the lever rule
A see-saw balances when the lighter child sits far from the pivot and the heavier one sits close. The lever rule uses exactly this balance to read amounts off a phase diagram. Land your overall composition in a two-phase field, draw the tie line (the horizontal line at that temperature), and treat your composition point as the pivot of a see-saw whose ends rest on the two phase boundaries. The amount of each phase is proportional to the length of the arm on the opposite side — the far phase is present in the smaller amount, just like the child who must sit far out to balance.
In numbers it is a clean fraction. If your overall composition is C0, the liquid at that temperature has composition C_L (read off the liquidus) and the solid has composition C_S (read off the solidus), then the fraction that is liquid equals (C0 - C_S) divided by (C_L - C_S), and the fraction solid is the leftover, (C_L - C0) divided by (C_L - C_S). Notice each fraction uses the opposite arm — that is the inverse in inverse lever rule. Work one: with C_S = 20, C0 = 50, C_L = 80 (percent of some component), the liquid fraction is (50 - 20)/(80 - 20) = 30/60 = one half. Half liquid, half solid.
This is the quantitative payoff of a phase diagram — not just which phases exist, but how much of each. A ceramist uses it constantly: how much glassy liquid has formed in a porcelain at its firing temperature (too much and it slumps, too little and it stays porous), how much primary mullite has crystallized, how much residual solid skeleton a refractory keeps at service heat. Given a diagram and a ruler, the lever rule turns a point on a map into a real recipe of phase amounts.
Overall composition sits at 50, tie line ends at solid 20 and liquid 80. Liquid fraction = (50 - 20)/(80 - 20) = 30/60 = 0.5. So exactly half the sample is molten at that temperature. Move your point closer to the liquid end and the arm flips: more liquid, less solid — the see-saw rebalances.
Fraction of a phase equals the opposite (far) tie-line arm over the whole tie line — an inverse, see-saw balance.
Use the opposite arm — the fraction of a phase is proportional to the distance to the other boundary, not to its own. Getting the arms backwards is the classic lever-rule error, and it also only works inside a two-phase field, never in a single-phase region.