Questions: A dairy farmer wants to mix a 65% protein supplement and a standard 15% protein ration to make 1200 pounds of a high-grade 30% protein ration. How many pounds of each should he use?

A dairy farmer wants to mix a 65% protein supplement and a standard 15% protein ration to make 1200 pounds of a high-grade 30% protein ration. How many pounds of each should he use?
Transcript text: A dairy farmer wants to mix a $65 \%$ protein supplement and a standard $15 \%$ protein ration to make 1200 pounds of a high-grade $30 \%$ protein ration. How many pounds of each should he use?
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Solution

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Solution Steps

Step 1: Define the equations based on the problem statement

We have two equations based on the given information:

  • Total weight equation: \(x + y = W_f\)
  • Protein balance equation: \(P_s \cdot x + P_r \cdot y = P_f \cdot (x + y)\)
Step 2: Solve the system of equations

Substitute \(y = W_f - x\) into the protein balance equation and solve for \(x\). This gives us \(x = \frac{P_f \cdot W_f - P_r \cdot W_f}{P_s - P_r}\) and \(y = W_f - x\).

Step 3: Calculation

After substituting the given values, we find that \(x = 360\) pounds of the protein supplement and \(y = 840\) pounds of the standard protein ration are needed.

Final Answer:

To achieve a 30% protein concentration in the final 1200 pounds mix, use 360 pounds of the high protein supplement and 840 pounds of the standard protein ration.

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