A dairy farmer wants to mix a 75% protein supplement and a standard 30% protein ration to make 1200 pounds of a high-grade 45% protein ration. How many pounds of each should he use?
Let x be the number of pounds of the 75% protein supplement.
Then, the number of pounds of the 30% protein ration would be 1200 - x.
We can set up the following equation:
0.75x + 0.3(1200 - x) = 0.45(1200)
Simplifying:
0.75x + 360 - 0.3x = 540
0.45x = 180
x = 400
Therefore, the dairy farmer should use 400 pounds of the 75% protein supplement and 800 pounds of the 30% protein ration to make 1200 pounds of a high-grade 45% protein ration.
To determine the number of pounds of each protein supplement the dairy farmer should use, we can set up a system of equations based on the given information.
Let's assume the dairy farmer uses x pounds of the 75% protein supplement and y pounds of the 30% protein ration.
From the given information, we know the following:
1. The total weight of the mixture should be 1200 pounds:
x + y = 1200
2. The desired protein percentage in the mixture should be 45%:
(0.75x + 0.30y) / 1200 = 0.45
Now, we can solve this system of equations to find the values of x and y.
Step 1: Solve the first equation for x:
x = 1200 - y
Step 2: Substitute the value of x in the second equation:
(0.75(1200 - y) + 0.30y) / 1200 = 0.45
Step 3: Simplify the equation:
900 - 0.75y + 0.30y = 0.45 * 1200
900 - 0.75y + 0.30y = 540
Step 4: Combine like terms:
900 - 0.45y = 540
Step 5: Move the constant term to the right side:
-0.45y = 540 - 900
-0.45y = -360
Step 6: Divide both sides by -0.45 to solve for y:
y = -360 / -0.45
y = 800
Step 7: Substitute the value of y into the first equation to solve for x:
x + 800 = 1200
x = 1200 - 800
x = 400
Therefore, the dairy farmer should use 400 pounds of the 75% protein supplement and 800 pounds of the 30% protein ration to make 1200 pounds of a high-grade 45% protein ration.