Denise needs to buy 25/3 gallons of 85% ethanol and 35/3 gallons of 25% ethanol to have 20 gallons of 50% ethanol.
x | y | 20 gal
use x and y because you don't know how much she needs.
0.85x | 0.25y | 20(0.5)
85% is 85/100 or 0.85, and you need that much of x, same goes for the 25% and 50% mixtures
so now you can make up 2 equations
1) x + y = 20
2) 0.85x + 0.25y= 10 (you get 10 when you multiply 20 by 0.5)
now you can solve for x or y using substitution. first rewrite 1) in terms of x or y:
x+ y= 20 ----> y= 20 - x
now you can substitute 20- x for y in the second equation..
0.85x + 0.25y= 10
0.85x + 0.25(20-x)= 10 distribute here..(0.25 * 20 and 0.25 * (-x) )
0.85x + 5 - 0.25x = 10 combine like terms
0.6x +5 = 10 move the 5 over to the other side
0.6x= 10 -5
0.6x = 5 divide both sides by 0.6
x= 25/3 or 8.3 now you know the amount of x so you can substitue this back into the first equation to find y.
0.85x + 0.25y= 10
0.85(25/3) +0.25y= 10
85/12 + 0.25y= 10
0.25y = 10- 85/12
0.25y= 35/12
y= 35/3 or 11.6
you can check by putting these values into the euations:
1) x+ y= 20
25/3 + 35/3 =20
20= 20 good so far
2) 0.85x + 0.25y= 10
0.85(25/3) + 0.25(35/3)=10
10 = 10 so our values for x and y work