Pearson's Square - Or Why Didn't My Chemistry Lecturers Teach Me This?

A common problem in chemistry (and feedlots, and home distilling, etc),

You have two solutions of different concentrations, let's call them H (high concentration) and L (low concentration), and you want to mix them to create a solution of concentration F (final concentration).

Using algebra, it takes a bit of time to get this.
xH + (1-x)L = C, etc, etc, etc, solve for x.

Using the Pearson's Square, it goes like this:

H - F = parts of solution L
F - L = parts of solution H

Gaaahhh.

Here's a worked example from home distillation (The most recent example I've come across).
You have a solution (H) of 95% ethanol straight from a still, and a wine (L) of 8% that you want to fortify up to 20% ethanol (F).

H - F = 95 - 20 = 75 parts of the wine.
F - L = 20 - 8 = 12 parts of the distillate.

Thus (simplifying our ratios), we can make our fortified wine up by mixing 4 parts distillate and 25 parts wine.

A feedlot example would be as follows. You have a cheap fodder crop with 2% protein by dry weight, and a more expensive high-protein grain with 10% protein by dry weight. You want to supplement the fodder crop with the grain, to get a protein concentration of 4%.

10 - 4 = 6 parts fodder.
  4 - 2 = 2 parts grain.

Simplifying, you would mix 1 part grain to 3 parts inexpensive fodder. (In reality feedlot calculations are more complex than this, balancing sources of non-protein nitrogen, protein, carbohydrate and mineral supplements, as well as issues of bio-availability, but Pearson's Square enables you to do quick back-of-envelope calculations on pairs of sources.)

Pearson's Square only works for binary solutions of course, but you'd be surprised how often this type of problem shows up.