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2. 计算下面各题。
$\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{7}$
$\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\dots+\frac{1}{2450}$
$\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{7}$
$\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\dots+\frac{1}{2450}$
答案
2. $\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{7}$
$=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}$
$=1$
$\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\dots+\frac{1}{2450}$
$=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\dots+\frac{1}{49}-\frac{1}{50}$
$=1-\frac{1}{50}$
$=\frac{49}{50}$
$=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}$
$=1$
$\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\dots+\frac{1}{2450}$
$=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\dots+\frac{1}{49}-\frac{1}{50}$
$=1-\frac{1}{50}$
$=\frac{49}{50}$
例 计算：$\frac{5}{6} - \frac{7}{12} + \frac{9}{20} - \frac{11}{30} + \frac{13}{42}$。
思路点拨 常用的分数巧算技巧如“拆”的方法，即先把某些分数拆分成两个分数相加或相减的形式，再相互抵消或合并成一个分数，达到简化运算的目的。此题可以先把每个分数拆分成两个分数的和的形式再进行巧算。
$\frac{5}{6} = \frac{1}{2} + \frac{1}{3}$ $\frac{7}{12} = \frac{1}{3} + \frac{1}{4}$ $\frac{9}{20} = \frac{1}{4} + \frac{1}{5}$ $\frac{11}{30} = \frac{1}{5} + \frac{1}{6}$ $\frac{13}{42} = \frac{1}{6} + \frac{1}{7}$
答案详解 $\frac{5}{6} - \frac{7}{12} + \frac{9}{20} - \frac{11}{30} + \frac{13}{42}$
$= ( \frac{1}{2} + \frac{1}{3} ) - ( \frac{1}{3} + \frac{1}{4} ) + ( \frac{1}{4} + \frac{1}{5} ) - ( \frac{1}{5} + \frac{1}{6} ) + ( \frac{1}{6} + \frac{1}{7} )$
$= \frac{1}{2} + \frac{1}{3} - \frac{1}{3} - \frac{1}{4} + \frac{1}{4} + \frac{1}{5} - \frac{1}{5} - \frac{1}{6} + \frac{1}{6} + \frac{1}{7}$
$= \frac{1}{2} + \frac{1}{7}$
$= \frac{9}{14}$
思路点拨 常用的分数巧算技巧如“拆”的方法，即先把某些分数拆分成两个分数相加或相减的形式，再相互抵消或合并成一个分数，达到简化运算的目的。此题可以先把每个分数拆分成两个分数的和的形式再进行巧算。
$\frac{5}{6} = \frac{1}{2} + \frac{1}{3}$ $\frac{7}{12} = \frac{1}{3} + \frac{1}{4}$ $\frac{9}{20} = \frac{1}{4} + \frac{1}{5}$ $\frac{11}{30} = \frac{1}{5} + \frac{1}{6}$ $\frac{13}{42} = \frac{1}{6} + \frac{1}{7}$
答案详解 $\frac{5}{6} - \frac{7}{12} + \frac{9}{20} - \frac{11}{30} + \frac{13}{42}$
$= ( \frac{1}{2} + \frac{1}{3} ) - ( \frac{1}{3} + \frac{1}{4} ) + ( \frac{1}{4} + \frac{1}{5} ) - ( \frac{1}{5} + \frac{1}{6} ) + ( \frac{1}{6} + \frac{1}{7} )$
$= \frac{1}{2} + \frac{1}{3} - \frac{1}{3} - \frac{1}{4} + \frac{1}{4} + \frac{1}{5} - \frac{1}{5} - \frac{1}{6} + \frac{1}{6} + \frac{1}{7}$
$= \frac{1}{2} + \frac{1}{7}$
$= \frac{9}{14}$
答案
$\frac{5}{6} - \frac{7}{12} + \frac{9}{20} - \frac{11}{30} + \frac{13}{42}$
$= ( \frac{1}{2} + \frac{1}{3} ) - ( \frac{1}{3} + \frac{1}{4} ) + ( \frac{1}{4} + \frac{1}{5} ) - ( \frac{1}{5} + \frac{1}{6} ) + ( \frac{1}{6} + \frac{1}{7} )$
