5.14问题解决计划，比例

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  问题解决计划，比例

  

  [Figure 1]

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  At the beginning of the school year, the 7 ^th  grade students decided to keep a tally of the books they read. At the end of the year, the students had read a total of 544 books and organized them into categories.
  ::在学年开始时,七年级学生决定对所读书籍进行统计,到年底,学生总共阅读了544本书,并将其分为几类。

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  History: 12 books
  ::历史:12本书

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  Adventure: 250 books
  ::冒险:250本书

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  Romance: 100 books
  ::浪漫:100本书

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  Mystery: 120 books
  ::神秘:120本书

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  Nature/Science: 62 books
  ::自然/科学:62本书

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  What is the ratio of mystery books to adventure books? Do these ratios form a  proportion ? Why or why not?
  ::神秘书与冒险书的比例是多少?这些比例是否占一定比例?为什么或为什么不是?

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  In this concept, you will learn how to solve problems involving ratios and proportions.
  ::在这个概念中,你们将学会如何解决涉及比例和比例的问题。

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  Solving Problems Involving Ratios and Proportions
  ::解决涉及比率和比例的问题

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  A   ratio   compares two quantities. 
  ::A比率比较了两个数量。

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  Ratios can be written as fractions, with a colon or with the word “to.”
  ::比率可以作为分数写成,用冒号或“to”一词写成。

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    $\begin{array}{r}\frac{2}{3}\end{array}$ , 2:3, and "2 to 3" are ratios.
  ::2,3,2,3,2,3 和"2,3"是比例。

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  A   proportion   is created when two ratios are found to be equivalent or equal. A proportion can be written
  ::当发现两个比例对等或相等时,就设定一个比例。 一个比例可以写出来。

  $$\begin{array}{r}\frac{1}{2}=\frac{3}{6}\end{array}$$

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  Proportional reasoning ,   or examining the relationship between two numbers, can be used to determine an unknown value.
  ::比例推理,或审查两个数字之间的关系,可用来确定未知值。

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  A proportion can also be written with colons.
  ::比例也可以用冒号书写。

  $$\begin{array}{r}\frac{a}{b}=\frac{c}{d}ora:b=c:d\end{array}$$

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  When a proportion is written with colons, in the form a:b = c:d, the two  terms  closest together, b and c, are called the “means.” The two terms on the ends, a and d, are called the “ extremes .”
  ::当比例以冒号写成时,以 a:b = c:d 的形式,两个最接近的词, b和c, 一起被称为“手段 ” 。 结尾的两个词, a 和 d, 被称为“ 极端 ” 。

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  The  Cross Products  Property  of Proportions  states that the product of the means is equal to the product of the extremes. 
  ::《比例制的交叉产品属性》指出,这些手段的产品与极端产品相同。

  $$\begin{array}{rl}\frac{a}{b}& =\frac{c}{d}\\ b\cdot c& =a\cdot d\end{array}$$

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  Let's look at this example.
  ::让我们来看看这个例子。

  $$\begin{array}{rl}\frac{1}{2}& =\frac{3}{6}\\ 2\times 3& =1\times 6\\ 6& =6\end{array}$$

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  If two ratios form a proportion, the means will be equal to the extremes.
  ::如果两个比率成比例,其手段将等于极端。

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  Some problems will be easier to solve using proportional reasoning. Others will be easier using cross  multiplication .
  ::有些问题比较容易通过比例推理解决,另一些问题比较容易采用交叉乘法解决。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about the 7 ^th  grade students and a tally of the books they read for a year.
  ::更早之前,你得到一个问题 关于七年级的学生 和统计他们读了一年的书。

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  At the end of the year, the students had read a total of 544 books and organized them into categories.
  ::年底,学生共阅读了544本书,并将其分为几类。

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  History: 12 books
  ::历史:12本书

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  Adventure: 250 books
  ::冒险:250本书

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  Romance: 100 books
  ::浪漫:100本书

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  Mystery: 120 books
  ::神秘:120本书

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  Nature/Science: 62 books  
  ::自然/科学:62本书

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  What is the ratio of mystery books to adventure books? What is the ratio of History to Romance? Do these ratios form a proportion? Why or why not?
  ::神秘书与冒险书之比是多少?历史与浪漫之比是多少?这些比例是否成比例?为什么或为什么不是?

