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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
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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.
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you know what the total income each year is.
it's $530.
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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%
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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
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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
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this is the same as (5%*y)/2 + (4%*y)/3 + (3.5%*y)/6 = $530.
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to get the interest rate you have to divide by 100%.
5% / 100% = .05
4% / 100% = .04
3.5% / 100% = .035
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you substitute interest rate for % interest and the equation becomes
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(.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.
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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.
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your answer is total money invested is $12000.
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