1、单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,*,北师大版 五年级上册 第五单元 分数的意义,问题,探究,练习,拓展,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,把一个整体平均分成若干份,其中的一份或几份,可以用,分数,表示。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profil
2、e 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Crea
3、ted with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3.圈一圈,填一填,再说一说。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.
4、2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,4.和同伴说一说下面每个分数表示的意思。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,5.为帮助灾区人民,奇思捐献了零花钱的 ,妙想捐献了,零花钱的 ,妙想捐的钱一定比奇思多?请说明理由。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profi
5、le 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,北师大版 五年级上册 第五单元 分数的意义,问题,探究,练习,拓展,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 As
6、pose Pty Ltd.,?,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,1,1,2,1,2,1,3,1,3,1,3,1,4,1,4,1,4,1,4,1,5,1,5,1,5,1,5,1,5,1,6,1,6,1,6,1,6,1,6,1,6,1,7,1,8,1,9,1,10,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profil
7、e 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,1.制作一张纸条,以它为单位测量教室中某些物品,的长度。测量前先估计,再记录实际测量的结果,并与同伴交流。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2.,10cm()dm,1cm是1dm的,(),(),3cm是1dm的,(),(),1,10,1,10,3,Evaluation only.,Created with
8、 Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2.,1元()角,1角是1元的,(),(),7角是1元的,(),(),10,10,1,10,7,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3.观察下图中的分数,在括号里填上适当的数。,3个 是(),里有()个 ,,()个 是1。,1,1
9、0,7,10,1,10,1,10,在这些分数中,最接近0的是(),最接近1,的是()。,3,10,1,10,7,10,9,10,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,4.下面哪些分数更接近0,哪些分数更接近1?分别,填入圈内,并尝试说明理由。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,C
10、opyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,北师大版 五年级上册 第五单元 分数的意义,问题,探究,练习,拓展,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.
11、1,4,1,一又四分之一,带分数,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,真分数,假分数,Evaluation only.,Created with Aspose.Slide
12、s for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,1.每只小狗分到几根香肠?,用长方形纸片代替香肠,画一,画,并与同伴交流你的想法。,用分数表示每只小狗分到的香,肠数。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2.以7为分母,分别写出3个真分数和3个假分数。,7,Evaluation only.,Create
13、d with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3.下面哪些分数比1大?把它们圈出来。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,4.平均每人分到多少块月饼?,画一画,分一分,并与同伴,交流。,Evaluation only.,Created with Aspose.Sl
14、ides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,5.如图,在上面的 里填上适当的假分数,在,下面的 里填上适当的带分数。,分别写出两个比1小的分数和两个比1大的分数。,分别写出两个比2大且比3小的带分数和假分数。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created
15、 with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,北师大版 五年级上册 第五单元 分数的意义,第一课时,第二课时,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,把1块蛋糕平均分给2个小朋友,每人可以分到几块蛋糕?,如果把7块蛋糕平均分给3个小朋友呢?,12,1,2,73,7,3,
16、Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,你发现分数与除法有什么关系?与同伴说一说。你能用字母表示分数与除法之间的关系。,12,1,2,73,7,3,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,举例说一说,假分数和带分数如
17、何进行互化?,1,3,2,7,3,2,1,3,1,3,6,3,61,3,7,3,61,3,1,3,6,3,2,1,3,1,3,2,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,1.这些茶叶平均装在4个小罐里,每小罐装多少千,克?平均装在5个小罐里呢?,想一想,算一算,并与同伴,交流。,请你再举一个例子,说明分,数与除法的关系。,Evaluation only.,Created with Aspose.Slide
18、s for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2.把10块巧克力平均分给3个人,每人分到几块?,平均分给4个人呢?5个人呢?画一画,分一分,,并与同伴交流你是怎么分的。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3.把下列假分数化成带分数或整数,把带分数化成,假分数。,Evaluation only.,Cr
19、eated with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,13,1,3,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,43,4,3,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.
20、2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,5.山娃家一共养了6只羊,8只鸭子,20只兔子。,羊的只数是鸭子的几分,之几?,羊的只数是兔子的几分,之几?兔子的只数是羊,的几分之几?,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5
21、2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,6.把3m彩带平均分给4个小朋友,每人分到几米?,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,7.在括号里填上适当的数,并与同伴交流。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004
22、2011 Aspose Pty Ltd.,8.这板药能吃多少天?,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,9.分桃子。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created wi
23、th Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,北师大版 五年级上册 第五单元 分数的意义,问题,探究,练习,拓展,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3,4,6,8,9,12,Evaluation only.,Created with Aspose.Slides for.
24、NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,分数的分子和分母同时乘(或除以)一个不为零的数,分数的大小不变。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Cop
25、yright 2004-2011 Aspose Pty Ltd.,1.涂一涂,填一填。,2,6,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2.请你用画图或列算式的方式说明分数的基本性质。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty
26、Ltd.,3.在括号里填上合适的数。,32,4,5,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3.在括号里填上合适的数。,32,4,5,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,4.,Evaluation only.,C
27、reated with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,5.说出相等的分数。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.
