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 14% protein?
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Let x = amount of soybean meal used
y = amount of cormeal used
x + y = 320
.16x + .08y = .15(x + y) = .15(320) = 48
.16x + .08y = 48
That's two equation and two unknowns:
x + y = 320
.16x + .08y = 48
Multiply the first equation by .08:
.08x + .08y = 25.6
.16x + .08y = 48
Subtract the first equation from the second equation:
.16x - .08x + .08y - .08y = 48 = 25.6
.08x = 22.4
Divide by .08:
x = 280
x + y = 320
280 + y = 320
y = 40
Use 280 lbs of soybean meal and 40 lbs of cornmeal.
To find the quantities of soybean meal and cornmeal that need to be mixed to achieve the desired protein percentage, we can set up an equation based on the proportions of protein in each ingredient.
Let's assume x represents the pounds of soybean meal and y represents the pounds of cornmeal in the mixture.
We know that soybean meal is 16% protein, so the amount of protein contributed by soybean meal is 0.16x.
Similarly, cornmeal is 8% protein, so the amount of protein contributed by cornmeal is 0.08y.
Since we want the final mixture to be 320 pounds and have a protein content of 14%, we can write:
0.16x + 0.08y = 0.14 * 320
Simplifying the equation:
0.16x + 0.08y = 44.8
Next, we add another equation based on the total weight of the mixture:
x + y = 320
Now, we have a system of two equations:
0.16x + 0.08y = 44.8
x + y = 320
We can use substitution, elimination, or matrix methods to solve this system of equations. Let's use the substitution method:
Rearrange the second equation to express x in terms of y:
x = 320 - y
Substitute this value of x in the first equation:
0.16(320 - y) + 0.08y = 44.8
Simplify:
51.2 - 0.16y + 0.08y = 44.8
Combine like terms:
-0.08y = 44.8 - 51.2
-0.08y = -6.4
Divide both sides by -0.08:
y = -6.4 / -0.08
y = 80
Substitute this value of y back into the second equation:
x + 80 = 320
x = 320 - 80
x = 240
Therefore, to get a 320-lb mixture that is 14% protein, you need to mix 240 pounds of soybean meal and 80 pounds of cornmeal.