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?
Alright, just for the record, I believe this is a question from a-o-p-s. But anyways, I'll still try and solve it.
We have the red potion: which is 75% syrup and 25% water.
We have the blue potion: which is 15% syrup and 85% water.
a) We have to figure out how much we are aiming for, and how much we already have. We already have:
15% of the 200
= 30mL of syrup in 200 milliliters.
And we need 25% which would mean we would need :
25% of the 200
= 50mL of syrup in the water.
Right? No, it's not, since we are actually pouring more potion and water into the blue potion, the volume is going to increase...
Hmm, how should we do this?
We could start by defining a variable :
which signifies the amount of red potion that we are adding:
(.75x + 30)/(200 + x) = 25/100
We multiply by 200 + x to both sides:
\((.75x + 30)= (25/100)(200 + x) \)
\((.75x + 30) = 50 + x/4\)
\(.50x = 20\)
\(x = 40\)
Let's try that out, just to be sure!
We are going to be adding 30 ml of syrup and 40ml of potion all together.
(30+30)/(200+40) = 25/100?
60/240 = 1/4 = 25/100
So, we would be adding 40ml of red potion for part a.
I'm going to leave the next two for you to do, because I practically made this amazing solution, so you should read it, and then you'll be able to easily figure out the other two!