Soybean meal is 16% protein; cornmeal is 8% protein. How many pounds of each should be mixed together in order to get 320-lb mixture that is 10% protein?
x + y = 320
.16x + .08y = .10(320)
From there, just use the addition/elimination principle to solve for x and y.
To solve this problem, we can use a method called the "mixture equation" or "allegation method." The idea behind this method is to find the right proportion of the two ingredients.
Let's solve this step-by-step:
Step 1: Assign variables to the unknown quantities:
Let's say x represents the pounds of soybean meal to be mixed, and y represents the pounds of cornmeal to be mixed.
Step 2: Set up the equation for the protein content:
Since soybean meal is 16% protein and cornmeal is 8% protein, the total protein in the mixture can be calculated as follows:
(16x + 8y)/(x + y) = 10
Step 3: Set up the equation for the total weight:
The total weight of the mixture is given as 320 pounds:
x + y = 320
Step 4: Solve the system of equations:
We now have two equations:
(16x + 8y)/(x + y) = 10
x + y = 320
We can solve this system of equations to find the values of x and y.
To do that, multiply both sides of the first equation by (x + y) to eliminate the denominator:
16x + 8y = 10(x + y)
Simplifying, we get:
16x + 8y = 10x + 10y
Rearranging the terms, we get:
6x = 2y
Dividing both sides by 2, we further simplify to:
3x = y
Now substitute this value of y into the second equation:
x + 3x = 320
Combining like terms, we get:
4x = 320
Dividing both sides by 4, we find:
x = 80
Substitute this value of x back into the second equation to solve for y:
80 + y = 320
Subtracting 80 from both sides, we have:
y = 320 - 80 = 240
So, you need 80 pounds of soybean meal and 240 pounds of cornmeal to get a 320-pound mixture that is 10% protein.