Why are ordinal numbers used to express fractions?

Solution 1:

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.