How many gallons of each should she use?

Question

A sparkling water distributor wants to make up 200 gal of sparkling water to sell for $5.00 per gallon. She wishes to mix three grades of water selling for $10.00, $1.00, and $4.50 per gallon, respectively. She must use twice as much of the $4.50 water as the $1.00, water. How many gallons of each should she use?

Answer 1

Let x=gallons of $10 water

Let y=gallons of $1 water

Let z=gallons of $4.50 water

1. Twice as much $4.50 water as $1 water
2. 2y=z
3. The purpose of this step is to streamline the number of variables from 3 to 2 temporarily
4. Mixture is 200 gallons @ $5 per gallon and is made up of x gallons @ $10 per gallon, y gallons @ $1 per gallon, and z gallons @ $4.50 per gallon.

(200)(5)=1000=10x+y+4.50z

x+y+z=200

2y=z, so substitute 2y for z:

1000=10x+y+4.50(2y)

200=x+y+2y

1000=10x+y+9y

1000=10x+10y

200=x+3y

1000=10x+10y

10[200=x+3y]

1000=10x+10y

2000=10x+30y

1000=20y

y=50

2y=z=2(50)=100

x+y+z=200

x+50+100=200

x=50

Let x=gallons of $10 water=50

Let y=gallons of $1 water=50

Let z=gallons of $4.50 water=100