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PetSmart.com sold the following varieties of dog food in June 2003.

Nature's Recipe Venison Meal & Rice Canine 20 percent protein, $21.99 per 20-lb bag

Nutro Max Natural Dog Food 27 percent protein, $12.99 per 17.5-lb bag

PetSmart Premier Oven Baked Lamb 25 percent protein, $22.99 per 30-lb bag

A dog breeder wants to make 300 pounds of a mix containing 22 percent protein. How many bags of each dog food variety should the breeder buy? (Hint: Note that each bag is a different weight and fractions of bags may not be purchased.)

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Let x  be the number of bags of Nature's Recipe Venison Meal & Rice Canine 20 percent protein, y be the number of bags of Nutro Max Natural Dog Food 27 percent protein and z be the number of bags of PetSmart Premier Oven Baked Lamb 25 percent protein a dog breeder buys. He  wants to make 300 pounds of a mix. Then

20x+17.5y+30z=300.

Now

  • in 20x pounds of Nature's Recipe Venison Meal & Rice Canine 20 percent protein is 0.2·20x=4x pounds of protein;
  • in 17.5y pounds of Nutro Max Natural Dog Food 27 percent protein is 0.27·17.5y=4.725y pounds of protein;
  • in 30z pounds of PetSmart Premier Oven Baked Lamb 25 percent protein is 0.25·30z=7.5z pounds of protein;
  • in 300 pounds of mix containing 22 percent protein is 0.22·300=66 pounds of protein.

Then 4x+4.725y+7.5z=66.

You get a system

[tex]\left\{\begin{array}{l}20x+17.5y+30z=300\\4x+4.725y+7.5z=66.\end{array}\right.[/tex]

From the first equation [tex]z=10-\dfrac{2}{3}x-\dfrac{7}{12}y.[/tex] Substitute it into the second equation:

[tex]4x+4.725y+7.5(10-\dfrac{2}{3}x-\dfrac{7}{12}y)=66.[/tex]

Simplify it:

[tex]-x+\dfrac{7}{20}y=-9,\\ \\ 20x-7y=180.[/tex]

y must be divided by 20, then

  • y=0, x=9, z=10-6=4;
  • y=20, x=16, z is not a whole number;
  • y=40, x=23,  z is not a whole number;
  • y=60, x=30, z=10-20-35<0;
  • thus, all other possible z will be <0.

Answer: 9 bags with 1st mix, 0 bags with 2nd mix and 4 bags with 3rd mix.

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