计算 $( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} ) × ( \frac{1}{3} + \frac{1}{4} + \frac{1}{5} ) - ( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} ) × ( \frac{1}{3} + \frac{1}{4} )$
算式中反复出现的分数组合有$\frac{1}{3} + \frac{1}{4}$和$\frac{1}{3} + \frac{1}{4} + \frac{1}{5}$这两组，设$\frac{1}{3} + \frac{1}{4} = a$，$\frac{1}{3} + \frac{1}{4} + \frac{1}{5} = b$。
① 用字母表示重复部分。
$\begin{aligned}&( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} ) × ( \frac{1}{3} + \frac{1}{4} + \frac{1}{5} ) - ( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} ) × ( \frac{1}{3} + \frac{1}{4} ) \\=& ( \frac{1}{2} + a ) × b - ( \frac{1}{2} + b ) × a \\=& \frac{1}{2}b + ab - \frac{1}{2}a - ab \\=& \frac{1}{2}b - \frac{1}{2}a \\=& \frac{1}{2}(b - a) \\=& \frac{1}{2} × ( \frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{3} - \frac{1}{4} ) \\=& \frac{1}{2} × \frac{1}{5} \\=& \frac{1}{10}\end{aligned}$
② 利用乘法分配律抵消相同部分。
③ 代入原数求值。

$\begin{aligned}&( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} ) × ( \frac{1}{3} + \frac{1}{4} + \frac{1}{5} ) - ( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} ) × ( \frac{1}{3} + \frac{1}{4} ) \\=& ( \frac{1}{2} + a ) × b - ( \frac{1}{2} + b ) × a \\=& \frac{1}{2}b + ab - \frac{1}{2}a - ab \\=& \frac{1}{2}b - \frac{1}{2}a \\=& \frac{1}{2}(b - a) \\=& \frac{1}{2} × ( \frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{3} - \frac{1}{4} ) \\=& \frac{1}{2} × \frac{1}{5} \\=& \frac{1}{10}\end{aligned}$

计算：$( \frac{1}{11} + \frac{1}{13} + \frac{1}{17} ) × ( \frac{1}{13} + \frac{1}{17} + \frac{1}{23} ) - ( \frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{23} ) × ( \frac{1}{13} + \frac{1}{17} )$。

设$\frac{1}{13}+\frac{1}{17}=a$，$\frac{1}{13}+\frac{1}{17}+\frac{1}{23}=b$
$\begin{aligned}原式&=(\frac{1}{11}+a)× b - (\frac{1}{11}+b)× a\\&=\frac{1}{11}b + ab - \frac{1}{11}a - ab\\&=\frac{1}{11}(b-a)\\&=\frac{1}{11}×\frac{1}{23}\\&=\frac{1}{253}\end{aligned}$