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（课本 P101 第 8 题改编题）
例 $1 - \frac{1}{2} = \frac{1}{2}$ $\frac{1}{2} - \frac{1}{3} = \frac{1}{6}$ $\frac{1}{3} - \frac{1}{4} = \frac{1}{12}$ $\frac{1}{4} - \frac{1}{5} = \frac{1}{20}$ $\frac{1}{5} - \frac{1}{6} = \frac{1}{30}$
你发现了什么？用你的发现计算下面这道题。
$\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}$
【分析与解答】
上面的算式分子是 1，分母互质，它们相减的差的分子是 1，分母是它们的乘积。
下面算式中的分数刚好是上面几个算式的差，很多分数可以消掉。
正确解答：$\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}$
$= (1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{5}) + (\frac{1}{5} - \frac{1}{6})$
$= 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + \frac{1}{5} - \frac{1}{6} = 1 - \frac{1}{6} = \frac{5}{6}$
例 $1 - \frac{1}{2} = \frac{1}{2}$ $\frac{1}{2} - \frac{1}{3} = \frac{1}{6}$ $\frac{1}{3} - \frac{1}{4} = \frac{1}{12}$ $\frac{1}{4} - \frac{1}{5} = \frac{1}{20}$ $\frac{1}{5} - \frac{1}{6} = \frac{1}{30}$
你发现了什么？用你的发现计算下面这道题。
$\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}$
【分析与解答】
上面的算式分子是 1，分母互质，它们相减的差的分子是 1，分母是它们的乘积。
下面算式中的分数刚好是上面几个算式的差，很多分数可以消掉。
正确解答：$\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}$
$= (1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{5}) + (\frac{1}{5} - \frac{1}{6})$
$= 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + \frac{1}{5} - \frac{1}{6} = 1 - \frac{1}{6} = \frac{5}{6}$
答案
$\frac{5}{6}$
解析
$\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}$
$=(1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{5}) + (\frac{1}{5} - \frac{1}{6})$
$=1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + \frac{1}{5} - \frac{1}{6}$
$=1 - \frac{1}{6}$
$=\frac{5}{6}$
$=(1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{5}) + (\frac{1}{5} - \frac{1}{6})$
$=1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + \frac{1}{5} - \frac{1}{6}$
$=1 - \frac{1}{6}$
$=\frac{5}{6}$
一、先计算前 5 题，再用你的发现计算第 6 题。
1. $\frac{1}{2} - \frac{1}{3} =$
6. $\frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} + \frac{1}{132}$
1. $\frac{1}{2} - \frac{1}{3} =$
$\frac{1}{6}$
2. $\frac{1}{3} - \frac{1}{4} =$$\frac{1}{12}$
3. $\frac{1}{4} - \frac{1}{5} =$$\frac{1}{20}$
4. $\frac{1}{5} - \frac{1}{6} =$$\frac{1}{30}$
5. $\frac{1}{6} - \frac{1}{7} =$$\frac{1}{42}$
6. $\frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} + \frac{1}{132}$
答案
1. $\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}$
2. $\frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}$
3. $\frac{1}{4} - \frac{1}{5} = \frac{5}{20} - \frac{4}{20} = \frac{1}{20}$
4. $\frac{1}{5} - \frac{1}{6} = \frac{6}{30} - \frac{5}{30} = \frac{1}{30}$
5. $\frac{1}{6} - \frac{1}{7} = \frac{7}{42} - \frac{6}{42} = \frac{1}{42}$
6.
$\frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} + \frac{1}{132}$
$=\frac{1}{5 × 6} + \frac{1}{6 × 7} + \frac{1}{7 × 8} + \frac{1}{8 × 9} + \frac{1}{9 × 10} + \frac{1}{10 × 11} + \frac{1}{11 × 12}$
$=(\frac{1}{5} - \frac{1}{6}) + (\frac{1}{6} - \frac{1}{7}) + (\frac{1}{7} - \frac{1}{8}) + (\frac{1}{8} - \frac{1}{9}) + (\frac{1}{9} - \frac{1}{10}) + (\frac{1}{10} - \frac{1}{11}) + (\frac{1}{11} - \frac{1}{12})$
$=\frac{1}{5} - \frac{1}{12}$
$=\frac{12}{60} - \frac{5}{60}$
$=\frac{7}{60}$
2. $\frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}$
3. $\frac{1}{4} - \frac{1}{5} = \frac{5}{20} - \frac{4}{20} = \frac{1}{20}$
4. $\frac{1}{5} - \frac{1}{6} = \frac{6}{30} - \frac{5}{30} = \frac{1}{30}$
5. $\frac{1}{6} - \frac{1}{7} = \frac{7}{42} - \frac{6}{42} = \frac{1}{42}$
6.
$\frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} + \frac{1}{132}$
$=\frac{1}{5 × 6} + \frac{1}{6 × 7} + \frac{1}{7 × 8} + \frac{1}{8 × 9} + \frac{1}{9 × 10} + \frac{1}{10 × 11} + \frac{1}{11 × 12}$
$=(\frac{1}{5} - \frac{1}{6}) + (\frac{1}{6} - \frac{1}{7}) + (\frac{1}{7} - \frac{1}{8}) + (\frac{1}{8} - \frac{1}{9}) + (\frac{1}{9} - \frac{1}{10}) + (\frac{1}{10} - \frac{1}{11}) + (\frac{1}{11} - \frac{1}{12})$
$=\frac{1}{5} - \frac{1}{12}$
$=\frac{12}{60} - \frac{5}{60}$
$=\frac{7}{60}$
二、观察与计算。
$\frac{5}{6} = \frac{1}{2} + \frac{1}{3}$ $\frac{7}{12} = \frac{1}{3} + \frac{1}{4}$ $\frac{11}{30} = \frac{1}{5} + \frac{1}{6}$
计算：$\frac{5}{6} - \frac{7}{12} + \frac{1}{20}$
$\frac{5}{6} = \frac{1}{2} + \frac{1}{3}$ $\frac{7}{12} = \frac{1}{3} + \frac{1}{4}$ $\frac{11}{30} = \frac{1}{5} + \frac{1}{6}$
计算：$\frac{5}{6} - \frac{7}{12} + \frac{1}{20}$
答案
$\frac{3}{10}$
解析
$\frac{5}{6}-\frac{7}{12}+\frac{1}{20}$
$=(\frac{1}{2}+\frac{1}{3})-(\frac{1}{3}+\frac{1}{4})+\frac{1}{20}$
$=\frac{1}{2}+\frac{1}{3}-\frac{1}{3}-\frac{1}{4}+\frac{1}{20}$
$=\frac{1}{2}-\frac{1}{4}+\frac{1}{20}$
$=\frac{2}{4}-\frac{1}{4}+\frac{1}{20}$
$=\frac{1}{4}+\frac{1}{20}$
$=\frac{5}{20}+\frac{1}{20}$
$=\frac{6}{20}$
$=\frac{3}{10}$
$=(\frac{1}{2}+\frac{1}{3})-(\frac{1}{3}+\frac{1}{4})+\frac{1}{20}$
$=\frac{1}{2}+\frac{1}{3}-\frac{1}{3}-\frac{1}{4}+\frac{1}{20}$
$=\frac{1}{2}-\frac{1}{4}+\frac{1}{20}$
$=\frac{2}{4}-\frac{1}{4}+\frac{1}{20}$
$=\frac{1}{4}+\frac{1}{20}$
$=\frac{5}{20}+\frac{1}{20}$
$=\frac{6}{20}$
$=\frac{3}{10}$
