When something is divided into 3 parts in the ratio a:b:c the respective
3 parts are these 3 fractions a/(a+b+c), b/(a+b+c), and c/(a+b+c) of the
whole. 

The sum of the 3 parts of the ratios are 1+4+5=10 and 2+4+3=9. Their LCM is
90, so let's let the length of the original rope be 90x.

We cut the original rope into pieces which are 1/10, 4/10, and 5/10 of the
length 90x, which are the lengths:

9x, 36x, 45x

Then we cut the 45x piece into smaller pieces which are 2/9, 4/9, and 3/9 of
the length 45x, which are the lengths:

10x, 20x, 15x.

The shortest rope is 9x.
The second longest piece is 36x.

So the ratio is 9:36 or 1:4.

The difference between our answers is the ambiguous "2nd longest piece". I
took it to mean the 2nd longest piece before we cut up what was originally
the longest piece.

Greenestamps took it to mean the 2nd longest after the longest was cut up
and what was the 2nd longest before the longest was cut up, was no longer
the 2nd longest.  I suppose he's right.  But 1:4 is a nicer answer. lol

Edwin