SOLUTION: Jill invested $100,000 in stocks and bonds. Equities earned a total return of 12%, and the fixed income component earned 8%. If she had invested twice as much in equities, she wo

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Question 1148351: Jill invested $100,000 in stocks and bonds. Equities earned a total return of
12%, and the fixed income component earned 8%. If she had invested twice
as much in equities, she would have made $1,800 more. How much was
invested in equities?
I don't need the answer as much as I need to understand the steps. Your help is greatly appreciated, thank you in advance!!!!
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
this one was tough, since i never saw one like this before, but i think i have it.

let x = the amount invested in stocks.
let y = the amount invested in bonds.

you get:

x + y = 100,000
.12 * x + .08 * y = I

I represents the interest on the investment.

if you invest twice as much in stocks, you will get I + 1800.

when you invest twice as much in stocks, the amount invested in bonds is less by the original amount invested in stocks.

you will get 2x + (y - x) = 100,000
simplify to get 2x + y - x = 100,000
combine like terms to get x + y = 100,000
bottom line here is that the equation of 2x + (y-x) = 100,000 doesn't tell me anything, by itself, since i simplifies to the original equation of x + y = 100,000.

i decided not to simplify the x + y equation and left it as 2 * x + (y - x) = 100,000.

as it turns out, i didn't have to use it, since i had enough without it.

the interest equation does tell me something.

i get .12 * x + .08 * y = I for the original equation.
i get .12 * (2 * x) + .08 * (y - x) = I + 1800 for the doubling of x equation.

simplify the second equation to get:

.24 * x + .08 * y - .08 * x = I + 1800
combine like terms to get:
.16 * x - .08 * y = I + 1800

there are two equations that need to be solved simultaneously.
they are:

.12 * x + .08 * y = I
.16 * x + .08 * y = I + 1800

subtract the first equation from the second to get:
.04 * x = 1800
solve for x to get:
x = 1800 / .04 = 45,000

this means that y = 55,000

i have x = 45,000 and y = 55,000
.12 * x + .08 * y = 9800

when i double x, i have to subtract that same amount from y.
i get 2 * x + (y - x) = 100,000 which becomes:
90,000 + 10,000 = 100,000
the interest becomes:
.12 * 90,000 + .08 * 10,000 = 11,600

the difference is 11,600 - 9,800 = 1,800.

your solution is that the original amount invested in equities is 45,000.

i have no idea if this is the preferred way to solve this, but it did work, so i'll stick with it for now.















Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
Jill invested $100,000 in stocks and bonds. Equities earned a total return of
12%, and the fixed income component earned 8%. If she had invested twice
as much in equities, she would have made $1,800 more. How much was
invested in equities?
I don't need the answer as much as I need to understand the steps. Your help is greatly appreciated, thank you in advance!!!!
Let amount invested in equities (stocks), be E

Then amount invested in bonds = 100,000 - E

With stocks' and bonds' RORs being 12% and 8%, respectively, we get the following interest amounts: .12E and .08(100,000 - E) = 8,000 - .08E

Therefore, total earnings = .12E + 8,000 - .08E = .04E + 8,000

We then get: .12(2E) + .08(100,000 - 2E) = .04E + 8,000 + 1,800

.24E + 8,000 - .16E = .04E + 9,800

.24E - .16E - .04E = 9,800 - 8,000

.04E = 1,800

E, or  

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