植树问题方阵问题

　 植树问题

　【含义】

　按相等的距离植树，在距离、棵距、棵数这三个量之间，已知其中的两个量， 要求第三个量，这类应用题叫做植树问题。

　【数量关系】

　1 非封闭线路上的植树问题主要可分为以下三种情形:

　⑴如果在非封闭线路的两端都要植树,那么:

　株数＝段数＋1＝全长÷株距+1棵数=全长÷间隔长+1

　全长＝株距×(株数－1)全长=间隔长×(棵数-1)

　株距＝全长÷(株数－1)间隔长=全长÷(棵数-1)

　⑵如果在非封闭线路的一端要植树,另一端不要植树,那么:

　株数＝段数＝全长÷株距棵数=全长÷间隔长

　全长＝株距×株数全长=间隔长×棵数

　株距＝全长÷株数间隔长=全长÷棵数

　⑶如果在非封闭线路的两端都不要植树,那么:

　株数＝段数－1＝全长÷株距－1棵数=全长÷间隔长-1　

　全长＝株距×(株数＋1)全长=间隔长×(棵数+1)

　株距＝全长÷(株数＋1)间隔长=全长÷(棵数+1)　

　2 封闭线路上的植树问题的数量关系如下 此题是植树问题中植树线路是封闭的一种.在圆、正方形、长方形、闭全曲线等上面植树,因为首尾相接,两端重合在一起.

　株数＝段数＝全长÷株距棵数=全长÷间隔长　

　全长＝株距×株数全长=间隔长×棵数

　株距＝全长÷株数间隔长=全长÷棵数线形植树 环形植树棵数＝距离÷棵距

　方形植树棵数＝距离÷棵距－4 三角形植树棵数＝距离÷棵距－3

　面积植树 棵数＝面积÷（棵距×行距）

　锯木头 总时间=每次用时×次数

　总台阶=每个楼梯的台阶数×楼梯数

　

　例1 一条河堤 136 米，每隔 2 米栽一棵垂柳，头尾都栽，一共要栽多少棵？

　 例2 一个圆形池塘周长为 400 米，在岸边每隔 4 米栽一棵白杨树，一共能栽多 少棵白杨树？

　 例3 一个正方形的运动场，每边长 220 米，每隔 8 米安装一个照明灯，一共 可以安装多少个照明灯？

　 例4 给一个面积为 96 平方米的住宅铺设地板砖，所用地板砖的长和宽分别是 60 厘米和 40 厘米，问至少需要多少块地板砖？

　例5 一座大桥长 500 米，给桥两边的电杆上安装路灯，若每隔 50 米有一个电 杆， 每个电杆上安装 2 盏路灯，一共可以安装多少盏路灯？ 解 （1）桥的一边有多少个电杆？ （2）桥的两边有多少个电杆？（3）大桥两边可安装多少盏路灯？

