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 $50\%$ magical syrup by volume (and the rest is just water), and blue potion, which is $20\%$ 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 $600$ mL of blue potion in order to produce potion that is $30\%$ magical syrup by volume.
(b) Find the amounts of red potion and blue potion (in mL) that can be combined in order to produce $150$ mL of a potion that is $42\%$ 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 number of milliliters of red potion that must be added to $600$ mL of blue potion. Since the red potion is $50\%$ magical syrup by volume, then it contains $\frac{50}{100}\cdot x = \frac{x}{2}$ mL of magical syrup. Similarly, the blue potion contains $\frac{20}{100}\cdot 600 = 120$ mL of magical syrup. Thus, the total amount of magical syrup in the mixture is $\frac{x}{2} + 120$ mL. The total volume of the mixture is $x + 600$ mL.
For the mixture to be $30\%$ magical syrup, we need to solve the equation $\frac{\frac{x}{2} + 120}{x + 600} = \frac{30}{100}$.
Cross multiplying gives $150(x + 600) = 30(\frac{x}{2} + 120)$. Expanding both sides gives $150x + 90000 = 15x + 3600$. Collecting like terms gives $135x = 86400$, so $x = \frac{86400}{135} = 640$.
Thus $\boxed{640}$ mL of the red potion must be added to $600$ mL of blue potion to produce a potion that is $30\%$ magical syrup by volume.
(b) Let $r$ and $b$ be the amounts of red potion and blue potion, respectively, that are combined. We are given that $r + b = 150$ mL and $\frac{r}{r+b} = \frac{42}{100}$.
Multiplying the second equation by $r+b$ gives $r = \frac{42}{100}\cdot (r+b) = \frac{42}{100}\cdot 150 = 63$.
Substituting $r = 63$ into the first equation gives $63 + b = 150$, so $b = 87$.
Thus, $\boxed{63}$ mL of red potion and $\boxed{87}$ mL of blue potion can be combined to produce $150$ mL of a potion that is $42\%$ magical syrup by volume.
(c) Assume that $r$ and $b$ are the amounts of red potion and blue potion, respectively, that are combined. We want the mixture to be $75\%$ magical syrup by volume.
Thus, $\frac{r}{r+b} = \frac{75}{100}$. Cross multiplying gives $100r = 75(r+b)$, so $25r = 75b$.
Dividing both sides by $25$ gives $r = 3b$.
Since $r$ and $b$ must be positive, the only possible solution is $r = 3$ and $b=1$. However, these values do not correspond to the given amounts of red potion and blue potion.
Therefore, there does not exist a combination of red potion and blue potion that can produce a potion that is $75\%$ magical syrup by volume.