SOLUTION: $27,090 is invested, part at 13% and the rest at 10%. If the interest earned from the amount invested at 13% exceeds the interest earned from the amount invested at 10% by $440.16,

Question 659032: $27,090 is invested, part at 13% and the rest at 10%. If the interest earned from the amount invested at 13% exceeds the interest earned from the amount invested at 10% by $440.16, how much is invested at each rate? (Round to two decimal places if necessary.
I've tried to work out this problem 2 different ways! Still, I have no luck. Please Help!
Found 2 solutions by Theo, amalm06:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
27090 is invested.
part at 13%
part at 10%
the interest earned at 13% exceeds the interest earned at 10% by 440.16
how much is invested at each rate.
let x = the amount of money invested at 13%
let y = the amount of money invested at 10%
so far you get 2 equations.
x + y = 27090
.13x + .10y = total interest earned.
unfortunately we don't know that, but we do know that .13x - .10y = 440.16
we should be able to used that equation to solve for x and y.
we have 2 equations that need to be solved simultaneously
they are:
x + y = 27090
.13x - .10y = 440.16
if we multiply both sides of the top equation by .10, then that equation becomes:
.10x + .10y = 2709
our 2 equations are now:
.10x + .10y = 2709.00
.13x - .10y = 440.16
if we add these equations together we will get:
.23x = 3149.16
if we divide both equations by 23 we will get:
x = 13692
this means that y must be equal to 27090 - 13692 = 13398
we have:
x = 13692
y = 13398
.13x = .13 * 13692 = 1779.96
.10y = .10 * 13398 = 1339.8
.13x - .10y = 440.16
this looks to be a good answer.
13,692 is invested at 13%
13,398 is invested at 10%
the simultaneous equations could also have been solved by substitution as follows:
start with:
x + y = 27090
.13x - .10y = 440.16
solve for y in the first equation to get:
y = 27090 - x
substitute for y in the second equation to get:
.13x - .10(27090-x) = 440.16
simplify to get:
.13x - .10*27090 + .10x = 440.16
simplify further to get:
.13x - 2709 + .10x = 440.16
combine like terms to get .23x - 2709 = 440.16
add 2709 to both sides of the equation to get:
.23x = 3149.16
divide both sides by .23 to get:
x = 13692
you get the same value for x which leads to the same value for y so both methods (elimination method was used first) give you the same result as they should.

Answer by amalm06(224) (Show Source): You can put this solution on YOUR website!
0.13x-0.10(27090-x)=440.16
0.13x-2709+0.10x=440.16
0.23x=3149.16
x=$13692 at 13%, $13398 at 10% (Answer)
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