A dairy farmer wants to mix 85% protein supplement and a standard 35% protein ration to make 1400 pounds of high grade 45% protein ration. How many pounds of each should he use?
Let x be the pounds of the 85% protein supplement and y be the pounds of the 35% protein ration.
We know that the total weight of the mixture is 1400 pounds, so x + y = 1400.
We also know that the protein content in the mixture is 45%, so the protein in the 85% supplement is 0.85x pounds and the protein in the 35% ration is 0.35y pounds.
Therefore, the protein in the final mixture is 0.85x + 0.35y pounds.
Since the total weight of the mixture is 1400 pounds, we can write the equation 0.85x + 0.35y = 0.45 * 1400.
Simplifying, we have 0.85x + 0.35y = 630.
Multiplying the first equation by 0.35 gives us 0.35x + 0.35y = 0.35 * 1400.
Simplifying, we have 0.35x + 0.35y = 490.
Subtracting this equation from the previous one, we get 0.85x - 0.35x = 630 - 490.
Simplifying, we have 0.5x = 140.
Dividing both sides by 0.5, we get x = 280.
Substituting this value back into the first equation, we have 280 + y = 1400.
Subtracting 280 from both sides, we get y = 1120.
Therefore, the farmer should use 280 pounds of the 85% protein supplement and 1120 pounds of the 35% protein ration.
To determine the amount of each protein supplement the dairy farmer needs to use, we can set up a system of equations.
Let x be the amount (in pounds) of the 85% protein supplement.
Let y be the amount (in pounds) of the 35% protein ration.
The total weight of the mixture is 1400 pounds, so we have the equation:
x + y = 1400
The protein content in the mixture is 45%, so we have another equation based on the protein content:
0.85x + 0.35y = 0.45 * 1400
Simplifying the second equation, we get:
0.85x + 0.35y = 630
Now we can solve the system of equations to find the values of x and y.
Using the substitution method, we can solve for x in terms of y:
x = 1400 - y
Substituting this value into the second equation, we have:
0.85(1400 - y) + 0.35y = 630
Simplifying further:
1190 - 0.85y + 0.35y = 630
Combining like terms:
0.50y = 560
Dividing both sides by 0.50:
y = 1120
Now substituting the value of y back into the equation x + y = 1400:
x + 1120 = 1400
x = 280
Therefore, the dairy farmer should use 280 pounds of the 85% protein supplement and 1120 pounds of the 35% protein ration to make 1400 pounds of high grade 45% protein ration.