Soybean meal is 12% protein: cornmeal is 6% protein. How many pounds of each should be mixed together in order to get 240-lb. mixture that is 8% protein?
solve :
.12x + .06(240-x) = .08(240)
To solve this problem, we can use a method called the mixture problem, which involves finding the ratio of the two ingredients.
Step 1: Assign variables
Let's assume the pounds of soybean meal to be x, and the pounds of cornmeal to be y. We need to find the values of x and y.
Step 2: Set up the equation
Since the concentration of protein in soybean meal is 12% and in cornmeal is 6%, we can set up the following equation based on the information given:
(0.12x + 0.06y) / (x + y) = 0.08
Step 3: Solve the equation
To get rid of the denominator, we can multiply both sides of the equation by (x + y):
0.12x + 0.06y = 0.08(x + y)
Next, distribute the 0.08 on the right side of the equation:
0.12x + 0.06y = 0.08x + 0.08y
Now, move all the terms containing x to one side and all the terms containing y to the other side:
0.12x - 0.08x = 0.08y - 0.06y
0.04x = 0.02y
Divide both sides by 0.02:
x = 0.5y
Step 4: Substitute the value of x in terms of y
Since x = 0.5y, we can substitute this value back into the original equation:
0.12(0.5y) + 0.06y = 0.08(0.5y + y)
Simplify the equation:
0.06y + 0.06y = 0.08(1.5y)
Combine like terms:
0.12y = 0.12y
Step 5: Solve for y
Since the equation simplifies to 0.12y = 0.12y, we realize that y can be any value.
Step 6: Determine the value of y
Since we can choose any value for y, let's assume y = 100 pounds.
Step 7: Calculate the value of x
Using the equation x = 0.5y, substitute the value of y:
x = 0.5(100)
x = 50
Therefore, to obtain a 240-lb. mixture with 8% protein, mix 50 pounds of soybean meal with 100 pounds of cornmeal.