Tenth and fifteenth (desime and quinsime) were the rates of taxation on movable goods levied from the 1330s onwards in England. They refer to proportions of the value of the goods: one-tenth (1/10) or one-fifeenth (1/15). The tenth was paid by towns represented in parliament, and the fifteenth by shires (not represented in parliament).

The respective rates were set as some fraction or multiple of the tenth and fifteenth: so here we have half a tenth and fifteenth (which amounts to a twentieth and a thirtieth) and one and a half tenths and fifteenths (three-twentieths and a tenth, respectively).

So a complete tenth and fifteenth just means exactly one whole tenth and fifteenth, neither a fraction (less) nor a multiple (more).

http://tudorhistory.org/glossaries/t/tenth_and_fifteenth.htm...
http://books.google.es/books?id=FSvAegniXsUC&pg=PA288&lpg=PA...
http://books.google.es/books?id=zhEn77SlKU4C&pg=PR25&lpg=PR2...

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Note added at 11 hrs (2012-02-03 09:14:48 GMT)
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I did not address the question of the allowances mentioned in the text, and how they were calculated. These allowances, also known as "reliefs" or "abatements", were sums distributed among impoverished towns, as the text implies. So when parliament agreed that the towns and shires would contribute more in tax (one and half tenths and fifteenths instead of a half: three times as much), the allowance was correspondingly raised, and was linked to the tax rate: £6000 per whole/complete tenth and fifteenth contributed. This sum (a great deal of money in that period) was a total to be distributed among towns according to need. But this way of calculating the allowance (a rebate, if you like) applies only to this particular period.

For further details on this, see http://www.british-history.ac.uk/report.aspx?compid=15722 .