I've been reading Volume 3 with a reading group far more quickly than is appropriate (I'm very busy), and having got a little frustrated with doing so I started going through the book again and making some general notes. In doing so I spotted (or at least I think I did) what seems to be the source of the debate about the transformation problem in the chapter on the formation of a general rate of profit. As I've said, I've not read this too carefully, and I'm not masochistic enought to want to immerse myself in great seas of equations, but from a fairly cursory look I can't really see what all the fuss is about. Marx has stated repeatedly that he's trying to penetrate 'surface' phenomena through abstraction, and the solution that he offers seems to be just that:
"...This statement seems to conflict with the fact that under capitalist production the elements of productive capital are, as a rule, bought on the market, and that for this reason their prices include profit which has already been realised, hence, include the price of production of the respective branch of industry together with the profit contained in it, so that the profit of one branch of industry goes into the cost-price of another. But if we place the sum of the cost-prices of the commodities of an entire country on one side, and the sum of its surplus-values, or profits, on the other, the calculation must evidently be right. For instance, take a certain commodity A. Its cost-price may contain the profits of B, C, D, etc., just as the cost-prices of B, C, D, etc., may contain the profits of A. Now, as we make our calculation the profit of A will not be included in its cost-price, nor will the profits of B, C, D, etc., be included in theirs. Nobody ever includes his own profit in his cost-price. If there are, therefore, n spheres of production, and if each makes a profit amounting to p, then their aggregate cost-price = k — np. Considering the calculation as a whole we see that since the profits of one sphere of production pass into the cost-price of another, they are therefore included in the calculation as constituents of the total price of the end-product, and so cannot appear a second time on the profit side. If any do appear on this side, however, then only because the commodity in question is itself an ultimate product, whose price of production does not pass into the cost-price of some other commodity."