A dog food manufacturer makes three types of dog food mix. Each type is solid in

Question

A dog food manufacturer makes three types of dog food mix. Each type is solid in bags containing 5 kg with the ratio of meat:rice:vegetables being as follows:

MIX A 5:3:2

MIX B 1:1:3

MIX C 0:3:2

The manufacturer orders 3350 kg of meat, 4850kg of vegetables and 4300kg of rice for a particular production run. The run involves no weight loss for any of the ingredients and all the quantities ordered are exactly used up. If the run produces x bags of MIX A, y bags of MIX B and z bags of MIX C write three equations that apply and solve to find x,y and z.

Explanation / Answer

x represents the number of bags of mix A, y mix B, and z mix C.

5x+y=3350 This represents the meat required.
3x+y+3z=4300 This represents the rice required.
2x+3y+2z=4850 This represents the vegetables required.

There are a number of ways to solve the sysytem including using matrices. Since you did not indicate a method, we can use substitution:

From the first equation y=3350-5x. Substitute the expression for y into the last two equations to get:

3x+(3350-5x)+3z=4300
2x+3(3350-5x)+2z=4850

-2x+3z=950
-13x+2z=-5200 Multiply the first equation by 2 and the second by 3 to get:

-4x+6z=1900
-39x+6z=-15600 Subtract the second equation from the first:

35x=17500

x=500

Then -4(500)+6z=1900 ==> 6z=3900 ==> z=650

y=3350-5(500)=850

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There should be 500 units of mix A, 850 units of mix B, and 650 units of mix C

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