The Sixteenths Ladder

Question

What is frac316 as a percentage, instantly? Build the halving ladder that makes every sixteenth immediate, plus cousins like frac140.

Accepted Answer

The halving ladder: each rung is half the one above: frac12 = 50%, quad frac14 = 25%, quad frac18 = 12.5%, quad frac116 = 6.25%, quad frac132 = 3.125%. Multiples come free: frac316 = 3 times 6.25% = mathbf18.75%. Or descend from a neighbor: frac316 = frac14 - frac116 = 25% - 6.25%, same answer, often faster. Rungs worth owning outright: frac516 = 31.25%, frac716 = 43.75%, frac58 = 62.5%, frac78 = 87.5%. The shift cousins: dividing any rung by 10 is a decimal shift, which unlocks the 40ths and 80ths: frac120 = 5%, frac140 = 2.5% (it is frac14 div 10), frac180 = 1.25%. Alongside: frac125 = 4%, frac150 = 2%, frac1200 = 0.5% = 50 bp. Why the drill is bidirectional: the payoff is recognizing these numbers in both directions. Seeing 6.25% and knowing it is frac116 turns "6.25% of 320" into 320 div 16 = 20; seeing 18.75% of 4,800 as frac316 gives frac480016 = 300, times 3 = 900. Odds convert through the same ladder: a 13-to-3 underdog wins with probability frac316 = 18.75%. Optiver, Flow Traders, and IMC test these conversions raw on their numerical screens, and market-making games reprice them constantly, the ladder is the difference between translating odds and computing them.

The Sixteenths Ladder

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