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# Algebra

+1
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Help can anyone explain this to me

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 28, 2023

#1
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(a)

Setting the amount of red potion to $$r\;\rm mL$$ and the amount of blue potion to $$b\;\rm mL$$ , we write the equation $$\dfrac{\frac12r+45}{r+300}=\dfrac15$$. Solving this, we have r=50 mL

Jan 28, 2023
#2
+79
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(b)

Using the same variables as part (a), we have two linear equations below:

$$r+b=180$$

$$\frac12r+\frac{3}{20}b=72$$

We can solve this using either substitution or elimination. Personally, I prefer elimination, so I'll show that work below, but both give the same answers.

$$r+\dfrac{3}{10}b=144$$               Multiply equation 2 by 2

$$\dfrac{7}{10}b=36$$                        Subtract above equation from equation 1

$$b=\dfrac{360}{7}=51+\dfrac{3}{7}$$       Solve for b

Next, we substitute that into the first equation to find that $$r=\dfrac{900}{7}=128+\dfrac47$$

In decimal form, that's

r=128.57

b=51.42

Jan 28, 2023
#3
+79
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(c)

Yes there does. That's between the 50% and 15% of the red and blue potions, so there is a way to get 25% syrup.

Jan 28, 2023