【小学奥数题库系统】1-3-2 多位数计算.教师版

【题型】计算
2 2 999 9 2 ＝ × 1000 0 − 1 2 × 222 222 2 × 222 2＝ × × 222 9 1998个9 1998个 2 9 1998个0 1998个 2 1998个 2 1998个 2 1 1 444 4 000 0 − 444 4 1000 0 − 1 4 ＝ × × 444 ＝ × 9 1998个 4 1998个0 9 1998个0 1998个 4 1998个 4 1 ＝ × 444 43555 56 、 9 1997 个 4 1997 个 5
2004 个 3 2004 个9
原式 = 333 3 × 59049 = 999 9 ÷ 3 × 59049 = 999 9 × 19683
2004 个 3 2004 个9 2004 个9
= (1000 0 − 1) × 19683 = 1968300...0 = 1968299...9980317 − 19683 ( (
知识点拨
一、 多位数运算求精确值的常见方法
9 10k − 1 ，进行变形 = 1. 利用 999
k 个9
2. “以退为进”法找规律递推求解
二、
多位数运算求数字之和的常见方法
k个 9 k个 9
M× 999...9 )．可以利用上面性质较快的获得结果． 的数字和为 9×k．(其中 M 为自然数，且 M≤ 999...9
多位数计算
教学目标
多位数的运算在奥数计算体系里面一般都扮演难题角色，因为多位数计算不仅能体现普通数字四则运 算的一切考法，还有自身的“独门秘籍”，那就是“数字多的数不出来”，只能依靠观察数字结构发现数字规 律的方式掌握多位数的整体结构，然后再确定方法进行解题。
多位数的主要考查方式有
1.用带省略号的描述方式进行多位数的具体值四则计算 2.计算多位数的各个位数字之和
2007 个 5 2007 个 9 2007 个 5 2007 个 0 2007 个 5 2007 个 0 2007 个 5
= 55 ⋅ ⋅ ⋅ 5 44 ⋅ ⋅ ⋅ 45 ÷= 3 185 ⋅ ⋅ ⋅ 1851848148 ⋅ ⋅ ⋅ 14815
2008个 9 2008个8 2008个 6
【考点】多位数计算之求精确值 【难度】3 星 【题型】计算 【解析】 本题着重是给大家一种凑的思想，除数是 66 ，所以需要我们的被除数也能凑出 ⋅ ⋅ ⋅ 6 66 ⋅⋅⋅ 6
2008个 6 2008个 6
这就需要我们根据乘法的性质来计算了。所以： 原式 = 3 × 33 ⋅ ⋅ ⋅ 3 × 4 × 22 ⋅ ⋅ ⋅ 2 ÷ 66 ⋅ ⋅ ⋅ 6 = 3 × 4 × 11 ⋅ ⋅ ⋅ 1 × 66 ⋅ ⋅ ⋅ 6 ÷ 66 ⋅⋅⋅ 6
k 个9
【题型】计算
原式= 333 3 ×2 × 3 × 999 3 ×2 × 3 × 3 × 333 9 3 = 333
2004 个 3 2008个3 2004 个 3 2008个9
= 1999 0 − 1 ）= 1999 0 - 1999 98 × （ 1000 98 × 1000 98
2004 个 0 2004 个 0 1999 个 9
【答案】 1968299...9980317
1999 个 9
【巩固】 计算 666 6 × 9 × 333...3 的乘积是多少？
2004 个6 2008个 3
【考点】多位数计算之求精确值 【难度】3 星 = 9 10k − 1 的变形： 【解析】 我们可以将原题的多位数进行 999
2008个 3 2008个 2 2008个 6 2008个1 2008个 6 2008个 6
32 = 3 × 44 ⋅ ⋅ ⋅ 4 = 133
2008个 4
2007 个3
【答案】 133 32
2007 个3
1-3-2.多位数计算.题库
教师版
page 2 of 8
【例 2】 请你计算 99 9 × 99 9 + 199 9 结果的末尾有多少个连续的零？
【答案】 296 ⋅ ⋅ ⋅ 296 ⋅ ⋅ ⋅ 037 2957 037 04
668个 296 668个 037
【巩固】 计算 333 3 × 59049
2004 个 3
【考点】多位数计算之求精确值 【难度】3 星 【题型】计算 【解析】 我们可以把 333 3 转化为 999 9 ÷ 3 ，进而可以进行下一步变形，具体为：
2007 个 3
出一个 99 ⋅ ⋅ ⋅ 9 ，然后在原式乘以 3 的基础上除以 3，所以
2007 个 3
原式 （1 00 = 55 ⋅ ⋅ ⋅5× ⋅ ⋅ ⋅ 0 -1） ÷= 3 （55 = 55 ⋅ ⋅ ⋅ 5 × 99 ⋅⋅⋅9 ÷ 3 ⋅ ⋅ ⋅ 500 ⋅ ⋅ ⋅ 0 - 55 ⋅ ⋅ ⋅ 5） ÷3
【巩固】 快来自己动手算算 （11 ⋅ ⋅ ⋅ 1 × 99 ⋅ ⋅ ⋅ 9 + 99 ⋅ ⋅ ⋅ 9 × 77 ⋅ ⋅ ⋅ 7） ÷ 3 的结果看谁算得准？
2007 个1 2007 个 9 2007 个 9 2007 个 7
【考点】多位数计算之求精确值 【难度】3 星 【题型】计算 【解析】 本题是提取公因数和凑整的综合。 原式 =[ 99 ⋅ ⋅ ⋅ 9 × 88 ⋅ ⋅ ⋅ 8 ÷ 3 =(1 00 ⋅ ⋅ ⋅ 0 − 1) × 88 ⋅ ⋅ ⋅8 ÷3 ⋅ ⋅ ⋅ 9 × (11 ⋅ ⋅ ⋅ 1 + 77 ⋅ ⋅ ⋅ 7)] ÷ 3 = 99
2008个 9 2008个 9 2008个 9
【考点】多位数计算之求精确值 【难度】3 星 【题型】计算 【解析】 同学们观察会发现，两个乘数都非常大，不便直接相乘，可以引导学生按照两种思路给学生展开 方法一：是学生喜欢的从简单情况找规律 9×9=81；99×99=9801 ；999×999=998001；9999×9999=99980001；…… 所以： 99 9 × 99 9 = 99 9800 01
2007 个8 2007 个 9 2007 个8 2007 个 0 2007 个8 2007 个 0 2007 个8
= ⋅ ⋅ ⋅ 296 ⋅ ⋅ ⋅ 037 3 296 = 88 ⋅ ⋅ ⋅ 8711 ⋅ ⋅ ⋅ 12 ÷ 2957 037 04
2006 个8 2006 个1 668个 296 668个 037
所以末尾有 4016 个 0 【答案】4016 个 0 【例 3】 计算 222 2 × 222 2 的积
1998个 2 1998个 2
【考点】多位数计算之求精确值 【难度】3 星 【解析】 我们先还是同上例来凑成 999 9；
k个 9
2003个9 2008个0 2003个9 2008个0 2003个9
= 1999 979998000 02 .
