Buckets to a Safe Bath: A Mixing Ratio for Bath Water Without a Thermometer

This study examines a practical question in everyday hygiene by explicit arithmetic from stated assumptions. At a cold supply of 15 C, the exact ratio is 4.9 buckets of hot per 10 buckets of mixed water, and whole-bucket filling gives a bath at 36.5 C, inside the band. At 8 C the hot share rises to 5.6 buckets in 10 and at 22 C it falls to 3.9 in 10. Hot-supply drift of plus or minus 3 C moves the finished bath by plus or minus 1.3 C at the middle ratio. Whole-bucket counting stays within the 36 to 38 C band for cold supplies between 10 and 20 C; outside that range a half-bucket final adjustment is required. Filling hot first exposes a peak intermediate temperature of 60 C, against 33 C for the cold-first alternating sequence at the same final temperature. The proposed arrangement is as follows. The deliverable is a counted filling rule: cold first to a stated depth, then alternating buckets in a stated ratio, finishing with a stated adjustment bucket chosen by the season's cold temperature. Cold first is a safety feature, not a mixing detail, since a tub that is momentarily too cold harms nobody while a tub momentarily too hot can scald in seconds. All results are derived from the accompanying simulation script, in which every assumption appears as a named, replaceable value. SIMULATED, NOT MEASURED.