1. Three pieces of fabric have a total length of $34 \mathrm{~m}$. If we cut off a quarter of the length of the first piece, half of the length of the second piece, and one-sixth of the length of the third piece, the remaining parts of the fabric will have equal lengths. What is the length of each piece of fabric before cutting?
First method:

Let $x$ be the length of the first, $y$ be the length of the second, and $z$ be the length of the third piece of fabric. Let $n$ be the length of the equal parts of fabric after cutting, i.e.,

$\frac{3}{4} x=\frac{1}{2} y=\frac{5}{6} z=n$

$\frac{3}{4} x=n \Rightarrow x=\frac{4}{3} n, \frac{1}{2} y=n \Rightarrow y=2 n, \frac{5}{6} z=n \Rightarrow z=\frac{6}{5} n$

$x+y+z=\frac{4}{3} n+2 n+\frac{6}{5} n=34$

$20 n+30 n+18 n=510$

$68 n=510$

$n=7.5$

Further, $x=\frac{4}{3} \cdot 7.5=10$, so the first piece of fabric was originally $10 \mathrm{~m}$ long, the second piece was originally $y=2 \cdot 7.5=15 \mathrm{~m}$ long, and the third piece was originally $z=\frac{6}{5} \cdot 7.5=9 \mathrm{~m}$ long.