + $160 + $70 = $530
which becomes
$530 = $530
which proves the equation is good.
-----
your answer is total money invested is $12000.

RELATED QUESTIONS
I need help to learn how to set up equations so that I can solve it; Because I am having... (answered by rfer)

Please help me solve, Iam unsure how to set up the problem... it shows in the book but I... (answered by Alan3354,solver91311,MathTherapy,stanbon)

Well today I missed school but Im trying to do the homework. Weve done this in the past... (answered by Alan3354)

Sin θ=4/5, cos θ=-3/5 I need to find θ but I don't understand how to solve for it. If (answered by MathLover1)

Please help me solve this word problem: A man drove his car 132 miles before the water... (answered by ankor@dixie-net.com)

Could you please tell me how to set up the correct equation for this word problem. "you... (answered by htmentor)

I know how to set the problem up but I do not understand how they the get the answer to... (answered by stanbon)

I would greatly appreciate some help with factoring. I understand that is basically... (answered by Nate)

I'm so confused on what my math teacher is asking... Sin θ=4/5, cos θ=-3/5 I need to... (answered by MathLover1,math_tutor2020,ikleyn,mccravyedwin)