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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 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?

Jul 14, 2024

#1
+129806
+1

(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.

Let x be the amount of red potion to  be added

.60x  + 500 * .30  =   (500 + x) *.40

.60x + 150  = .200 + .40x

,60x -.40x = 200 - 150

.20x = 50

x = 50 /.20  = 250ml  of red must be added

Jul 15, 2024
#2
+129806
+1

(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.

Let x be the amount of red potion and 100-x be the amount of blue

.60x + .30 (100-x)  = .54 * 100

.60x + 30  - .30x  = 54

.30x + 30 = 54

.30x = 24

x = 24/.30  = 80

80ml of red must be added

20 ml of blue must be added

Jul 15, 2024
#3
+129806
+1

(c) Does there exist a combination of red potion and blue potion that can produce a potion that is 75% magical syrup by volume?

Not possible since neither potion exceeds 60 %  magical syrup

Jul 15, 2024