Blending Two Feeds to a Protein Target

The Pearson square is a piece of nineteenth-century arithmetic pedagogy that has outlived most of its contemporaries because it works and it can be drawn on a feed bag. Put the target protein in the middle of a square, the two ingredient analyses on the left corners, subtract diagonally and ignore the signs, and the numbers on the right corners are the parts of each ingredient. It is taught as a geometric trick, and it is exactly the algebra $x = (a_t - a_2)/(a_1 - a_2)$.

Blending 44% soybean meal with 9% maize to a 16% ration gives $(16-9)/(44-9) = 7/35 = 20\%$ soybean meal and 80% maize. The square makes one constraint visually obvious in a way the algebra does not: a blend always lies between its ingredients. No mixture of a 44% and a 9% feed reaches 50% or drops to 5%, and if the target sits outside the pair, the arithmetic returns a negative or greater-than-one share — which is the equation reporting that the blend is impossible rather than quietly returning a wrong answer.

The same square blends fertiliser, and it is worth noticing that the equation is identical rather than merely similar. Feed formulators and fertiliser blenders both need one ingredient's share of a two-part mixture hitting a target concentration, and the mathematics does not care which trade is asking. Anyone who has learned it in one setting already knows it in the other.

What the square cannot do is balance more than one nutrient at a time, and that is its real limit. A ration blended to a protein target lands wherever the two ingredients happen to put its energy, fibre, calcium, phosphorus and vitamins, and those are rarely where they should be. Real ration formulation is a constrained optimisation over a dozen nutrients and as many ingredients, solved by linear program to find the cheapest mix that satisfies all of them at once. The square remains the right tool for a single nutrient and a quick check on a bag, and it stops there — a two-ingredient blend that meets a protein target while sitting short on energy will not grow anything at the rate the protein figure implies.