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六上周周练 第十二周---认识比1

六上周周练  第十二周---认识比1
六上周周练  第十二周---认识比1

【同步教育信息】

一、本周主要内容:

比的意义和基本性质、按比例分配问题

二、本周学习目标:

1、了解比的意义,掌握比的读、写方法,知道比的各部分名称以及比与分数、除法的关系。

2、理解并掌握比的基本性质,能应用比的意义和基本性质求比值、化简比,能应用比的知识解答按比例分配的实际问题。

3、经历比的概念的抽象过程,经历探索比与分数、除法的关系以及比的基本性质的过程,积累数学活动的经验,进一步体会数学知识之间内在联系,培养观察、比较、抽象、概括以及推理的能力。

三、考点分析:

1、两个数相除又叫做两个数的比。如:3÷2也就是3:2。比的前项除以后项所得的商叫做比值。比值通常用分数表示,也可以用小数表示,有时也可以是整数。3:2的比值是1.5。

2、同除法比较,比的前项相当于被除数,后项相当于除数,比值相当于商;同分数比较,比的前项相当于分子,后项相当于分母,比值相当于分数值。

3、比的基本性质相当于除法中的商不变性质和分数中的基本性质。因此应用比的基本性质可以将比进行化简。比的前项和后项为互质数时,这个比就是最简整数比。

4、求比值和化简比的核心区别在于结果的表达形式不同,求比值的结果一定要是一个数,化简比的结果一定要是一个比。

5、把一个数量按照一定的比来进行分配,这种分配的方法叫做按比例分配。

四、典型例题

例1、(重点展示)从甲地到乙地共300千米,甲车要行8小时,乙车要行6小时。甲车所行的路程与所用时间的比是(),比值是();乙车所行的路程与所用时间的比是(),比值是()。

分析与解:求哪两个量的比就把这两个量按先后顺序写下来,再在中间添上比号。求比值,就用前项除以后项。

从甲地到乙地共300千米,甲车要行8小时,乙车要行6小时。甲车所行的路程与所用时间的比是(300:8),比值是(37.5);乙车所行的路程与所用时间的比是(300:6),比值是(50)。

点评:比与除法、分数之间有着密切的联系。但不不是说,它们之间是等同的。它们之间的区别是:比是两个量之间的关系,除法是一种运算,而分数是一个数。在理解意义的时候要注意区分。

例2、(重点展示)化简。(1)20:25 (2)0.3:0.27 (3)

43:3

2

分析与解:根据比的基本性质,第(1)题比的前项和后项直接除以5;第(2)题要先把前项和后项同时乘100,再化简;第(3)题要将比的前项和后项同时乘12,再化简。 正确解答:

(1)20:25 = (20÷5):(25÷5)= 4:5 (2)0.3:0.27 = (0.3×100):(0.27×100)= 30:27 =(30÷3):(27÷3)=10:9

(3)

43:32= (43×12):(3

2

×12)= 9:8 点评:在化简过程中,如果比的前项和后项都是整数,那就同时除以它们的最大公约数;

如果前项和后项是小数或是分数,先将它们同时乘一个数化成整数,再化简。要注意:最后化简到比的前项和后项是互质数的比是最简整数比。

例3、(误点诊所)化简。 (1)0.4:0.16 (2)

43:5

2

错误解法:(1)0..4 : 0.16 (2)

43:52 = 4 : 16 = 43×52

= 1 : 4 = 10

3

分析与解:将比的前项和后项同时乘或除以同一个数,进行化简。

(1)0.4 : 0.16 (2)

43:52

= 40 : 16 = (43×20):(5

2

×20)

= 5 : 2 = 15 : 8

点评:第(1)题两个小数,一个是两位小数,一个是一位小数,要将两个数同时乘100化成整数,再化简,而不能一个乘10,一个乘100,那样比的大小就改变了;第(2)题不能为了约分而用乘法,应该将比的前项和后项同时乘12,化成整数比,再化简。

例4、(难点突破)15

8

的前项加上8,如果要使比值不变,后项应该加上( )。 分析与解:把

15

8

的前项增加8,之后前项就变成了16,相当于前项乘了2,要使比值不变,后项也应当乘2,变成30,后项应该加上15。

15

8

的前项加上8,如果要使比值不变,后项应该加上( 15 )。

点评:比的基本性质中讲的是前项和后项同时乘或除以相同的数(0除外),比值不变。加上8,就要把这种加法之间的关系转化为乘法,再去判断。

例5、(重点展示)公园里柳树和杨树的课数比是5:3,柳树和杨树共40棵。柳树和杨树各多少棵?

分析与解:公园里柳树和杨树的课数比是5:3,也就是40棵树中,柳树占5份,杨树占3份,一共是(5+3)份,即柳树占总棵数的

355+,杨树占总棵数的3

53+。 柳树的棵数:40×

355

+ = 25(棵) 杨树的棵数:40×3

53

+ = 15(棵)

答:柳树有25棵,杨树有15棵。

点评:在解答按比例分配应用题时,还可以直接用份数来解。这道题目通过分析,已经知道柳树和杨树共8份,就可以用40÷8,求出每份有5棵,柳树有5份,用5×5=25(棵),求出柳树的棵数。同样,用5×3=15(棵),求出杨树的棵数。

例6、(误点诊所)商店运来一批洗衣机,卖出24台,卖出的台数与剩下的台数的比是3:5,这批洗衣机一共有多少台?

