10000 kg of whole milk of 5% fat is used to prepare standard milk using skim milk of 0.5% fat. Calculate the amount of skim milk and standard milk.
Understand the Problem
The question asks us to calculate the amount of skim milk needed and the amount of standard milk produced when preparing standard milk from 10000 kg of whole milk with 5% fat, using skim milk with 0.5% fat. To do this, we'll need to set up a system of equations based on the conservation of mass and the conservation of fat content.
Answer
Skim milk needed: $20000$ kg Standard milk produced: $30000$ kg
Answer for screen readers
Skim milk needed: $20000$ kg Standard milk produced: $30000$ kg
Steps to Solve
- Define variables
Let $x$ be the amount of skim milk (0.5% fat) needed, and $y$ be the amount of standard milk (2% fat) produced. All amounts are in kg.
- Set up the mass balance equation
The total mass of the skim milk and whole milk must equal the total mass of the standard milk produced: $$ x + 10000 = y $$
- Set up the fat balance equation
The total amount of fat in the skim milk plus the total amount of fat in the whole milk must equal the total amount of fat in the standard milk. This gives us: $$ 0.005x + 0.05(10000) = 0.02y $$
- Simplify the fat balance equation
$$ 0.005x + 500 = 0.02y $$
- Solve the system of equations
We can solve the system of equations using substitution. From the mass balance equation, we have $x = y - 10000$. Substitute this into the fat balance equation: $$ 0.005(y - 10000) + 500 = 0.02y $$
- Solve for y
Expand and simplify the equation: $$ 0.005y - 50 + 500 = 0.02y $$ $$ 450 = 0.015y $$ $$ y = \frac{450}{0.015} = 30000 $$
- Solve for x
Now substitute the value of $y$ back into the equation $x = y - 10000$: $$ x = 30000 - 10000 = 20000 $$
- State the results
The amount of skim milk needed is 20000 kg, and the amount of standard milk produced is 30000 kg.
Skim milk needed: $20000$ kg Standard milk produced: $30000$ kg
More Information
This problem demonstrates a common application of mixture problems, often encountered in chemical engineering, food science, or even basic algebra courses. It showcases how mass balance and component balance (in this case, fat content) can be used to solve for unknown quantities in a system.
Tips
A common mistake is to forget to convert the percentages to decimals when setting up the equations, or to mix up the variables. Also, students often make errors in the algebraic manipulation of the equations, particularly when distributing and combining like terms. Carefully double check the equations and the algebra.
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