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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 75% magical syrup by volume (and the rest is just water), and blue potion, which is 15% 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 200 mL of blue potion in order to produce potion that is 25% magical syrup by volume.

(b) Find the amounts of red potion and blue potion (in mL) that can be combined in order to produce 400 mL of a potion that is 30% magical syrup by volume.

(c) Does there exist a combination of red potion and blue potion that can produce a potion that is 35% magical syrup by voume?

 Dec 11, 2022
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Hello! 

 

a.) We can use a system here to solve this. 

Let x=amount of red potion used

Let y=total amount of potion in solution

 

\(200+x=y \)

\(.15(200)+.75x=.25y\)

.15(200) because 15% of the blue potion=.15 and there is 200mL. 

.75x because 75% of the red potion=.75 and there is x amount of it 

.25y because we want 25% of the total potion=.25 and there is y amount of it

 

We can substitute for y in the second equation:

\(.15(200)+.75x=.25(200+x)\)

 

Simplify: 

\(30+.75x=50+.25x\)

\(-20=-.5x\)

\(x=40\)

 

Knowing that x=40, we know that y=240, meaning our answer is 240.

 

b.) We can again use a system for this problem

Let x=Amount of red potion

Let y=Amount of blue potion

 

\(x+y=400 \)

\(.3(400)=.15y+.75x\)

.3(400)=30% of the 400 mL 

.15y=15% of the magic syrup

.75x=75% of the magic syrup

 

We can substitute for x in this equation: 

\(120=.15y+.75(400-y)\)

 

Simplify: 

\(120=.15y+300-.75y\)

\(-180=-.6y\)

\(y=300, x=100\)

 

There are 300 mL of blue potion used and 100 mL of red potion used. 

 

c.) Let x=amount of red potion

Let y=amount of blue potion

 

\(.15x+.75y=.35(x+y)\)

.35(x+y) because the total amount of the potion is x+y

 

Simplify: 

\(.15x+.75y=.35x+.35y\)

\(.4y=.2x\)

\(2y=x\)

 

Whenever there is a ratio of 2 (blue potion):1 (red potion), there will be a 35% amount of magical syrup.

For example, 200 mL of blue potion and 100 mL of red potion would work. 

 

Hope this helps!

 Dec 11, 2022

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