A botanist can purchase plant food of four different types, I, II, III, and IV.

Question

A botanist can purchase plant food of four different types, I, II, III, and IV. Each food comes in the same size bag, and the following table summarizes the number of grams of each of three nutrients that each bag contains.

The botanist wants to use a food that has these nutrients in a different proportion and determines that he will need a total of 10,000 g of A, 20,000 g of B, and 20,000 g of C. Find the number of bags of each type of food that should be ordered. (Let x = Type I bags, y = Type II bags,z = Type III bags, and w = Type IV bags. If there are infinitely many solutions, express your answers in terms of w as in Example 3.)

Looking for the answer to (x,y,z,w)= ( , , , )

Food (grams) I II III IV Nutrient A     5 5 10 5 Nutrient B 10 5 30 10 Nutrient C 5 15 10 25

Explanation / Answer

we can formulate the above information in system of equations as given below:

5x + 5y + 10z + 5w = 10000 -----(1)

10x + 5y + 30z + 10w = 10000 -----(2)

5x + 15y + 10z + 25w = 10000 -----(3)

using 1 and 2 we get:

5x + 20z = 10000 - 5w ------(4)

using 3 and 2 we get:

-25x -80z = -40000 - 55w ------(5)

using 4 and 5 we get:

20z = 10000 -80w

that is : z = 500 - 4w ------(6)

Substituting 6 in 4, we get:

x = 15w -------(7)

Substituting 6 and 7 in 1, we get:

5(15w) + 5y +10(500-4w) = 10000 -5w

75w + 5y + 5000 - 40w = 10000 - 5w

5y = 5000 - 40w

y = 1000 - 8w    -------(7)

NOTE: Here we have infinitely many solutions because we have only three equations provided with four variables (x,y,z and w). Hence the final answer will be in form of any one variable (here we have taken w as that variable).

Final answer: (x,y,z,w) = ( 15w , 1000 - 8w , 500 - 4w , w )

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