The nutritional content per ounce of three foods is presente

The nutritional content per ounce of three foods is presented in the table on the right. If a meal consisting of the three foods allows exactly 1600 calories, 78 grams of protein, and 3450 milligrams of vitamin C. how many ounces of each kind of food should be used? Food A: square oz Food B: square oz Food C: square oz

Solution

Let the quantity of the foods A, B, and C be x, y, and z oz respectively.

Then 100x+200y+400z = 1600, or, on dividing both the sides by 100, we have x+2y +4z = 16...(1)

8x+12y+13z = 78 ...(2) and

350x +750y + 250z = 3450 or, on dividing both the sides by 50, we have 7x+15y+ 5z = 69...(3)

On multiplying both the sides of the 1st equation by 7 and 8 respectively, we get 7x+14y +28z = 112...(4) and

8x+16y+32z = 128...(5)

Now, on subtracting the 3rd equation from the 4th equation, we get -y+23z = 43...(6)

Similarly, on subtracting the 2nd equation from the 5th equation, we get 4y+19z = 50...(7)

On multiplying both the sides of the 6th equation by 4, we get -4y+92z = 172...(8)

Now, on adding the 7th and the 8th equations, we get 111z = 222 so that z = 2.

Then on substituting z = 2 in the 6th equation, we get -y+46 = 43 so that y = 46-43 or, y = 3.

Finally, on substituting y = 3 and z = 2 in the 1st equation, we get x+6+8 = 16 or, x +14 = 16 so that x = 16-14 or, x = 2.

Thus the answer is x = 2,y = 3 and z = 2.