1.) Calculate a new field to hold the starting area of each polygon in Layer B2.) Use the Identity tool (requires ArcInfo) to split the polygons in Layer B by the borders in Layer A3.) Calculate a new field to get the percentage of each divided polygon4.) Select based on the percentages, then Export to a new layer and Dissolve the pieces back together
Possibly Intersect...? It will split Layer B by Layer A, although you'll lose areas in Layer B that don't overlap with Layer A.
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