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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?

 Jun 29, 2024
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(a) Let's first find out how much magical syrup is in 300 mL of blue potion. Since the blue potion is 15% magical syrup by volume, the amount of magical syrup in 300 mL of blue potion is 0.15 x 300 = 45 mL.

 

Now, let's assume that we add x mL of red potion to this mixture. The resulting potion will have a total volume of 300 + x mL, and its magical syrup content will be (45 mL + 0.5x mL) / (300 mL + x mL). We want this to be equal to 20% (or 0.2), so we can set up the following equation:

 

(45 mL + 0.5x mL) / (300 mL + x mL) = 0.2

 

Solving for x, we get:

 

x = 150 mL

 

Therefore, 150 mL of red potion must be added to 300 mL of blue potion to produce a potion that is 20% magical syrup by volume.

 

(b) Let's assume that we mix x mL of red potion with y mL of blue potion to produce 180 mL of a potion that is 40% magical syrup by volume. We can set up the following two equations based on the amount of magical syrup and the total volume:

 

Amount of magical syrup: 0.5x mL + 0.15y mL = 0.4(180 mL)

 

Total volume: x mL + y mL = 180 mL

 

Simplifying the first equation, we get:

 

0.5x + 0.15y = 72

 

We can now use substitution to solve for x and y. Solving the second equation for y, we get:

 

y = 180 - x

 

Substituting this into the first equation, we get:

 

0.5x + 0.15(180 - x) = 72

 

Solving for x, we get:

 

x = 120 mL

 

Substituting this into the equation y = 180 - x, we get:

 

y = 60 mL

 

Therefore, 120 mL of red potion and 60 mL of blue potion can be combined to produce 180 mL of a potion that is 40% magical syrup by volume.

 

(c) Let's assume that we mix x mL of red potion with y mL of blue potion to produce a potion that is 25% magical syrup by volume. We can set up the following two equations based on the amount of magical syrup and the total volume:

 

Amount of magical syrup: 0.5x mL + 0.15y mL = 0.25(x+y) mL

 

Total volume: x mL + y mL = some value

 

We have two equations and two unknowns, so we can solve for x and y. Simplifying the first equation, we get:

 

0.25x - 0.1y = 0

 

Substituting the second equation into this equation, we get:

 

0.25x - 0.1(x + y) = 0

 

Simplifying, we get:

 

0.15x - 0.1y = 0

 

We can now use substitution to solve for x and y. Solving the second equation for y, we get:

 

y = some value - x

 

Substituting this into the first equation, we get:

 

0.15x - 0.1(some value - x) = 0

 

Simplifying, we get:

 

0.25x - 0.1(some value) = 0

 

Therefore, we can see that there is no solution that satisfies these equations for any positive values of x and y. So there is no combination of red

 Jul 5, 2024

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