　【知识运用】

　一、直线型植树问题

　（一）两端都种： I 求全长 1、 在一条小路的一侧， 每隔 10 米种一棵柳树， 从头到尾共种 20 棵， 则小路全长多少米？

　在一条小路的一侧，从头到尾共安装 10 根电线杆，每隔 10 米安装一根，则小路全长 多少米？

　3、10 路共公汽车从起点到终点共有 13 的车站，每两个车站相距 2 千米，则 10 路汽车 全程多少千米？

　4、时钟报时，5 时敲 5 下，每两下之间间隔 2 秒，则一共用了多少时间？

　6、小明家住在 6 层，他每上一层需要 10 秒种，则他从一楼到家需要多少秒？

　7、小明家住在 6 层，每个楼梯上有 16 级台阶，则他从一楼到家需要走多少个台阶？

　II 求棵数 1、在一条小路的一侧，每隔 10 米种一棵柳树，如果小路全长 100 米，则可种柳树多少 棵？

　2、在一条小路的一侧，从头到尾每隔 10 米安装一根电线杆，如果小路全长 100 米，则 可以安装电线杆多少根？

　3、10 路共公汽车从起点到终点全长 24 千米，每两个车站相距 2 千米，则 10 路汽车全 程共有多少个车站？

　4、一根木料锯成若干段需要 40 分钟，每锯一下需要 4 分钟，则可以把它锯成多少段？

　5、小明从一楼到家需要 60 秒，他每上一层需要 10 秒种，则他家住在多少层，？

　6、小明从一楼到家需要走 80 个台阶，每个楼梯上有 16 级台阶，则家住在几层？

　III 求间距 1、在一条小路的一侧从头到尾共种 11 棵树，小路全长 100 米，则每两棵树之间相距多 少米？

　2、在一条小路的一侧，从头到尾共安装 10 根电线根，如果小路全长 90 米，每两根电线 杆之间相距多少米？

　3、10 路共公汽车从起点到终点全长 24 千米，10 路车从头到尾共有 13 个车站，那么每 两个车站之间相距多少千米？

　4、一根木料锯成 5 段需要 40 分钟，每锯一下需要多少分钟？

　5、小明从一楼到六楼需要 60 秒，则他每上一层需要多少秒

　6、小明从一楼到六楼要走 80 个台阶，那么每两层之间有多少个台阶？

　（二）只种一端 I 求全长

　1、在教学楼前小路的一侧，每隔 10 米种一棵柳树，共种 20 棵，则小路全长多少米？

　2、 在校门前小路的一侧， 共安装 10 根电线杆， 每隔 10 米安装一根， 则小路全长多少米？

　II 求棵数 1、在教学楼前小路的一侧，每隔 10 米种一棵柳树，如果小路全长 100 米，则可种柳树 多少棵？

　2、在校门前小路的一侧，每隔 10 米安装一根电线杆，如果小路全长 200 米，则可以安 装电线杆多少根？

　III 求间距 1、在教学楼前一侧共种 11 棵树，小路全长 100 米，则每两棵树之间相距多少米？

　2、在校门前小路的一侧，共安装 10 根电线根，如果小路全长 90 米，每两根电线杆之间 相距多少米？

　（三）两端都不种

　I 求全长 1、在教学楼与图书馆之间小路的一侧，每隔 10 米种一棵柳树，共种 20 棵，则小路全长 多少米？

　2、在校门前至公共汽车站的小路一侧，共安装 10 根电线杆，每隔 10 米安装一根，则小 路全长多少米？

　II 求棵数 1、在教学楼与图书馆之间小路的一侧，每隔 9 米种一棵柳树，如果小路全长 100 米，则 可种柳树多少棵？

　2、，在校门前至公共汽车站的小路一侧，每隔 10 米安装一根电线杆，如果小路全长 200 米，则可以安装电线杆多少根？

　III 求间距 1、在教学楼与图书馆之间小路的一侧共种 9 棵树，小路全长 100 米，则每两棵树之间相 距多少米？

　2、在校门前至公共汽车站的小路一侧，共安装 9 根电线根，如果小路全长 90 米，每两根 电线杆之间相距多少米？

　（四）特别问题：锯木头 数量关系式： 锯的次数＝间隔数－1 其他的一般都是干扰条件 1、一根木料锯成 7 段，每锯一下需要 4 分钟，则一共需要多少分钟？

　2、一根木料平均锯成 4 段，用时 12 分钟，如果平均锯成 6 段，需要多少分钟？

　二、封闭型植树问题与只种一头相同

　棵数=间隔数

　1、一个池塘的周长为 240 米，沿池塘周围每隔 4 米载一棵柳树，可以植树多少棵？

　2、一个池塘的周长为 240 米，沿池塘周围共种树 40 棵，每两棵树相距？

　3、一个池塘每隔 4 米种一棵树，共种 60 棵，则这个池塘的周长是多少米？

　三、方阵问题

　一周总数＝每边数量×边数－边数 每边数量＝一周总数÷边数＋1 （一）求一周的总数量 1、正方形花坛，每边摆 6 盆花（每个顶点摆一盆），一周可以摆多少盆？ 一周总数＝（每边数量－1）×边数

