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I don’t recall hearing that there was ever any serious demand on the part of the 13 colonies that they have representatives in Parliament—it wouldn’t have done them a lot of good (they’d be greatly outvoted, and the people they sent could easily be co-opted by the other members of Parliament and forget the wishes of the people back home)...and it would justify Parliament laying taxes on the colonists.

The whole objection to continued British rule was based on being taxed without representation in Parliament. The colonists generally rejected the idea that money should be collected to be sent to London without their having a say in its use.

They called it "taxation without representation". It's a fantastic phenomenon, well worth discussing at length later ~ a google.com search on that phrase alone yields 343,000 separate entries on the net.

Now, regarding relative populations, part of the UK problem entering into the war was UK had maybe 7 million people. (SEE: http://chartsbin.com/view/28k ) and the colonies had between 2 million and 7 million.

Both parties ~ the Americans and the Ukian people, found it necessary to "bring in help" ~ the Americans had the use of French troops and the Ukians used German troops.

I suspect the population information in both main areas of the Kingdom were not all that good ~ and later on in the 1800s the Brits would address the issue of "the rotten boroughs" by holding their own census. In the new United States the first census was held before the century was out in 1790. Imperfect as it was (given that it was the first really serious census anyone had taken anywhere) they found 3,929,214 people in the United States.

The Brits had obviously made a mistake in fighting a war so far from home against such a large number of heavily armed people. In just a few decades the USA population far outstripped that of the UK.