Mavericks installed new Numbers and Pages, but did not remove the old versions. Can I remove them or is there a reason to keep them.

The Mac App Store kept the old iWork versions in case you need to use them.

The new iWork version drops some features respecting to the old iWork version, so I would recommend you to keep the old applications if you don't know if you will need it or not. If you don't need it, you can remove the old iWork applications from the Applications folder without any issue, or move them to an external drive

You may want to not rush to delete them (they can both co-exist on your Mac) as there is a lot of negative press about the new versions having had much of their functionality removed, although Apple did announce yesterday that some of the lost features will be re-introduced over the next six months.

I only use the apps for simple stuff and have no problem with the new versions, but if you use them at their full potential you may find the new versions a disappointment.

You are welcome

Apple finally released some information on these new, but unimproved apps.