SOLUTION: the research department of a company that manufactures children's fruit drinks is experimenting with a new flavor. a 17.5% fructose solution is needed but only 10%and 20% solution

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Question 767867: the research department of a company that manufactures children's fruit drinks is experimenting with a new flavor. a 17.5% fructose solution is needed but only 10%and 20% solutions are available. How many Galons of the 10% fructose solution should be mixed with the 20% fructose solution to obtain 20 gallons of 17.5% fructose solution?
(please put the problem in system equation form which is {Ax+By=z
Gx+Hy=c before you compute them. thank you

Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Account for the percent fructose wanted and used:
x for gallons 10% fructose, y for gallons of 20% fructose.
10x%2B20y=20%2A17.5
Notice the left and right sides being equated are the amount of fructose, as their mass (or weight). Simplifies if desired to :
x%2B2y=2%2A17.5
highlight%28x%2B2y=35%29

Account for the amounts of solutions: Wanted is 20 gallons of fructose solution, so highlight%28x%2By=20%29.

The highlighted equations give you your system. Next is solve the system for x and y. One way is subtract the solution sum members from the percentage account members and solve first for y...
%28x%2B2y%29-%28x%2By%29=35-20
.
.


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ANOTHER DESCRIPTION TO DEVELOP THE PERCENT FRUCTOSE ACCOUNTING EQUATION:

The available fructose solutions to be mixed are 10% and 20%, and what is wanted is 17.5% fructose.
Using x for the quantity of 10% fructose, and y for the quantity of 20% fructose, both x and y being in gallons, we can make a rational expression to represent the desired concentration.

The amount of FRUCTOSE would be 10%2Ax%2B20%2Ay, and this is for only the fructose, as if without other materials, without any of the water.
The amount of mixed solution would be %28x%2By%29; in whatever units we have, in this case, %28x%2By%29 GALLONS. The concentration in percent is then:

%2810x%2B20y%29%2F%28x%2By%29=17.5

We already know that we want 20 gallons of this 17.5% fructose solution, and that x+y=20. The rational equation is then this:

%2810x%2B20y%29%2F20=17.5

That should be simplified before working the steps for solving.
Instead of being in rational form, this should be transformed to an easier, more typical linear equation format. Other simplifications, too.

10x%2B20y=%2817.5%29%2820%29________multplied left and right by 20.
x%2B2y=%2817.5%29%282%29___________divide left and right by 10.
highlight%28x%2B2y=35%29__________________simple multiplication computation.

...and highlighted here is the best, simplest form of that equation to use.