通分典型例题

例1  已知http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image002.gif ，a、b最小各是多少？

　　分析：根据题意，可把45分解质因数，看组成“http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image004.gif ”缺哪一个质因数，这是约分所致，应设法补上．把45分解质因数是：45＝3×3×5，要把3×3×5变换成“http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image005.gif ”的形式，必须补上质因数“5”．

　　解：因为：http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image007.gif

　　　　所以：http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image009.gif

例2  一位同学是这样通分的，对不对？如果不对，错在哪里？

http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image011.gif

　　分析：通分时，用两个异分母分数的两个分母的最小公倍数作公分母．而上面的两个分数的分母分别扩大了7倍、5倍，但分子未变，因此，两个分数的大小都改变了．这样通分是不对的.

　　解：这位同学这样通分是不对的．正确的通分是：

　　http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image013.gif

例3 把下面的分数约分后，再按照从小到大的顺序排列起来

　　http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image015.gif

　　分析：根据分数的基本性质，先把http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image017.gif 约分得http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image019.gif ，观察发现http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image021.gif ，原http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image023.gif 再约分得http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image025.gif 比，http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image027.gif ，把http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image029.gif 和http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image031.gif 比，因为；http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image033.gif ，所以http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image035.gif ，此题可解．

　　解：http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image037.gif

　　　　http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image039.gif

　　　因为：http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image043.gif

例4  甲、乙两工人做同样的机器零件，甲2小时做3个，乙5小时做7个，他们做一个零件需多少时间？谁做得快？

　　分析：谁做得快一些，实质是比大小．比大小需根据题意理解，此题所求的是时间，做得快慢是指效率．因此，时间用得少的工作效率应该快．

　　解：http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image045.gif （小时）      http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image047.gif （小时）

　　　http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image049.gif          http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image051.gif

　　   http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image053.gif ，甲做一个零件比乙做一个零件的时间少，所以甲比乙做得快些，

　　答：甲做一个零件需http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image055.gif 小时，乙做一个零件需http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image057.gif 小时，甲做得快些．

例5  约分http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image059.gif

　　分析：分子199619961996＝1996×100010001，分母199719971997＝1997×100010001

　　解：http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image061.gif

例6  分母是100的最简真分数有多少个？

　　分析：因为http://resource.ahedu.cn/statics/jspx/xxpd/xkjx/x5sx/x5sx31/02/dxlt/image063.gif ，即100的质因数只有2和5．100以内2的倍数有50个，5的倍数有20个，共有70个，其中10的倍数有10个是重复的，所以与100有公约数的数共有20＋50－10＝60（个），与100没有分约数的数共有100－60＝40（个）

　解：分母是100的最简真分数有40个．