28、0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,北师大版 五年级上册 第五单元 分数的意义,问题,探究,练习,拓展,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pt
29、y Ltd.,12的因数,18的因数,1,12,2,6,3,4,1,18,2,9,3,6,12和18的公因数有,。,12和18的最大公因数有,。,1,2,3,6,6,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,12的因数,18的因数,1,12,2,6,3,4,,18,9,12和18的公因数,1,2,6,3,Evaluation only.,Created with Aspose.Slides for.NET
30、3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,1.找出9和15的所有因数及最大公因数,并与同伴,交流你是怎么找的。,9的因数有,;,1,3,9,15的因数有,;,1,3,5,15,9和15的最大公因数有,。,3,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2.填一填,与同伴交流。,1,2,3,6,6的因数,8的因数,6和8的公因
31、数,1,2,4,8,1,2,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3.找出下面各组数的最大公因数。,2和4,9和8,9和18,6和9,9和12,8和18,5和10,14和7,20和15,2,1,9,3,3,2,5,7,5,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 20
32、04-2011 Aspose Pty Ltd.,4.写出下列分数分子和分母的最大公因数。,4,8,12,(),15,21,(),12,15,(),6,18,(),3,3,6,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,分别写出1,2,,,20各数和4的最大公因数。,根据上表完成下图。,1,2,1,4,1,2,1,4,1,2,1,4,Evaluation only.,Created with Aspose.Sl
33、ides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,找一找1,2,,,20各数和10的最大公因数,,你有什么发现?与同伴交流。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profil
34、e 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,北师大版 五年级上册 第五单元 分数的意义,问题,探究,练习,拓展,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,8,24,4,12,2,6,1,3,最简分数,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,
35、Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,1.胜的场数、平的场数、,输的场数各占总场数的,几分之几?,用最
36、简分数表示,说一,说你是怎么约分的。,请你再举一个例子说明,约分的过程,并与同伴,交流。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2.圈出最简分数,并把其余的分数约分。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3.连
37、谜底。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,4.在 里填上,“,”“,”,或,“,”,。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,5.明明的头长是20cm,身高是120cm,明明头长,是身高的几分之几?,Eval
38、uation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,6.在下图中表示下列分数,你发现了什么?,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET
39、 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,北师大版 五年级上册 第五单元 分数的意义,问题,探究,练习,拓展,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,3
40、2,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,4和6的公倍数有,。,4和6的最小公倍数是,。,12,24,36,48,12,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright
41、 2004-2011 Aspose Pty Ltd.,1.在下表中用,“,”,标出3的倍数,用,“,”,标,出5的倍数。,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,哪些数上既标有,“,”,又标有,“,”,?这些数有什么特点?,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Prof
42、ile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2.,1,2,3,4,5,6,7,8,9,10,8,8,16,6,6,12,50以内6和8的公倍数有,,,最小公倍数是,。,24,32,40,48,56,64,72,80,18,24,30,36,42,48,54,60,24,48,24,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3.想一想,填一填。6和9的最小
43、公倍数是多少?,18,36,6,12,24,30,42,48,9,27,45,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,4.求下面各组数的最小公倍数。,3和6,8和10,2和7,6和10,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty
44、Ltd.,18,24,2,9,12,3,4,3,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,北师大版 五年级上册 第五单元 分数的意义,问题,探究,练习,拓展,Evaluation
45、 only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2,7,1,6,1,6,2,7,2,7,12,42,1,6,7,42,12,42,7,42,所以,2,7,1,6,所以,2,7,1,6,Evaluation
46、 only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3,10,1,6,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,1,6,10,60,3,10,18,60,10,60,18,60,所以,1,6,3,10,3,10,1,6,1,6,5,30,3,10,9
47、30,5,30,9,30,所以,1,6,3,10,6和10的,公倍数,做分母,6和10的,最小公倍数,做分母,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,1.淘气、笑笑和奇思看一本同样的数学故事书。,淘气和笑笑谁看的页数多?说一说你是怎么想的。,怎样比较两个分数的大小?与同伴交流。,比一比,笑笑和奇思谁看的页数多?,Evaluation only.,Created with Aspose.Slides fo
48、r.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,2.比较下面各组分数的大小,并与同伴交流。,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,3.我国现已建立两千多个自然保护区,各级保护区,个数占所有保护区总数的几分之几?,1,10,19,50,13,25,Evaluation only.,Created with Aspo
49、se.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,4.森林运动会上,小兔和小山羊进行跑步比赛。在,相同时间内,小山羊跑了全程的 ,小兔跑了,全程的 。谁跑得快呢?,5,6,5,6,6,7,6,7,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,5.小小食品店有三种数量相同的冷饮,星期五的销,售情况如下。,如果这个食品店要进货,应该多进哪种冷饮?为什么?,5,7,1,2,2,9,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,Evaluation only.,Created with Aspose.Slides for.NET 3.5 Client Profile 5.2.0.0.,Copyright 2004-2011 Aspose Pty Ltd.,






