Why do I need systems of equations to mix solutions?
When you mix two strengths of something, two facts must hold at once: the amounts add up to the total, and the active ingredient adds up to the target strength. That's a system of two equations. Solving it tells you exactly how much of each to pour, whether it's fertilizer, cleaner, paint, or medicine.
- Use this much A
- 10 L
- Use this much B
- 10 L
Challenge: Find a mix that uses equal amounts of A and B. Done!
Play
Pick two strengths and the one you want. The beakers show how much of each to pour.
Challenge: Find a mix that uses equal amounts of A and B. The box under the picture turns green when you get it.
Stuck? Pick one of the examples from the “Try an example” menu, or press “New example.”
Understand
Two unknowns, and , need two equations:
The first says the amounts add up. The second says the active ingredient adds up. Substitute into the second and solve:
If , the denominator is zero, which is the math saying that two equal strengths can't make a different one.
Use
Every input has a unit menu, so you can type values in the units you already have. Results follow your units.
Show the work
- Total amount
x + y = 20 - Amount of the active ingredient
0.1x + 0.4y = 0.25 \times 20 - Solve for x
x = V\frac{c - b}{a - b} = 20 \times \frac{25 - 40}{10 - 40} = 10 - Then y
y = 20 - 10 = 10
Export
Enter strengths as percentages and the total you need. For dilution with plain water, set one strength to 0%.
- Add the stronger part to the weaker part, especially with acids.
- Measure by volume unless the label gives percent by weight.
For learning and estimation. Verify with applicable codes, standards, and a qualified professional before using in design, construction, or safety-critical work.
Cheat card
| Symbol | Meaning | Unit |
|---|---|---|
| amounts of A and B | L | |
| strengths of A and B | % | |
| target strength | % | |
| total amount | L |
- The target must lie between the two strengths. You can't mix two weak solutions into a strong one.
- Water counts as a 0% solution when diluting.
- Check by computing the active ingredient in the final mix.
Where it’s used
- Chemistry
Chemists and nurses dilute stock solutions to exact concentrations. - Home Projects
Mixing cleaning concentrate or paint tints to a target strength. - Gardening
Blending two fertilizers to get the nitrogen percentage a lawn needs.
Related exhibits
Start hereSale price and tax: what you really pay
30% off, then tax on top. Percent math decides the number at the register.
Go furtherCooling coffee: when is it drinkable?
Hot things cool fast, then slower and slower. Exponential decay predicts the sip.
Also tryPhone plans: finding the break-even point
Two plans, two lines. Where they cross, the price is the same.
Questions people ask
How do I dilute a solution to a lower strength?
Use water as the second solution (0%). To make 4 L of 10% from 60% bleach, use about 0.67 L of bleach and 3.33 L of water.
Why can't I reach any strength I want?
A mix is always between its ingredients. From 10% and 40% you can make anything from 10% to 40%, but not 60%.
What happens if both solutions are the same strength?
Every mix has that same strength, so the system has no solution for any other target. The site shows