$= \frac{1}{2} + \frac{1}{3} - \frac{1}{3} - \frac{1}{4} + \frac{1}{4} + \frac{1}{5} - \frac{1}{5} - \frac{1}{6} + \frac{1}{6} + \frac{1}{7}$
$= \frac{1}{2} + \frac{1}{7}$
$= \frac{9}{14}$
$= ( \frac{1}{2} + \frac{1}{3} ) - ( \frac{1}{3} + \frac{1}{4} ) + ( \frac{1}{4} + \frac{1}{5} ) - ( \frac{1}{5} + \frac{1}{6} ) + ( \frac{1}{6} + \frac{1}{7} )$
$= \frac{1}{2} + \frac{1}{3} - \frac{1}{3} - \frac{1}{4} + \frac{1}{4} + \frac{1}{5} - \frac{1}{5} - \frac{1}{6} + \frac{1}{6} + \frac{1}{7}$
$= \frac{1}{2} + \frac{1}{7}$
$= \frac{9}{14}$
3. 计算下面各题。
$\frac{9}{20} - \frac{11}{30} + \frac{13}{42} - \frac{15}{56} + \frac{17}{72}$
$\frac{5}{6} - \frac{7}{12} + \frac{9}{20} - \frac{11}{30} + \frac{13}{42} - \frac{15}{56} + \frac{17}{72} - \frac{19}{90} + \frac{21}{110} - \frac{23}{132} + \frac{25}{156}$
$\frac{9}{20} - \frac{11}{30} + \frac{13}{42} - \frac{15}{56} + \frac{17}{72}$
$\frac{5}{6} - \frac{7}{12} + \frac{9}{20} - \frac{11}{30} + \frac{13}{42} - \frac{15}{56} + \frac{17}{72} - \frac{19}{90} + \frac{21}{110} - \frac{23}{132} + \frac{25}{156}$
答案
3. $\frac{9}{20} - \frac{11}{30} + \frac{13}{42} - \frac{15}{56} + \frac{17}{72}$
$=(\frac{1}{4}+\frac{1}{5})-(\frac{1}{5}+\frac{1}{6})+(\frac{1}{6}+\frac{1}{7})-(\frac{1}{7}+\frac{1}{8})+(\frac{1}{8}+\frac{1}{9})$
$=\frac{1}{4}+\frac{1}{5}-\frac{1}{5}-\frac{1}{6}+\frac{1}{6}+\frac{1}{7}-\frac{1}{7}-\frac{1}{8}+\frac{1}{8}+\frac{1}{9}$
$=\frac{1}{4}+\frac{1}{9}$
$=\frac{13}{36}$
$\frac{5}{6} - \frac{7}{12} + \frac{9}{20} - \frac{11}{30} + \frac{13}{42} - \frac{15}{56} + \frac{17}{72} - \frac{19}{90} + \frac{21}{110} - \frac{23}{132} + \frac{25}{156}$
$=(\frac{1}{2}+\frac{1}{3})-(\frac{1}{3}+\frac{1}{4})+(\frac{1}{4}+\frac{1}{5})-(\frac{1}{5}+\frac{1}{6})+(\frac{1}{6}+\frac{1}{7})-(\frac{1}{7}+\frac{1}{8})+(\frac{1}{8}+\frac{1}{9})-(\frac{1}{9}+\frac{1}{10})+(\frac{1}{10}+\frac{1}{11})-(\frac{1}{11}+\frac{1}{12})+(\frac{1}{12}+\frac{1}{13})$
$=\frac{1}{2}+\frac{1}{13}$
$=\frac{15}{26}$
$=(\frac{1}{4}+\frac{1}{5})-(\frac{1}{5}+\frac{1}{6})+(\frac{1}{6}+\frac{1}{7})-(\frac{1}{7}+\frac{1}{8})+(\frac{1}{8}+\frac{1}{9})$
$=\frac{1}{4}+\frac{1}{5}-\frac{1}{5}-\frac{1}{6}+\frac{1}{6}+\frac{1}{7}-\frac{1}{7}-\frac{1}{8}+\frac{1}{8}+\frac{1}{9}$
$=\frac{1}{4}+\frac{1}{9}$
$=\frac{13}{36}$
$\frac{5}{6} - \frac{7}{12} + \frac{9}{20} - \frac{11}{30} + \frac{13}{42} - \frac{15}{56} + \frac{17}{72} - \frac{19}{90} + \frac{21}{110} - \frac{23}{132} + \frac{25}{156}$
$=(\frac{1}{2}+\frac{1}{3})-(\frac{1}{3}+\frac{1}{4})+(\frac{1}{4}+\frac{1}{5})-(\frac{1}{5}+\frac{1}{6})+(\frac{1}{6}+\frac{1}{7})-(\frac{1}{7}+\frac{1}{8})+(\frac{1}{8}+\frac{1}{9})-(\frac{1}{9}+\frac{1}{10})+(\frac{1}{10}+\frac{1}{11})-(\frac{1}{11}+\frac{1}{12})+(\frac{1}{12}+\frac{1}{13})$
$=\frac{1}{2}+\frac{1}{13}$
$=\frac{15}{26}$
