What is Density, by formula?

Example 46.2 (Reading the letters)

A formula is a sentence in shorthand. $\rho = m/V$ reads: “to find the density, divide the sample’s mass by its volume.” The letters are placeholders — for this pebble, $m$ becomes $75\,\mathrm{g}$ and $V$ becomes $30\,\mathrm{cm}^{3}$:

Same division as last year — but the formula remembers the recipe for every pebble to come.

Example 46.4 (The recipes at work)

Mass from volume: what does $250\,\mathrm{cm}^{3}$ of iron ($\rho = 7.9$) weigh? $m = 7.9 \times 250 = 1975\,\mathrm{g}$ — nearly two kilograms in a coffee-mug’s bulk. Volume from mass: what room does $540\,\mathrm{g}$ of oak ($\rho = 0.6$) take? $V = 540 \div 0.6 = 900\,\mathrm{cm}^{3}$. Sanity partner: iron answers should be heavy-and-small, wood answers light-and-large — the density table is the judging step’s best friend.

Example 46.8 (Alloys and in-betweens)

Blend two substances and the blend’s density lands between their signatures — nearer the more generous ingredient. Bronze (copper $9.0$ with a little tin) signs near $8.8$; the crown’s gold-and-silver blend signed between $10.5$ and $19.3$; sea water, salt dissolved in water, edges up to about $1.03$. Between-ness is itself a clue: a reading of $12$ from a “pure gold” bar is a confession of company.