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# Algebra

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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 50% 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 300 mL of blue potion in order to produce potion that is 20% magical syrup by volume.

(b) Find the amounts of red potion and blue potion (in mL) that can be combined in order to produce 180 mL of a potion that is 40% magical syrup by volume.

(c) Does there exist a combination of red potion and blue potion that can produce a potion that is 25% magical syrup by voume?

Jan 8, 2023

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Hmmm...

I'm going to try and solve this in the best way possible (although I might get the answer wrong, so I advise you to read through my explanation first)

Before I start my answer, I want to clarify what you should know:

- how percents work

- how to define a variable

- how to solve a simple algebraic expression

If you don't know any of these, than I advise you to learn as these are major concepts which will help you later on, to make sure that you understand and read my explanation, I will only be giving you the answer to a) so that you can work out (hopefully) the other ones.

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Let's start by clarifying the question, we have:

red potion  contains 50% of the syrup

blue potion contains 15% of the syrup

We want to know how much red potion should be added to 300 ml of blue potion to get a potion which is 20% syrup.

We have 300 ml of blue potion. Let's start by figuring out how much syrup is in the 300 ml.

We have 15% of 300 = 45 ml of syrup

We want 20% of the total to be syrup

We define x as the amount of red potion that goes in to be 20%

Our equation is:

(.50x + 45)/(300 + x) = 20/100

We multiply by 300 + x

(.50x + 45)(100) = (6000+20x)

50x + 4500 = 6000 + 20x

30x = 1500

x = 50

So our equation says that we will add 50ml of red potion, let's confirm:

50 ml of red potion has 25 ml of syrup:

(45 + 25)/(300+50)
70/350 = 20/100 = 1/5

So for part a) we need 50ml of red potion, hope you can solve the other ones because I think I gave a pretty solid solution! (If it is wrong please tell me, as I will be happy to fix it!)

~ Hmmmm

Jan 9, 2023