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.
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
- Define the Variables Let:
- $X$ = quantity of whole milk (kg)
- $Y$ = quantity of skim milk powder (SMP) (kg)
- $Z$ = quantity of water (kg)
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Set Up the Mass Balance Equation The total mass equation is given by: $$ X + Y + Z = 500 $$
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Set Up the Fat Balance Equation The fat contribution from each component should equal the desired fat percentage: $$ 0.05X + 0.005Y = (0.03) * 500 $$
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Set Up the SNF Balance Equation The SNF contribution from each component should equal the desired SNF percentage: $$ 0.09X + 0.96Y = (0.085) * 500 $$
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Solve the Mass Balance Equation for Z Rearranging gives: $$ Z = 500 - X - Y $$
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Substitute Z into the Equations Using the value of $Z$ from step 5 in the other balance equations.
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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.
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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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