A magician makes potions by combining maple syrup from a magical maple tree with ordinary water. The magician starts with a large supply of two potions: a red potion, which is 60% magical syrup by volume (and the rest is just water), and blue potion, which is 30% magical syrup by volume. (Perhaps you're wondering how the same syrup can produce both red and blue potions. That's why it's magic syrup!)
(a) Find the amount of red potion (in mL) that must be added to 500 mL of blue potion in order to produce potion that is 40% magical syrup by volume.
(b) Find the amounts of red potion and blue potion (in mL) that can be combined in order to produce 100 mL of a potion that is 54% magical syrup by volume.
(c) Does there exist a combination of red potion and blue potion that can produce a potion that is 75% magical syrup by volume?
(a) Let x be the amount of red potion that must be added to 500 mL of blue potion. The total volume of the mixture will be x+500 mL. The total amount of magical syrup in the mixture will be 0.6x+0.3(500)=0.6x+150 mL. We want the mixture to be 40% magical syrup, so we have the equation:
0.6x+150 = 0.4(x+500)
1.2x = 200
x = 166.67 mL
Therefore, 166.67 mL of red potion must be added to 500 mL of blue potion in order to produce a potion that is 40% magical syrup by volume.
(b) Let x be the amount of red potion and y be the amount of blue potion that are combined to produce 100 mL of a potion that is 54% magical syrup by volume. The total volume of the mixture will be x+y mL. The total amount of magical syrup in the mixture will be 0.6x+0.3y mL. We want the mixture to be 54% magical syrup, so we have the equation:
0.6x+0.3y = 0.54(x+y)
0.06x = 0.18y
x = 3y
Since we want the total volume of the mixture to be 100 mL, we have the equation:
x+y = 100
Substituting x=3y into the second equation, we get:
3y+y = 100
4y = 100
y = 25
Since x=3y, we have x=3(25)=75.
(c) I'll leave you to think about this one.