　2、一个正五边形花坛，每边摆 6 盆花（每个顶点摆一盆），一周可以摆多少盆？

　（二）求每边数量 1、正方形花坛一周共摆放 12 盆花（每个顶点摆一盆），那么每边可以摆多少盆？

　2、一个正五边形花坛一周共摆放 30 盆花（每个顶点摆一盆），每边可以摆多少盆？

　小结：解决植树问题和方阵问题，关键要与图结合，根据题目的特点画出草图，可以帮助 我们分析，从而选择适当的方法解决。

　综合题 1、有一条长 1200 米的公路，在公路的两旁载树绿化，现从头到尾每隔 4 米载一棵梧桐 树，共需多少棵梧桐树。

　2、一个池塘的周长为 240 米，沿池塘周围每隔 4 米载一棵柳树，每两棵柳树之间等距离 种了 3 棵杨树，一共载了多少棵柳树，多少棵杨树。

　3、小军乘电梯从 1 楼到 5 楼需 20 秒钟，那么从 1 楼到 20 楼需多少秒钟？

　4、挂钟 3 点敲 3 下，共需 4 秒钟，那么 10 点敲 10 下需多少秒钟？

　5、植树节到了，同学们在长 60 米的围墙边，从头到尾每隔 4 米挖好了树坑，后来发现树 坑太少了，改成每隔 3 米挖一个树坑，问共需新挖多少个树坑？填掉多少个树坑？

