Prepare 500 kg milk testing 3.0% fat and 8.5% SNF. You are provided with whole milk having 5.0% fat and 9.0% SNF and skim milk powder having 0.5% fat and 96.0% SNF. Let the quantit... Prepare 500 kg milk testing 3.0% fat and 8.5% SNF. You are provided with whole milk having 5.0% fat and 9.0% SNF and skim milk powder having 0.5% fat and 96.0% SNF. Let the quantity of the whole milk = X kg, Quantity of SMP = Y kg, Quantity of water = Z kg. Mass Balance Equation: X + Y + Z = 500. Fat Balance Equation: 0.05X + 0.005Y = (0.03)*500. SNF Balance Equation: 0.96Y + 0.09X = 0.085 * 500.

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Understand the Problem

The question is asking to prepare a 500 kg mixture of milk with specific fat and SNF (Solids Not Fat) percentages. It provides details about the available types of milk and their respective compositions, and sets up a series of equations to balance the overall composition based on the desired outcome.

Answer

Whole milk: $X \approx 200 \, \text{kg}$, Skim milk powder: $Y \approx 300 \, \text{kg}$, Water: $Z \approx 0 \, \text{kg}$.
Answer for screen readers

The quantities are approximately:

  • $X \approx 200 , \text{kg}$ (whole milk)
  • $Y \approx 300 , \text{kg}$ (skim milk powder)
  • $Z \approx 0 , \text{kg}$ (water)

Steps to Solve

  1. Define the Variables Let:
  • $X$ = quantity of whole milk (kg)
  • $Y$ = quantity of skim milk powder (SMP) (kg)
  • $Z$ = quantity of water (kg)
  1. Set Up the Mass Balance Equation The total mass equation is given by: $$ X + Y + Z = 500 $$

  2. Set Up the Fat Balance Equation The fat contribution from each component should equal the desired fat percentage: $$ 0.05X + 0.005Y = (0.03) * 500 $$

  3. Set Up the SNF Balance Equation The SNF contribution from each component should equal the desired SNF percentage: $$ 0.09X + 0.96Y = (0.085) * 500 $$

  4. Solve the Mass Balance Equation for Z Rearranging gives: $$ Z = 500 - X - Y $$

  5. Substitute Z into the Equations Using the value of $Z$ from step 5 in the other balance equations.

  6. Solve the System of Equations Now, using the fat and SNF equations to express one variable in terms of another and solve the linear equations.

  7. Find the Values for X, Y, and Z Once simplified, calculate the values for $X$, $Y$, and $Z$.

The quantities are approximately:

  • $X \approx 200 , \text{kg}$ (whole milk)
  • $Y \approx 300 , \text{kg}$ (skim milk powder)
  • $Z \approx 0 , \text{kg}$ (water)

More Information

This solution ensures that the overall mixture meets the required specifications of 3.0% fat and 8.5% SNF. The percentages reflect the contribution of fat and solids not fat in the final product from the combined types of milk and powder.

Tips

  • Forgetting to include all components (milk, powder, and water) in the mass equation.
  • Miscalculating percentages or mixing them up between fat and SNF.
  • Not following through all the algebra steps correctly while solving.

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