SOLUTION: a person invested $8000 for 1 year, part at 8%, part at 11%, and the remainder at 12%. The total annual income for these investments was $876. The amount of money invested at 12%

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Question 887182: a person invested $8000 for 1 year, part at 8%, part at 11%, and the remainder at 12%. The total annual income for these investments was $876. The amount of money invested at 12% was $200 more than the amounts invested at 8% and 11% combined. Find the amount invested at each rate.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
let x = amount invested at 8%
let y = amount invested at 11%
let z = amount invested at 12%

you get 2 equations:

x + y + z = 8000 and .08x + .11y + .12z = 876

the mount invested at 12% was $200 more than the amount invested at 8% and 12%.

this means that z = 200 + x + y

replace z with 200 + x + y in both your equations to get:

x + y + (200 + x + y) = 8000 and .08x + .11y + .12 * (200 + x + y) = 876

simplify both equations to get:

2x + 2y + 200 = 7800 and .08x + .11y + .12*200 + .12x + .12y = 876

simplify further to get:

2x + 2y + 200 = 7800 and .2x + .23y + 24 = 876

subtract 200 from both sides of the first equation and subtract 24 from both sides of the second equation to get:

2x + 2y = 7800 and .2x + .23y = 852

multiply both sides of the second equation by 10 to get:

2x + 2y = 7800 and 2x + 2.3y = 8520

subtract the first equation from the second equation to get:

.3y = 720

divide both sides of this equation by .3 to get:

y = 2400

replace y with 2400 in the equation of 2x + 2y = 7800 to get:

2x + 2(2400) = 7800
simplify this to get 2x + 4800 = 7800
subtract 4800 from both sides of this equation to get 2x = 3000
divide both sides of this equation by 2 to get x = 1500

you have x = 1500 and y = 2400

go back to your original equation of x + y + z = 8000 and replace x with 1500 and replace y with 2400 and solve for z to get z = 4100.

now you have:
x = 1500
y = 2400
z = 4100

those should be your solutions.
all that is left is to confirm that they are good solutions.

x + y + z = 8000 becomes 1500 + 2400 + 4100 = 8000 which becomes 8000 = 8000 so that part is good.

.08x + .11y + .12z = 876 becomes .08*1500 + .11*2400 + .12*4100 = 876 which becomes 120 + 264 + 492 = 876 which becomes 876 = 876 so that part is good.

looks like all parts are good so the solution is confirmed as good.

the amount invested at 8% is 1500
the amount invested at 11% is 2400
the amount invested at 12% is 4100