A grower does not want seeds; a grower wants plants, at a target density, on a date. Everything between the two is loss, and every loss divides.

Geometry first. $P = \dfrac{1}{w\,s}$ — P equals one over w s. $P$ is the plant population per unit area, $w$ the row spacing centre to centre, and $s$ the in-row seed spacing. Each plant owns a rectangle $w$ by $s$, and the population is one plant per rectangle — so the two spacings multiply, never add. 50 cm rows with a seed every 20 cm is 0.1 m² per plant, which is 10 plants/m² and 100,000 per hectare.

The tag. $PLS = p \times g$ — P-L-S equals p times g. $p$ is purity, the share of the bag by weight that is the species you bought; $g$ is laboratory germination, the share of those seeds that grow on wet paper in the warm. $PLS$ is pure live seed. They are two independent screens acting one after the other on the same bag, so they multiply — averaging them always flatters the lot, and forage seed is sold by weight, so purity is money.

The field's own verdict. $E = \dfrac{P}{S}$, where $P$ is plants counted per unit area and $S$ the seeds sown per unit area. Count PLANTS, not tillers: one cereal plant carries three to six stems, and counting stems is how emergence comes back above 100%. Measured once on your own farm, this is the single most useful number to carry into next spring.

And the calculation it all feeds. $S = \dfrac{P \cdot TKW}{1000\,G\,E}$ — S equals P T-K-W over one thousand G E. $S$ is the seeding rate by weight, $P$ the target established population, $TKW$ the thousand-kernel weight in grams from the tag, $G$ the tag germination and $E$ the expected field emergence, both bare fractions. The 1000 is not a fudge: $TKW$ weighs a thousand seeds, so dividing by 1000 gives the weight of one.

The shape is the lesson, and it is the same shape as gross-over-net two lessons back: both survival fractions sit on the bottom, so both push the rate up. Multiply by them instead and poorer seed gets sown thinner — which is exactly backwards, and expensive in the direction nobody notices until the stand is counted.