Write the following Roman numerals in ascending order:

$CXIX,XCIX,CXXI,CIX,CXX$

Converting the given Roman numbers to decimal numbers:

$CXIX=100+10+9=119$

$XCIX=90+9=99$

$CXXI=100+21=121$

$CIX=100+9=109$

$CXX=100+20=120$

Now, arranging these numbers in ascending order:

$99<109<119<120<121$

$\therefore XCIX<CIX<CXIX<CXX<CXXI$ is the ascending order of the given Roman numbers.