+40 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?
could you please explain the steps you took to get to the answer? just so i can understand. thanks!
+1926 (a) Let's make some obersvations about this problem first.
First off, if there's 500 mL, then we have \(500 * 30\% = 150 mL\) of magic maple syrup.
Setting this up as a fraction, we have \(\text{Percentage of Syrup} = \frac{150}{500}\)
Now, since 60% of red potion is syrup, then we have the equation
\(\frac{40}{100} = \frac{150+\frac{6}{10}x}{500+x}\) where x is the amount of mL of red potion we must add.
Thus, now we solve for x.
Cross multiplying the equations, we get
\(100000+200x=75000+300x\\-100x=-25000\\x=250\)
Thus, she must add 250mL of red potion.
Thanks! :)
+1926 (b) First, we let's set an equation.
Since they both add up to 50, we have the equation
\(\:\frac{30}{100}x+\frac{60}{100}y=54\) where x is the amount of blue potion and y is the amount of red potion.
Since they add up to 100mL, we also have the equation \(x+y=100\)
From the second equation, we isolate x to find that \(x=100-y\)
Plugging this value of x into the first equation, we get
\(\:\frac{30}{100}(100-y)+\frac{60}{100}y=54\)
Now, I won't show the steps, but solving this equation, we get that
\(y=80\)
This means that \(x=20\)
Thus, we find that we need 80mL of red potion and 20mL of blue potion.
Thanks! :)