　6、 一本画有插图的故事书共 301 页， 每两页插图之间有 2 页文字， 如果开头一页是插图， 那么这本故事书共有多少页文字？

　7、两根钢筋各长 15 米和 18 米，现打算分别把他们都锯成 3 米长的一段，每锯开一段需 3 分钟，全部锯完需多少分钟。

　8、把一根木头锯成 5 段要 8 分钟，如果把这根木头锯成 8 段，则需要多少分钟。

　9、时钟 4 点敲 4 下，用 12 秒敲完，那么 8 点敲 8 下，多少秒钟敲完。

　10、某人到一座大厦的 10 楼去办事，他从１楼到５楼用了 64 秒，用同样的速度走到 10 楼，还要多少秒钟。

　方阵问题公式

　　（1）实心方阵：（外层每边人数）2=总人数。

　　（2）空心方阵：

　　（最外层每边人数）2-（最外层每边人数-2×层数）2=中空方阵的人数。

　　或者是

　　（最外层每边人数-层数）×层数×4=中空方阵的人数。

　　总人数÷4÷层数+层数=外层每边人数。

　　例如，有一个3层的中空方阵，最外层有10人，问全阵有多少人？

　　解一先看作实心方阵，则总人数有

　　10×10=100（人）

　　再算空心部分的方阵人数。从外往里，每进一层，每边人数少2，则进到第四层，每边人数是

　　10-2×3=4（人）

　　所以，空心部分方阵人数有

　　4×4=16（人）

　　故这个空心方阵的人数是

　　100-16=84（人）

　　解二直接运用公式。根据空心方阵总人数公式得

　　（10-3）×3×4=84（人）

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The Czech high poopThe establishment of government support, multi party participation, rural credit loan guarantee mechanism of market operation.The formation of East and interaction, complementary advantages, mutual promotion and promote common development of the new pattern.I go to uphold and improve the "governor rice bag", in charge of the system, "food basket" the mayor is responsible for the system.The promoting agricultural technology integration, labor process machine, production and management of information technology.The increase of the old revolutionary base areas, minority areas, areas, poor areas development support.The whole city urban and rural social management, promote the reform of the household registration system and relax the small city settled conditions.That the majority of farmers with education, employment, medical care, medical, and housing.The development to help the old, disabled, save the orphans alone, poor relief, as the focus of social welfare and charity.The guides farmers orderly employment, encourage farmers to encourage employment, entrepreneurship support agricultural migrant workers.I go to carry out thorough and wide coverage, basic, multi-level, sustainable principle, accelerate the establishment of rural social security system.There must be in accordance with high yield, high-quality, efficient, ecological, safety requirements, accelerate the transformation of agricultural development mode change.The rapid construction of stable supply, reserves sufficient, effective adjustment, efficient operation of the food safety guarantee system.The real effective implementation of agricultural production, acquisition, processing, storage and transportation, quality safety supervision and responsibility of each link of sales.Allow the farmers to subcontract, lease, exchange, transfer, share cooperation and other forms of flow circulation of land contract and management rights.? To clear objectives, to develop planning, increase investment, focus on the overall strength of the relationship between the Department of the overall situation, the impact of long-term major events.The construction of production development, well-off life wide, rural civilization, clean village, new socialist countryside village and democratic management.According to the step by step, is about to land, intensive development, reasonable arrangement of distribution principle, promote the urbanization healthy development.There must have a clear goal, and planning, increase investment, focus in the power of good relations overall, long-term effects of long event.I go to market demand, technological innovation as a means, quality and efficiency as the goal, to build a modern agricultural industrial system.The reasonable arrangement of county towns, agricultural farmland protection, industrial agglomeration, village distribution, ecological conservation and space layout.According to the individual contributions, collective body grants, requires a combination of government subsidies, the establishment of new rural social endowment insurance system for the old.The gradually realize migrant workers labor remuneration, children's education, public health, housing, and urban residents enjoy the same treatment.The form is conducive to the protection of arable land, water, forest, grassland, wetland, and natural resources and agricultural species resources incentive mechanism.According to the service of farmers, and freedom, equal rights, democratic management, support agricultural farmer cooperatives to accelerate development.According to the property rights, use control system, economical and intensive, the original principle of strict management, further improve the rural land management system.I go to speed up the implementation of grain food production, increase farmers' income, enhance financial coordination, fully mobilize the enthusiasm of local farmers all sorts of grain, to catch food.The agricultural foundation is still weak, the need to strengthen the development of rural areas still lag behind; and most in need of support; farmers income is difficult, the need to speed up.Transfer of the right to land contractual management, shall not change the nature of collective ownership of land, shall not change the land use, shall not harm the interests of farmers' land contract.The product of hundreds of millions of farmers greatly mobilized the enthusiasm, the great liberation and development of rural social productivity, greatly improving the farmers' material and cultural life.I go to establish a fair, just, inclusive and orderly, the new international financial order rank direction, adhere to a comprehensive, are balanced, progressive, principle of effectiveness.The search for new selling points, refined provided a new theme, the development of new products, design new line, add new contents to the Pearl River Delta, Gong consolidate the traditional market, the development of long triangle in emerging markets.The grain grain production, farmers income, farmland protection, environmental governance, harmony and stability as a special place assessment is an important content of the leadership of the county class team performance.I go to speed up the establishment of commercial finance, financial cooperation, the combination of policy finance, capital adequacy, function perfect, perfect service, safe operation of the rural financial system.The innovation system mechanism, strengthen the agricultural infrastructure, increase farmers income, protect the rights and interests of farmers, promoting rural