+35 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 magical syrup by volume (and the rest is just water), and blue potion, which is 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 mL of blue potion in order to produce potion that is magical syrup by volume.
(b) Find the amounts of red potion and blue potion (in mL) that can be combined in order to produce mL of a potion that is magical syrup by volume.
(c) Does there exist a combination of red potion and blue potion that can produce a potion that is magical syrup by volume?
How to start to solve this question?
+121 To solve this question, you need to use the concept of mixing solutions with different concentrations to achieve a desired concentration. Here's how to start approaching each part:
(a) The goal here is to find the amount of red potion that must be added to a certain amount of blue potion to achieve a specific concentration. Let's denote the amount of red potion to be added as \(x\) mL. The concentration of the red potion is 1 magical syrup per volume, and the concentration of the blue potion is \(\frac{1}{2}\) magical syrup per volume. The final concentration we want is 1 magical syrup per volume.
Set up the equation based on the concentration formula:
\[\text{Concentration} \times \text{Volume} = \text{Concentration} \times \text{Volume}.\]
For the red potion:
\[1 \cdot x = \text{magical syrup in red potion}.\]
For the blue potion:
\[\frac{1}{2} \cdot 100 = \text{magical syrup in blue potion}.\]
Since the total volume after mixing is \(100 + x\) mL, and we want the concentration to be 1 magical syrup per volume, the equation becomes:
\[1 \cdot (100 + x) = (\text{magical syrup in red potion} + \text{magical syrup in blue potion}).\]
Solve for \(x\) using this equation.
(b) Similar to part (a), you need to find the amounts of red potion and blue potion to be combined to achieve a certain concentration. Let \(x\) be the amount of red potion (in mL) and \(y\) be the amount of blue potion (in mL). The concentration of the red potion is 1 magical syrup per volume, the concentration of the blue potion is \(\frac{1}{2}\) magical syrup per volume, and the final concentration is 1 magical syrup per volume.
Set up an equation for the amount of magical syrup:
\[1 \cdot x + \frac{1}{2} \cdot y = 1 \cdot (x + y).\]
Solve for \(y\) in terms of \(x\) using this equation, and then substitute this into the total volume equation to solve for \(x\).
(c) Think about the concentrations involved. If you have a red potion with a concentration of 1 magical syrup per volume and a blue potion with a concentration of \(\frac{1}{2}\) magical syrup per volume, is there any way to combine them to achieve a concentration of 1 magical syrup per volume?
Remember that concentrations represent the amount of magical syrup in a certain volume, and you're trying to mix these concentrations to achieve a desired result.
(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.
