Question 652559
Let the number of gallons of 20% solution of fertilizer to be used be {{{x}}}.
Let the number of gallons of 80% solution of fertilizer to be used be {{{y}}}.

As you mix the solutions, the volumes add up,
so the final volume (in gallons) is {{{x+y=120}}} .
That is your first equation.
{{{x}}} gallons of a 20% solution contain {{{0.20x}}} gallons of fertilizer.
{{{y}}} gallons of a 80% solution contain {{{0.80y}}} gallons of fertilizer.
When you mix them, the final mix will contain {{{0.20x+0.80y}}} gallons of fertilizer,
but that must be the amount of fertilizer in 120 gallons of 60% solution of fertilizer,
which is {{{0.60*120=72}}} gallons of fertilizer.
So {{{0.20x+0.80y=72}}} or {{{0.2x+0.8y=72}}} (which means the same in math class).
That is your second equation.
You now have a system of 2 equations linear in two variables:
{{{system(x+y=120,0.2x+0.8y=72)}}}
You can solve it by whatever method you chose.
For example, to solve by elimination, you may chose to multiply the first equation by {{{(-0.2)}}}, getting
{{{-0.20x-0.20y=24}}},
and add it to the second equation, getting
{{{0.2x+0.8y-0.20x-0.20y=72-24}}} --> {{{0.6x=48}}} --> {{{x=48/0.6))) --> {{{highlight(x=80)}}}
Now, from {{{x=80}}} and {{{x+y=120}}} , you get
{{{80+y=120}}} --> {{{y=120-80}}} --> {{{highlight(y=40)}}}
 
NOTES FROM A CHEMIST
In real life, volume is often additive, so if you mix {{{x}}} gallons of a solution with {{{y}}} gallons of another solution, you get {{{x+y}}} gallons of mixture. (In math problems, volume is always additive).Chemists (like me) specify very clearly the concentration of a solution.
If we say 20% weight in weight (w/w), we mean that 100 g of solution contains 20 g of the item of interest.
If we say 20% weight in volume (w/v), we mean that 100 milliliters of solution contains 20 g of the item of interest.
If we say 20% volume in volume (v/v), we mean that 100 gallons of solution contains 20 gallons of a liquid of interest.