There is double-sided "rug gripper" tape available at Target and Home Depot (probably most other stores, but I know those two carry it).
We use it to keep our entry rugs stuck to the tiled floor. It holds well enough that we can vacuum the rugs.
Disclaimer: Not sure how well it will stick to a carpet vs a hard floor.
Alternatively you could use rug pins/rug anchors to hold the area rugs to your carpet.
It's totally normal. Carpet comes on rolls in widths from 11'6" to 13'. They need to cut off a chunk of that roll to fit the room.
It's the reason why if you're carpeting a 10x10' room, you can't buy 100sqft of carpet: they'll charge you for more like 120sqft (12' roll, 10' long).
It also depends on the layout of the room, and where they put seams (if necessary). Eg: an L-shaped room has a couple different ways to lay out the carpet, and depending the layout and where the seams are, you'll use a different amount.
A good installer will also consider traffic: if there are seams, you don't want them right in front of a door way, or preferably not running down the middle of a high traffic pathway. I think they also avoid putting seams within a foot or so of walls (so there isn't a narrow strip of carpet running along the wall). All of these things may end up using more carpet than a layout optimized just to minimize scrap, but will result in a much better looking and longer lasting install.
The other factor is in patterned carpets, they will use a bit more since the patterns need to line up. The bigger the pattern, the more is scrap needed to keep it in line.
The important bit is to keep the scrap carpet, unless you really don't need it, since you
paid for it. Often there is a good size chunk, which can be used for many things:
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If anything, I would expect the opposite problem: the area under the rug would wear less, and be noticeable when you take up the rug for that reason (if at all)!
Assuming the rug is color-fast, of course.