Blending Two Fertilisers to a Target

This is the Pearson square, written as the algebra it always was. Draw a square, put the target in the middle, the two ingredient analyses on the left corners, subtract diagonally, and read the parts off the right corners. It is a lovely piece of nineteenth-century arithmetic pedagogy, and it is exactly $x = (a_t - a_2)/(a_1 - a_2)$.

Blending 46% urea with an inert filler to reach 30% takes $30/46 = 65.2\%$ urea by weight. The same square blends two real fertilisers: a 46% and a 21% product to reach 30% needs $(30-21)/(46-21) = 36\%$ of the stronger one.

The constraint the square makes obvious is that a blend always lies between its ingredients. No mixture of a 46% and a 21% product can reach 50%, and if the target falls outside the pair, the answer comes back as a negative or greater-than-one fraction — which is the algebra telling you the blend is impossible rather than giving a wrong answer.

What the square does not handle is more than one nutrient at a time. Real blending to an N-P-K target is a system of simultaneous equations, and there is usually no exact solution with only two ingredients — which is why commercial blends use three or four and a linear program. The square remains the right tool for a single nutrient, and the same equation reappears unchanged in feed formulation, where it blends two ingredients to a protein target.