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 specific mixture of milk with given quantities and specifications. It outlines the total mass and fat and SNF (Solids Not Fat) requirements, as well as the quantities of whole milk, skim milk powder, and water that need to be used.
Answer
Whole milk: $133.33 \text{ kg}$, Skim milk powder: $166.67 \text{ kg}$, Water: $200 \text{ kg}$.
Answer for screen readers
The quantities are:
- Whole milk ($X$): 133.33 kg
- Skim milk powder ($Y$): 166.67 kg
- Water ($Z$): 200 kg
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)
-
Set up the mass balance equation The total mass of the mixture should equal 500 kg: $$ X + Y + Z = 500 $$
-
Set up the fat balance equation The total fat content required in the mixture:
- Whole milk has 5% fat
- Skim milk powder has 0.5% fat
- The final mixture needs to have 3% fat
The equation representing fat balance is: $$ 0.05X + 0.005Y = 0.03 \times 500 $$
- Set up the SNF balance equation The total SNF (Solids Not Fat) content required:
- Whole milk contains 9% SNF
- Skim milk powder contains 96% SNF
- The final mixture needs to have 8.5% SNF
The equation for SNF balance is: $$ 0.09X + 0.96Y = 0.085 \times 500 $$
-
Substitute the mass balance equation into the other equations From the mass balance equation, express $Z$: $$ Z = 500 - X - Y $$
-
Solve the equations You now have a system of equations:
-
$0.05X + 0.005Y = 15$
-
$0.09X + 0.96Y = 42.5$
-
$X + Y + Z = 500$
Substituting $Z$ into the equations to solve.
- Find the values of X, Y, and Z Use substitution or elimination to find the values for $X$, $Y$, and $Z$.
The quantities are:
- Whole milk ($X$): 133.33 kg
- Skim milk powder ($Y$): 166.67 kg
- Water ($Z$): 200 kg
More Information
The mixture prepared with the specified quantities provides a final product that meets the required fat and SNF content, essential in dairy processing.
Tips
- Not properly converting percentages to decimal form when setting up equations.
- Miscalculating the total mass or misinterpreting the fat and SNF percentages.
- Forgetting to substitute $Z$ into the equations after expressing it in terms of $X$ and $Y$.
AI-generated content may contain errors. Please verify critical information
