So, if you multi-class to Thief/Fighter/Monk (and don't get to 3rd level or choose not to take the Archetype that offers spell-casting), you maintain standard Ranger progression, but if you choose Paladin, you actually lose spells every time you take an odd level of either.
This does not happen if you start as a straight spellcaster class and then multi-class to Ranger or Paladin, but will happen again if you multi-class to both Ranger and Paladin at odd levels of each.
What I believe the text was intended to state (and what could actually be interpreted from the verbiage as written) is that you take half of your (Paladin + Ranger) levels round down and a third of your (Figher-EldritchKnight + Thief-ArcaneTrickster) levels round down. That would achieve the result of never having you actually lose any spell slots you had already achieved while not allowing you to become overly powerful vs. base class only by giving you more spell slots than a full caster of your same level.
If you were to take RoundDown(.5*(Paladin + Ranger)), then Ranger 3/ Paladin 1 would still have 3 spell slots of 1st Level (which is exactly what you would have as Ranger 3/ Fighter 1, or just Ranger 3, or Ranger 2/ Paladin 2 which is what happens currently under the character sheet).
I agree that it should be Paladin+Ranger and Fighter+Rogue (assuming you took the appropriate sub-classes) but I think it should be rounding up as otherwise, you still lose spell slots (Fighter 4/Rogue 4, Ranger 3/Rogue 4, etc). The rules say otherwise but it makes no sense - clearly (based on this and class attribute pre-reqs being retroactive), someone at WotC hates multi-classing.