Mixing Problems
Rate in minus rate out for concentrations.
Key idea
Rate in minus rate out for concentrations.
In depth
For a tank, rate of change of salt = (rate in) − (rate out). The concentration out depends on the current amount, giving a linear first-order equation: A′ = r_in c_in − (A/V)r_out.
Worked example
A 100 L tank with brine at 2 g/L entering at 5 L/min and leaving at 5 L/min: A′ = 10 − A/20.
Real-world uses
- Working out salt in a tank when brine flows in and out
- Drug levels in the bloodstream
Common mistake
Forgetting that the volume changes if inflow and outflow rates differ.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Mixing Problems?
Rate in minus rate out for concentrations. It is a topic at the advanced level in Differential Equations, Abstract Algebra & Applied Math.
Where is Mixing Problems used in real life?
Real-world examples: Working out salt in a tank when brine flows in and out; Drug levels in the bloodstream.
What should I learn before Mixing Problems?
Make sure you are comfortable with Newton’s Law of Cooling, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Mixing Problems?
Forgetting that the volume changes if inflow and outflow rates differ.
How can I practise Mixing Problems?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Mixing Problems?
Move on to Exact Equations next.
Related advanced topics in Differential Equations, Abstract Algebra & Applied Math
- What Is a Differential Equation? – An equation relating a function and its derivatives.
- Order and Linearity of ODEs – Classify equations by highest derivative and linearity.
- Direction Fields – Slope fields show solution curves visually.
- Separable Equations – Separate variables and integrate both sides.
- First-Order Linear Equations – Use an integrating factor.
- Exponential Growth and Decay Models – dy/dt = ky gives y = y₀eᵏᵗ.