harmony, fully mobilize the enthusiasm of the majority of farmers, initiative, and creativity.The great practice of reform and development of the rural development, Yiyiwan greatly mobilized the enthusiasm of farmers, and the development of the great liberation of productive forces in rural areas, farmers very greatly improve the material and cultural life.The hard truth is to adhere to the development, adhere to grasp the development of the party in governing and rejuvenating the country first, adhere to the economic construction as the center, adhere to the development and reform of methods to solve the problems in the forward.The Chinese will continue to promote the construction of the sustainable development of the world economy, inclusive and orderly in the international financial system, a fair and reasonable international trade system, a fair and effective global development system.I go to the construction of a new socialist countryside as a strategic task to take the road of Ren, modern agricultural modernization Chinese characteristics as the basic direction, and to speed up the formation of a new pattern of urban and rural economic and social development integration as a fundamental requirement.In the revolution, construction and Reform in every historical period, our party adhere to the basic principle of the Marx doctrine with the concrete practice of China, has always been high attention, seriously, to solve the solution of agriculture, rural and farmer problems.The agriculture is the world, strategic industry stability in China people, no agricultural modernization is no national modernization, no rural prosperity and stability there is no national country's prosperity and stability, no comprehensive well-off farmers no national people well-off Kang.I go to ensure national food security, promote the strategic adjustment of agricultural structure adjustment, accelerate the innovation of agricultural science and technology, and strengthen the construction of agricultural infrastructure, the establishment of new agricultural social service system, and promote the sustainable development of agriculture and agriculture to expand opening to the outside world.That should adapt to the agricultural scale, precision, facilities and other requirements, accelerate the development of multi-functional, intelligent, and economic agricultural equipment preparation facilities, focus in the field work, facilities cultivation, aquaculture, intensive processing, storage and transportation and other aspects of preservation has made new progress.The more attention to strengthen and improve macro-control, more attention as reform and opening up and promoting independent innovation, and pay more attention to the adjustment of economic structure and improve the quality of development, pay more attention to saving resources and protect the environment, improve people's livelihood as more attention and promote social harmony.The "11th Five-Year" period, will have to maintain stable and rapid economic development, we must accelerate the transformation of economic growth mode, we must improve the ability of independent innovation, and promote the coordinated development of urban and rural areas must, must strengthen the construction of harmonious society, we must continue to deepen reform and opening up.? In order to accelerate the construction of public service mechanism based on cooperative economic organizations as the base, leading enterprises as the backbone, other social forces as supplement, public services and business services combined, special special services and integrated services to coordinate the new agricultural social service system.The past five years of reform and opening up and the comprehensive construction of a well-off society to have significant progress in five years, is the social production forces and overall national strength significantly increased in 55 years, is the all-round development of social undertakings and people get more affordable for five years, is China's international standing and influence is not definitely improved for five years.I go to speed up the transformation of agricultural development, promote agricultural scientific and technological progress and innovation, strengthen the material and technical equipment of agricultural industry, agricultural industry system, improve land productivity, resource utilization, labor productivity, enhance the anti risk ability of agriculture, international competition ability, the ability of sustainable development.- adhere to industry nurturing agriculture, city support rural and less take free policy, institutional innovation, and strengthen the foundation of agriculture, increase farmer income, safeguard the rights and interests of farmers, promoting agricultural rural harmony, fully mobilize the enthusiasm of farmers, the people's initiative, creativity, promote rural economic and social good and and fast development.Only to put the solution of agriculture, rural areas and farmers to ask questions as the priority among priorities, and adhere to the basic position of agriculture, adhere to the socialist market economy reform direction, adhere to the road of agricultural modernization Chinese characteristics, adhere to the protection of peasants' material interests and democratic rights Lili, in order to liberate and develop rural social solutions productivity.That must be in accordance with high yield, high quality, efficient, ecological, safety requirements, accelerate the transformation of agricultural development, and promote the progress of agricultural science and technology and innovation, strengthen the agricultural material and technical equipment, health and improve the agricultural industrial system, improve the rate of land, resource utilization, labor productivity, enhance the anti risk ability of agriculture and the international competition ability, sustainable development ability.Fan Wenwen: all kinds of documents writing parallel sentences highlightsYibin city part evolved from Highway urban road, only road; it from plain City Road junction, often design does not match, caused a decline in capacity and traffic chaos. Direct contact channel between the lack of network layout for a group significantly affected

