Fractions were written, as you might expect, using Roman numerals. This wasn't particularly elegant for anything more complex than adding and subtracting, but it worked great for commerce, and that was where it was most often used.
The units worked sort of like this:
- I = 1 (as) [vertical line]
- S = 1/2 (semis) [letter S]
- (or •) = 1/12 (uncia = ounce) [horizontal line or dot]
- (or ••) = 2/12 (sextans) [two horizontal lines or dots]
- And so on for multiples of 1/12
- = 1/24 (semuncia = half an ounce) [either capital sigma, or British pound sign without the crossbar]
- Ɔ = 1/48 (sicilius) [backward C]
- = 1/72 (sextula = 1/6 ounce) [backward S]
- = 1/144 (half a sextula) [backward S with a line through it]
- ℈ = 1/288 (scripulum = scruple) [backward C with a line through it]
- = 1/1728 (siliqua) [double backward C]
Two things to notice here!
One, the fraction system works sort of in base-12, because it turns out it's a lot easier to divide things when you're in base-12.
And two, most of these names are units of currency, which isn't a coincidence. Many coins were equivalent to weights (since their value came from their weight in precious metal), and were given names either from their value (sextans = "one-sixth") or from how much they weighed (siliqua = "carob seed"). Since these names meant fractions of currency, or fractions of weight, they got applied to fractions of other things too.
Here's an engineering example, from Frontinus's De Aquibus ("About Aqueducts"), with two translations (both mine). This is in the middle of a list of specifications of pipes:
Fistula quadragenaria: diametri digitos septem ℈III, perimetri digitos XXII, capit quinarias XXXIIS.
The forty-pipe: diameter seven digits, plus one ounce, plus a half-ounce, plus three scruples. Circumference 22 digits, plus five ounces. It holds 32 quinariae, plus a half, plus an ounce.
Pipe #40. Diameter: 7 + 1/12 + 1/24 + 3/288. Circumference: 22 + 5/12. Cross-sectional area: 32 + 1/2 + 1/12.
Frontinus uses a shorthand here: a full number after a fractional unit means "multiply that unit by this number". So ℈III means "three scruples". I've never seen this used with ounces, only smaller amounts.
So to answer your examples:
- "One-half" is S
- "Three-fourths" is S•••
I'm not entirely sure how these numbers would have been spoken out loud, since the main sources we have are written, and numerals were pretty much always used in writing (like how in English we'd write "69105.1" instead of "sixty-nine thousand, one hundred and five, and one-tenth"). But based on the currency, here's my best guess, using the example from Frontinus again:
septem ℈III
Septem, et uncia, et semuncia, et scripuli tres
In other words, join the units together with et, using normal numbers for multipliers.
Google also leads me to this table on Wikipedia, which gives special names for each integer multiple of one-twelfth.
EDIT: This is how the list of symbols looks for me, using Google Noto fonts. Unfortunately I can't find a way to align it with the actual list, and turning the text of the answer into an image makes it useless for screen readers. 