2003个 9 2003个 0
【答案】 1999 979998000 02
2003个 9 2003个 0
2007 个8 2007 个 0 2007 个8 2006 个8 2006 个1 668个 296 668个 037
【答案】 296 ⋅ ⋅ ⋅ 296 ⋅ ⋅ ⋅ 037 2957 037 04
668个 296 668个 037
【巩固】 计算 99 ⋅ ⋅ ⋅ 9 × 88 ⋅ ⋅ ⋅ 8 ÷ 66 ⋅⋅⋅ 6
例题精讲
模块一、多位数求精确值运算
【例 1】 计算： 55 ⋅ ⋅ ⋅ 5 × 33 ⋅ ⋅ ⋅3
2007 个5 2007 个 3
【考点】多位数计算之求精确值 【难度】3 星 【题型】计算 【解析】 这道题目，你会发现无规律可循.这时我们就要从找规律这个思想里走出来，将 33 ⋅ ⋅ ⋅ 3 乘以 3 凑
2008个 9 2008个 9 2007 个 9 2007个0
原式 = 99 9800 01+199 9 = 100 0
2007 个 9 2007个0 2008个 9 4016 个 0
方法二： 观察一下你会发现， 两个乘数都非常大， 不便直接相乘， 其中 999 很接近 1 000 ， 于是我们采用添项凑整，简化运算。 原式 =（ = 99 ⋅ ⋅ ⋅ 9 00 ⋅ ⋅ ⋅ 0 − 99 ⋅ ⋅ ⋅ 9 + 100 ⋅ ⋅ ⋅ 0 + 99 ⋅⋅⋅9 100 0 −1 × 99 9 + 100 0 + 99 9 ）
2008个 0 2008个 9 2008个 0 2008个 9 2008个 9 2008个 0 2008个 9 2008个 0 2008个 9
= 99 ⋅ ⋅ ⋅ 9 00 ⋅ ⋅ ⋅ 0 + 100 ⋅ ⋅ ⋅= 0 100 ⋅⋅⋅ 0
2008个 9 2008个 0 2008个 0 4016 个 0
1-3-2.多位数计算.题库
教师版
page 1 of 8
原式 = 88 ⋅ ⋅ ⋅ 8 × 99 ⋅⋅⋅9 ÷ 3 = 88 ⋅ ⋅ ⋅8× ⋅ ⋅ ⋅ 0 -1） ÷= 3 （88 ⋅ ⋅ ⋅ 800 ⋅ ⋅ ⋅ 0 - 88 ⋅ ⋅ ⋅ 8） ÷3 （1 00
2006 个 5 2007 个 4 668个185 668个148
【答案】 185 ⋅ ⋅ ⋅ 1851848148 ⋅ ⋅ ⋅ 14815
668个185 668个148
【巩固】 计算： 88 ⋅ ⋅ ⋅ 8 × 33 ⋅ ⋅ ⋅3
2007 个8 2007 个 3
来自百度文库
【考点】多位数计算之求精确值 【难度】3 星 【题型】计算 【解析】 这道题目，你会发现无规律可循.这时我们就要从找规律这个思想里走出来，将 33 ⋅ ⋅ ⋅ 3 乘以 3 凑
2007 个 3
出一个 99 ⋅ ⋅ ⋅ 9 ，然后在原式乘以 3 的基础上除以 3，所以
2007 个 9
2007 个 9 2007 个1 2007 个 7 2007 个 9 2007 个8 2007 个 0 2007 个8
⋅ ⋅ ⋅ 800 ⋅ ⋅ ⋅ 0 − 88 ⋅ ⋅ ⋅ 8) ÷ 3 = 88 ⋅ ⋅ ⋅ 8711 ⋅ ⋅ ⋅ 12 ÷ =( 88 = 3 296 ⋅ ⋅ ⋅ 296 ⋅ ⋅ ⋅ 037 2957 037 04