　responsibilities, agricultural administrative system coordinated operation.The solution of emancipating the mind, seeking truth from facts, and when all the times, continue to promote rural reform and development.The relevant departments at all levels should further emancipate the mind, deepen the reform of reform, opening up.I go to tell the truth, do practical work, seek practical results, resolutely opposed the formalism and bureaucracy.Leading cadres at all levels should adhere to the right of the people, love the people, and work for the people.The ideological emancipation, adhere to reform and opening up, promote scientific development, promote social harmony.The construction of a faith, dedication dedication, skills, re conduct basic rural grassroots cadres.The comprehensive implementation through Scientific Outlook on Development, uplifting spirit, solid work, forge ahead, pioneering and innovative.I go to the solution to continue emancipating the mind to insist on the implementation of reform and opening, promote scientific development, promote social harmony.The construction of tourism personnel training, introduction, selection, use, management, assessment, evaluation, incentive mechanism and so on.The socialist economic construction, political construction, cultural construction, social construction and Party construction has made new progress.I go Gaga more consciously implement to continue to emancipate the mind persist in reform and opening up, promote scientific development, promote social harmony.In the policy, the Ministry of finance work, put on the work of the whole Party cadres to embody in heavy strategic thinking.- adhere to education, system, supervision, reform, rectification rectification, combination of punishment, to promote rural village construction of punishing and preventing corruption.I go to expand the orderly participation, promote information disclosure, improve business negotiation procedure, strengthen the powers of supervision as the focus, and strengthen the building of grassroots political power.I go to the party construction has become the development of promoting science, led farmers to get rich, close ties with the masses, safeguard the stability of the strong strong core of leadership.The first priority to be more conscious, more firmly grasp the development of the party's affairs, more conscious, more firmly push to promote scientific development.- strengthen the unified leadership of the party committees, government together tube, comprehensive rural work department organization and coordination, relevant departments of agricultural and rural work leadership system and working mechanism.? In the revolution, construction and reform of each historical period, our Party firmly adhere to the basic principle of the Marx doctrine with the concrete practice of China, has always attached great importance to the end, seriously, to solve agriculture, rural and farmers problems.I go to the leadership construction as the focus, improve the party organization as the guarantee, level three, LIAN activities as the carrier, the construction of Party organization has become to promote scientific development, led farmers to get rich in close contact with the masses, strong leadership to maintain stability in rural areas.The majority of Party members, cadres should adhere to the right to use, for the people, love the people, and work for the people, concerned about the sufferings of the masses, listen to their whirring sound, on the wisdom of the masses, to tell the truth, do practical work, seek practical results, resolutely opposed the formalism, bureaucracy, efforts to create tangible in the performance.According to the construction of democracy and the rule of law, fair justice, honesty and friendship, full of life vitality, stability and order, and harmony between man and nature of the socialist harmonious society will be required to correctly handle the contradictions among the people under the new situation, solve the serious concern, the people most directly, the most realistic interests.That I worry we must be prepared, enhanced sense of urgency, and always maintain the Marxism of Marx, the meaning of socialist China characteristics, faith in the great rejuvenation of the Chinese nation Xing; must quit arrogance and rashness, hard work, keep in mind the socialist primary stage of the basic national conditions, for the party and the people's cause of unremitting efforts sure; to study hard, work hard, do not continue to create a stand the practice, people and history of test results; and must strengthen the unity of the overall situation, and consciously maintaining the unity of the whole party, the party to maintain close ties with the masses, Gong Gong solid people of all nationalities unity.The need to consolidate and strengthen the fundamental position of agriculture industry, has always been to solve the problem of population Shishijiyi priority rule in Ampang must safeguard the rights and interests of farmers; and has always been to realize, safeguard and develop the fundamental interests of the majority of farmers in rural areas as well all the work of the starting point and end point; we must continue to emancipate and the development of rural social productivity, has always been to reform and innovation as the fundamental driving force for rural development; must urban and rural economic and social development, it has always been the focus on building a new type of city workers and peasants, the relationship between urban and rural areas as a major strategy to accelerate the modernization of the rural work; we must adhere to the party tube, beginning Finally to strengthen and improve the party's leadership of the rural work as a political guarantee for promoting the development of rural reform.Economy classThe change in speed, change the kitchen, lavatories, changing circle.The rural land ownership, registration, certification work.The promotion of agricultural production, farmers income, rural prosperity and prosperity.T

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