Mixture Problem in the Food

¶ … mixture problem in the food process manufacturing industry where there are inputs with different compositions that are to be combined to meet a different composition requirement. Our problem example is as follows:

How many gallons of 30% milk and how many of 60% milk must be mixed together to produce 18 gallons of a 50% milk solution?

To solve this problem, we will let x be the number of gallons of 30% milk solution and we let y be the number of gallons of 60% milk solution. Thus, we can construct the following two equations:

Total number of gallons = x + y = 18

Gallons of all milk solution =.3x +.6y = 9 (Note:.3 = 30%,.6 = 60% and This equation is solved using the substitution method where one of the equations is first manipulated such that one variable is defined in terms of the other as shown below:

x

Next, the new definition of one variable, in this case y, is substituted for the same variable in the other equation. In this instance, the definition of y, which is 18 - x is substituted for the y term found in the other equation to obtain:

3x +.6 (18 - x) = 9

First, using the distributed property, we get:

3x + 10.8 -.6x = 9

Combining like terms we obtain:

3x + 10.8 = 9

Subtracting 10.8 from each side, we have:

3x = -1.8

Dividing both sides by -.3 we now solve for x:

Now that x is known, we can plug this value into any of the original equations and obtain a value for y:

y = 18