In which case will the gap be greater?

Question

Mathematics

Hello. Completely confused in an elementary task. The bottom line is:
Two balls of different radii were wrapped around the equator with wire. In both cases, the length of the wire was increased by exactly 1 meter. In which case will the gap between the surface of the ball and the wire be larger?
For the first ball, the length of the wire is 2pr, for the second - 2pR. After increasing the wire by 1 meter, the new length of the first wire is 2p (r + l), where l is the additional radius (gap), for the second - 2p (R + L), where L is the additional radius (gap). At the same time, since the new length of the wire differs from the old one by only 1 meter, we get:
2pr+1 = 2p(r+l) - for the first ball 2pR
+1 = 2p(R+L) - for the second ball => 2pl = 1 for the first 2pR + 1 = 2pR + 2pL => 2pL = 1 for the second.
It turns out that in both cases the gap will be the same and does not depend on the initial radii. Show me where is the mistake.