10000 kg of whole milk of 0.5% fat is used to prepare standard milk of 3.5% fat. Calculate the skim milk required and standard milk produced.
Understand the Problem
The question is asking to calculate the amount of skim milk needed and the amount of standard milk produced when using 10000 kg of whole milk with 0.5% fat to prepare standard milk of 3.5% fat.
Answer
Skim milk needed: $\frac{30000}{7}$ kg $\approx 4285.71$ kg Standard milk produced: $\frac{100000}{7}$ kg $\approx 14285.71$ kg
Answer for screen readers
Amount of skim milk needed: $x = \frac{30000}{7} \approx 4285.71$ kg
Amount of standard milk produced: $y = \frac{100000}{7} \approx 14285.71$ kg
Steps to Solve
- Define variables
Let $x$ be the amount (in kg) of skim milk needed, and $y$ be the amount (in kg) of standard milk produced.
- Set up the total mass equation
The total mass of the mixture is the sum of the skim milk and whole milk, which equals the amount of standard milk produced.
$$ x + 10000 = y $$
- Set up the fat content equation
The fat content of the whole milk plus the fat content of the skim milk equals the fat content of the standard milk. Skim milk has approximately 0% fat.
$$ (10000 \times 0.005) + (x \times 0) = y \times 0.035 $$
- Simplify the fat content equation
$$ 50 = 0.035y $$
- Solve for y
Divide both sides of the fat content equation by 0.035 to find the value of $y$.
$$ y = \frac{50}{0.035} = \frac{50000}{3.5} = \frac{100000}{7} \approx 14285.71 \text{ kg} $$
- Solve for x
Substitute the value of $y$ back into the total mass equation to solve for $x$.
$$ x + 10000 = \frac{100000}{7} $$ $$ x = \frac{100000}{7} - 10000 = \frac{100000 - 70000}{7} = \frac{30000}{7} \approx 4285.71 \text{ kg} $$
Amount of skim milk needed: $x = \frac{30000}{7} \approx 4285.71$ kg
Amount of standard milk produced: $y = \frac{100000}{7} \approx 14285.71$ kg
More Information
This calculation helps in standardizing milk fat content, an important process in the dairy industry. By adding skim milk, the fat percentage of whole milk can be precisely adjusted to meet specific product requirements.
Tips
A common mistake is forgetting to convert percentages to decimals when setting up the fat content equation. For example, using 0.5 instead of 0.005 and 3.5 instead of 0.035. Another mistake is incorrectly setting up the mass balance equation. Ensure that the total mass is conserved during the mixing process.
AI-generated content may contain errors. Please verify critical information
