How many gallons of each should she use?

Question

Mixing water A sparkling spring-water distributor want to make up 300 gal of sparkling water to sell for $6.00 per gallon. She wishes to mix three grades of water selling for $9.00, $3.00, and $4.50 per gallon, respectively. She must use twice as much of the $4.50 water as of the $3.00 water. How many gallons of each should she use?

Answer 1

Basically we can approach this question as a weighted average.

to start off will we need to build some equations/expressions form the word problem. 

1st we know that the sum of each type of water will equal 300 gallons

 if we have

X gallons at $9.00

Y gallons at $3.00

and Z gallons at $4.50

then X+Y+Z=300

also the problem tell us that there is twice the gallons at $4.50 as $3.00

therefore 2Y =Z

so our equation becomes

X+Y+2Y = 300  or X+3Y = 300

this means that  X = 300-3Y

know we have gotten our equation down to 2 unknowns that means we need to find another equation in order to solve the problem.  this is where our prices will come in to play, if we approach the final price per gallon of the mixture as a weighted average then.

($9.00*X + $3.00*Y +$4.50*Z)/300 = $6.00

substitute the fact that Z = 2Y

($9.00*X + $3.00*Y +$4.50*2Y)/300 = $6.00

simplify

($9.00*X +$3.00Y + $9.00Y)/300 = $6.00

($9.00*X +$12.00*2Y)/300 = $6.00

multiply both sides by 300

$9.00*X +$12.00Y = $1800.00

substitute 300-3Y in for X

$9.00*(300-3Y) +$12.00Y = $1800.00

distribute the $9.00

$9.00*300 - $9.00*3Y +$12.00Y = $1800.00

simplify

$2700.00 - $27.00Y +$12.00Y = $1800.00

combine like terms

$2700.00 - $15.00Y  = $1800.00

subtract $2700.00 from both sides

- $15.00Y = -$900.00

divide both sides by -$15.00

 Y = 60 gallons at $3.00

Z = 2Y = 2*60 + 120 gallons at $4.50

and

X = 300 - 3Y = 300 - 3*60 = 120 gallons at $9.00

good luck