Let the length of one of the shorter pieces be [tex]\ell[/tex]. Then, we know that two of the pieces are that length, and the third piece is [tex]3 \ell[/tex]. Since these are all pieces of a rope, all we need to do to figure out the lengths is add all those together, solve for [tex]\ell[/tex], and then plug that back in to the desired lengths:
[tex]3 \ell + \ell + \ell = 76 ft.[/tex]
[tex]5 \ell = 76 ft.[/tex]
[tex]\ell = 15.2 ft.[/tex]
Shorter pieces: [tex]\ell = \bf 15.2 ft.[/tex]
Longer piece: [tex]3 \ell = \bf 45.6 ft.[/tex]