错误解法:24÷3×5=40(台)

分析与解:卖出的台数与剩下的台数的比是3:5,即卖出的台数是3份,剩下的台数是5份,这批洗衣机的总台数是8份。24台对应的份数是3份,可以先求出每份是多少台,再求8份是多少台。

24÷3×(5+3)=64(台) 答:这批洗衣机一共有64台。

点评:在用份数来解按比例分配应用题时,要注意份数和量之间的对应关系。这道题目,是求的总台数,就要用每份数乘总台数所对应的份数。而3份和5份分别是卖出的份数和剩下的份数,它们的和才是总台数所对应的份数。在解题时要仔细思考。

例7、(难点突破)已知A 、B 、C 三个数的比是2:3:5,这三个数的平均数是90,这三个数分别是多少?

分析与解:三个数的平均数是90,那么这三个数的和是90×3=270,A 、B 、C 三个数的比是2:3:5,那么A 是2份,B 是3份,C 是5份,总共是2+3+5 = 10(份),那么就可以求出每份是多少,再逐步求出A 、B 、C 分别是多少? 90×3=270 270÷10 = 27

27×2=54 27×3=81 27×5 = 135 答:这三个数分别是54,81,135。

点评:按比例分配应用题的特点是告诉几个量的比,及这几个量的和或差,或其中的一个量,求出每个量或其中的某些量。但是有些题目,并不是直接知道和或差,而是通过间接条件求出和或差。

【模拟试题】

一、基础巩固题

1、一辆汽车上午3小时行驶96千米,下午4小时行驶140千米。 (1)上、下午行车时间的比是( )。 (2)上、下午所行路程的比是( )。 (3)下午与上午行驶速度的比是( )。

2、在括号里填上适当的数。

5 : 4 = ( ): 24 1.5 : 0.18 = ( ): 18 8 : 15 = 24 : ( ) 3

6 : 12 = 9 :( ) ( ): 0.5 = 9 : 5 14 : ( )=

7 : 1.6

4、甲数与乙数的比是5 : 4,甲数是乙数的

()(),乙数是甲数的()(),甲数是甲乙和的()

()

,乙数是甲乙和的

()

()

。 5、一批货物按3:4:5分配给甲、乙、丙三个队去运。甲队运了这批货物的

()

()

,乙队运

了这批货物的

()(),丙队运了这批货物的()

()。 二、思维拓展题

6、桃树有48棵,梨树是桃树的

8

7

,桃树和梨树的比是( )。 7、被减数是150,减数与差的比是3:2,减数是( ),差是( )。 8、分别写出下列正方体棱长的比和体积的比,并化简。

2厘米 4厘米

9、图书室买来540本新书,其中三分之一是连环画,其余的是文艺书和科技书,文艺书和科技书的比是3:2。三种书各是多少本? 三、自主探索题

10、把10克糖放入100克水中,糖与糖水的比是多少?

11、把2:5的前项增加4,要使比值不变,后项应增加多少? 12、一个长方形操场,周长76米,长与宽的比是10:9。这块操场的面积是多少平方米?

【试题答案】

一、基础巩固题

1、一辆汽车上午3小时行驶96千米,下午4小时行驶140千米。 (1)上、下午行车时间的比是( 3 : 4 )。 (2)上、下午所行路程的比是( 96 : 140 )。 (3)下午与上午行驶速度的比是( 35 : 32 )。

2、在括号里填上适当的数。

5 : 4 = ( 30 ): 24 1.5 : 0.18 = ( 150 ): 18 8 : 15 = 24 : ( 45 ) 3

6 : 12 = 9 :( 3 ) ( 0.9 ): 0.5 = 9 : 5 14 : ( 3.2 )=

7 : 1.6

4、甲数与乙数的比是5 : 4,甲数是乙数的

)4()5(,乙数是甲数的)

5()

4(,甲数是甲乙和的

)9()5(,乙数是甲乙和的)

9()4(。 5、一批货物按3:4:5分配给甲、乙、丙三个队去运。甲队运了这批货物的

)

4()

1(,乙队运了这批货物的

)3()1(,丙队运了这批货物的)

12()5(。 二、思维拓展题

6、桃树有48棵,梨树是桃树的

8

7

,桃树和梨树的比是( 8:7 )。 7、被减数是150,减数与差的比是3:2,减数是( 90 ),差是( 60 )。 8、分别写出下列正方体棱长的比和体积的比,并化简。

2厘米 4厘米

棱长的比 2:4 = 1:2 体积的比 8:64 = 1:8

9、图书室买来540本新书,其中三分之一是连环画,其余的是文艺书和科技书,文艺书和科技书的比是3:2。三种书各是多少本? 连环画 540×

31

= 180(本) 540 – 180 = 360(本) 文艺书 360×233

+ = 216(本)

科技书 360×2

32

+ = 144(本)

三、自主探索题

10、把10克糖放入100克水中,糖与糖水的比是多少?10:110 = 1:11 11、把2:5的前项增加4,要使比值不变,后项应增加多少?10 12、一个长方形操场,周长76米,长与宽的比是10:9。这块操场的面积是多少平方米? 76÷2 = 38(米) 38×

91010+= 20(米) 38×9

109

+= 18(米) 20×18 = 360(平方米)

数学趣味园

数字趣联

宋代大诗人苏东坡年轻是与几个学友进京考试。他们到达试院时为时已晚。考官说:"我出一联,你们若对得上,我就让你们进考场。"考官的上联是:一叶孤舟,坐了二三个学子,启用四桨五帆,经过六滩七湾,历尽八颠九簸,可叹十分来迟。

苏东坡对出的下联是:十年寒窗,进了九八家书院,抛却七情六欲,苦读五经四书,考了三番两次,今日一定要中。

考官与苏东坡都将一至十这十个数字嵌入对联中,将读书人的艰辛与刻苦情况描写得淋漓尽致。

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