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For fractional numbers other than n/2 and n/4 (which can also be expressed as n fourth/s), such as 1/3 (one third) and 7/10 (seven tenths), the denominator is expressed in the ordinal form of the number.

An explanation given on Quora is that it comes from "'a fourth part of a whole' (or, in general, ' part of a whole') which was a turn of speech that communicated clearly enough that you divided the whole into four (or n) parts and then you only took one of them." However this is unsourced.

French uses the same system but I did not find any source on any potential connection, and I have no idea where the French got the way of expression from.

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    Actually, "second" is the second minute time unit.
    – Hot Licks
    Commented Jan 1, 2020 at 23:21
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    Note also that 3/32 is "three thirty-seconds" (the "half/halves" only applies in n/2 situations). I suspect that we use what appear to be ordinals in English fractions, they probably come from phrases like "the fourth part of n," where n is any divisible object or quantity. You've made me curious about this as well.
    – Robusto
    Commented Jan 2, 2020 at 0:19
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    @RMac My question would then be why English constructs the denominator in a fractional number the same way, with a couple exceptions, as it constructs ordinal numbers, or vice-versa.
    – xngtng
    Commented Jan 2, 2020 at 0:24
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    @HotLicks It's common to have different rules for different ranges of numbers. E.g. eleven and twelve aren't oneteen and twoteen, and the teens aren't tenty-something.
    – Barmar
    Commented Jan 3, 2020 at 1:59
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    Just a thought... I grew up (in the 1950s in NZ) and remember it as "three thirty-secondth" or "three thirty-sec'nths" or sometimes "three thirty-two" when people spoke (I first remember this with spanner sizes). "Three thirty-seconds" sounded wrong to my ears when I first heard it some years later. The workshop had Kiwi and English and Scottish people working in it. I think it's possible that different dialects had slightly different ways of expressing fractions and over the last century or so of mathematics being taught in schools the language has become standardised to what we know now.
    – PeterH
    Commented Mar 6, 2020 at 9:18

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All quotes from OED (on line - subscription)

th, suffix2

Pronunciation: Primary stress is retained by the usual stressed syllable of the preceding element; see e.g. hundredth adj. and n.

[Used for] Forming ordinal numbers; in modern literary English used with all simple numbers from fourth onward; representing Old English -þa, -þe, or -oða, -oðe, used with all ordinals except fífta, sixta, ellefta, twelfta, which had the ending -ta, -te; in Scottish, northern English, and many midland dialects the latter, in form -t, is used with all simple numerals after third (fourt, fift, sixt, sevent, tent, hundert, etc.). In Kentish and Old Northumbrian those from seventh to tenth had formerly the ending -da, -de. All these variations, -th, -t, -d, represent an original Indo-European -tos (cf. Greek πέμπ-τος, Latin quin-tus), understood to be identical with one of the suffixes of the superlative degree. In Old English fífta, sixta, the original t was retained, being protected by the preceding consonant; the -þa and -da were due to the position of the stress accent, according to Verner's Law.

Note: The ordinals from twentieth to ninetieth have -eth, Old English -oða, -oðe. In compound numerals -th is added only to the last, as 1/ 1345, the one thousand three hundred and forty-fifth part; in his one-and-twentieth year.

Taking the example pf One hundred", hundredth, is both an adjective and a noun:

Etymology: < hundred n. and adj. + -th suffix2...

The ordinal numeral corresponding to the cardinal numeral hundred n. and adj.

A. adj. 1. Coming last in order of a hundred successive individuals.

1630 M. Drayton Noahs Floud in Muses Elizium 115 On the six hundred yeere of that iust man The second month, the seuenteenth day began, That horrid Deluge.

1841 W. Spalding Italy & Ital. Islands I. 103 Extending to the hundredth milestone.

2. hundredth part n. one of a hundred equal parts into which a whole is or may be divided.

a1300 Cursor Mundi 23140 Þe hundret [Gött. hundreth, F. hundre, Tr. hundride] part i mai noght mele.

1833 N. Arnott Elements Physics (ed. 5) I. 39 Compressed..so as to have bulk about a hundredth part less.

B. n. 1. A hundredth part.

1774 C. J. Phipps Voy. N. Pole 124 Divided..by a Vernier division into hundredths of an inch.

1861 J. S. Mill Utilitarianism ii. 26 Ninety-nine hundredths of all our actions are done from other motives.

You will see that the fraction/ordinal is used as a noun.

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