Two consecutive steps in transformation: from Values to Prices and from Sectoral Rates to a Weighted Common Rate

Karl Marx in chapter 9 of Capital, Volume III, remove profits from the cost-price of sectors so as to calcultate the production prices. If instead of doing so, we keep them we can obtain a set of production prices at the existing sectoral rates of profit in value terms which are proportional to values. This proportionality, based on the rate of surplus value, is only lost when trying to apply the resulting Marxian (wheigthed) rate of profit to the employed sectoral capitals measured in this proportional production prices. The common rate can only be applied by artificially establishing a price that modifies the common unit of the sectoral values.