How much movement would a wood elf have left if they spent half of their 35 feet of movement standing up from being prone?
Do they still have 15 feet of movement remaining? Or do they have 20 feet left?
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Nowhere in the rules does it state that movement has to be an integer multiple of 5 feet. There is nothing wrong with moving 17.5 feet, i.e. 17' 6".
The problem only comes in when you play on a grid. The variant rules for playing on a grid are found on page 192 of the PHB, and they state:
Rather than moving foot by foot, move square by square on the grid. This means you use your speed in 5 foot segments.
Hence, 35 feet correspond to 7 squares. 7 divided by 2 equals 3.5, but half squares are not accounted for in these variant rules. Therefore, you will have to round. By default, you always round down, unless a specific rule says differently. Hence, standing up from prone costs 3 squares of movement, leaving you with 4 squares of movement for your turn.
The "Variant: Playing on a Grid" sidebar says of speed/movement (PHB, p.192):
Rather than moving foot by foot, move square by square on the grid. This means you use your speed in 5-foot segments.
The rules for moving on a grid only provide guidance for how the player expends their speed, and does not say that all movement distances/requirements/penalties must be phrased in units of 5 feet (1 square).1 The relevant passage also recommends that players translate their speed into squares for convenience, but certainly does not mandate that all distances and movement be expressed in squares.
Therefore: A wood elf has a speed of 35 feet. They are prone, and would need to expend half that speed to stand up, which is 17.5 feet. (Possibly rounded down to 17.)
There are two ways to interpret this, both of which give (basically) the same answer. The first is that the "5 feet rule" applies to *all expenditure of movement", in which case the elf must expend 20 feet of movement to stand up, resulting in 15 feet remaining. (3 squares.)
Alternatively, the above rule only applies to moving around, not other expenditures of movement. Therefore, the elf may expend 17.5 (or 17) feet of movement to stand up, resulting in 17.5 (or 18) feet of movement remaining, which translates into 3 squares.
The distinction does matter in edge cases, (for example, if the elf immediately falls prone again after standing up), but the rules are ambiguous on which case is correct, so I won't make an argument for either over the other (considering how trivial the stakes are.)
1 If you don't see this, consider: An object is 17.5 feet away from a PC. A player cannot just claim that the distance gets rounded down to 15 feet; rather, they must expend a full 20 feet of movement to reach it.