Question

How much 90 percent vinegar do you have to add to a gallon of 5 percent vinegar to get 20 percent vinegar?
_________gallons.
Hint: You want the final percentage of pure vinegar to be 20%. You have 1+x gallons of fluid where x is the number of gallons you add. Think about how much pure vinegar is in those 1+x gallons.

Answer 1

To derive the formula, just go step by step...
If you add 1 gallon of 90%, the final percentage would be (0.9+0.05)/2.
If you add 2 gallons of 90%, the final percentage would be (0.9+0.9+0.05)/3.
If you add 3 gallons of 90%, the final percentage would be (0.9+0.9+0.9+0.05)/4.
So...
If you add x gallons of 90%, the final percentage would be (0.9x+0.05)/(x+1).
To get 20%, (0.9x+0.05)/(x+1)=0.2
0.9x+0.05=0.2x+0.2
0.7x=0.15
x=3/14
So, you need to add 3/14 gallons of 90% vinegar to 1 gallon of 5% vinegar to get 20% vinegar...