Why is the 会 in 会计 pronounced kuài?
In Old Chinese, it is generally thought that some words followed regular morphological alternations (which are preserved in a few places in MSM, but &quot;frozen&quot;, i.e., no longer productive). For instance:

Verb/Noun
处 chu3 &quot;to dwell&quot; / chu4 &quot;a place&quot;
数 shu3 &quot;to count&quot; / shu4 &quot;a number&quot;
知 zhi1 &quot;to know&quot; / (also 智) zhi4 &quot;knowledge&quot;

Similarly:

Active/Inactive Verb
好 hao4 &quot;to like&quot; / hao3 &quot;to be good&quot;
系 ji4 &quot;to tie&quot; / xi4 &quot;to be connected&quot;
会 kuai4 &quot;to bring together&quot; / hui4 &quot;to come together&quot;

Phonetic reconstructions for these alternations generally follow a regular pattern (e.g., voiced/voiceless, 去声/other tone), although specific reconstructions differ.
会 is a somewhat odd example, in that the hui4 reading has displaced the kuai4 reading in almost all places except proper nouns and 会计 = accounting. Branner gives the example from Confucius:

君子以文会友，以友辅仁

The customary reading of 会 in this passage is hui4, but etymologically, it really should be kuai4.
My references are taken from Branner's excellent essay, &quot;On Early Chinese Morphology and its
Intellectual History&quot;
EDIT:
The Middle Chinese readings corresponding to &quot;hui4&quot; and &quot;kuai4&quot; are &quot;hwajH&quot; and &quot;kwajH&quot; in Baxter's notation; the initials being 匣 and 見. Based on this, one would expect the MSM pronunciation of the latter to be &quot;gui4&quot; rather than &quot;kuai4&quot;. The common Japanese On reading of かい &quot;kai&quot; (attested earlier as くわい &quot;kwai&quot;) is actually a kan-on reading. According to wikipedia, the k- initial in a kan-on reading could correspond to either a 匣 or a 見 MC initial. In any event, the &quot;kai&quot; reading is used in both the 'meeting' and 'accounting' meanings, regardless of which MC reading it corresponds to. Some speculate (e.g., Victor Mair, p31) that the MSM &quot;kuai4&quot; may have been influenced by the Japanese &quot;kai&quot; (i.e., that the etymologically expected &quot;gui4&quot; was displaced by &quot;kuai4&quot; when 会计 was re-borrowed from Japanese to mean &quot;accounting&quot;). It is not the case, however, that the hui4/kuai4 distinction is an artifact of Japanese borrowing--a meaning of &quot;settling accounts&quot; is attested dating back well before the Japanese neologism for accounting (see halfway down random blogpost quoting from dictionary).