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  First, write the ratios.
  ::首先,写比率。

  $$\begin{array}{r}\frac{mystery}{adventure}=\frac{120}{250}\\ \frac{history}{romance}=\frac{12}{100}\end{array}$$

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  Next, write a proportion.
  ::下一个,写一个比例。

  $$\begin{array}{r}\frac{120}{250}=\frac{12}{100}\end{array}$$

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  Then, cross multiply.
  ::然后,交叉倍增。

  $$\begin{array}{rl}120\times 100& =12\times 250\\ 1,200& \ne 3,000\end{array}$$

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  The answer is that the ratios do not form a proportion because they are not equal.
  ::答案是,比率不构成一个比例,因为它们不相等。

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  Example 2
  ::例2

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  Solve two ways:
  ::解决两条路:

  $$\begin{array}{r}\frac{a}{4}=\frac{6}{8}\end{array}$$

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  First, use proportional reasoning. Ask what can be done to 8 to make it equal to 4.
  ::首先,使用比例推理。问对8可以做些什么来使其等于4。

  $$\begin{array}{r}8÷2=4\end{array}$$

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  Next, divide both numerator and denominator of the given ratio by 2.
  ::其次,将给定比率的分子数和分母除以2。

   

  $$\begin{array}{r}\frac{6÷2}{8÷2}=\frac{3}{4}\end{array}$$

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  The answer is 3.
  ::答案是3

  $$\begin{array}{r}\frac{6}{8}=\frac{3}{4}\end{array}$$

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  Then, solve the problem using cross products.
  ::然后,用交叉产品解决问题。

  $$\begin{array}{rl}a\cdot 8& =4\cdot 6\\ 8a& =24\\ a& =3\end{array}$$

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  The answer is the same, a = 3 
  ::答案是一样的,a=3

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  Example 3
  ::例3

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  Sonia measured the straight-line  distance  between Baltimore, MD, and Washington, D.C., to be 2 centimeters on a map. The scale on the map shows that 1 centimeter = 28 kilometers. What is the actual straight-line distance between Baltimore and Washington D.C.?
  ::索尼娅测量了巴尔的摩、麦地那和华盛顿特区之间的直线距离,在地图上为2厘米。地图上的尺度显示1厘米=28公里。巴尔的摩和华盛顿之间的实际直线距离是多少?

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  First, choose a method.
  ::首先,选择一种方法。

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  This problem involves a map, which is a type of scale drawing. It makes sense to use proportions to solve it. 
  ::这个问题涉及地图, 地图是一种比例图。 使用比例来解决这个问题是有道理的 。

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  Next, write the ratios for the unit scale and the scale distance compared to actual distance.
  ::下一步,写出单位比例和比例距离相对于实际距离的比率。

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    $\begin{array}{r}\frac{scale}{actual}\\ Unitscale& =\frac{1cm}{28km}\\ Distance& =\frac{2}{d}\end{array}$
  ::c c c = 1 c 28 km D = 1 c 28 km D = 2 d

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  Then, set the ratios equal to one another to form a proportion.
  ::然后,设定一个比例。

  $$\begin{array}{r}\frac{1cm}{28km}=\frac{2}{d}\end{array}$$

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  Next, cross multiply and solve.
  ::下一个,交叉倍数和解答。

  $$\begin{array}{rl}2\times 28& =1\times d\\ D& =56km\end{array}$$

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  The answer is that the actual distance from Baltimore to Washington is 56 km.
  ::答案是,从巴尔的摩到华盛顿的实际距离是56公里。

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  Example 4
  ::例4

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  A baker uses 22 cups of flour to make 4 loaves of bread. How many cups of flour will he need to use to make 31 loaves of bread?
  ::面包师用22杯面粉做4大面包。他需要用多少杯面粉做31大面包?

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  First, write the ratios.
  ::首先,写比率。

  $$\begin{array}{r}\frac{flour}{loaves}\frac{22}{4}\frac{x}{31}\end{array}$$

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  Next, write a proportion.
  ::下一个,写一个比例。

  $$\begin{array}{r}\frac{22}{4}=\frac{x}{31}\end{array}$$

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  Then, determine a method. Since the relationship between the terms in the denominators, 4 and 31, is not immediately obvious, cross multiplication is easier.
  ::然后,确定一种方法。由于分母(4)和(31)这两个词之间的关系不十分明显,相互乘法比较容易。

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  Next, cross multiply and solve.
  ::下一个,交叉倍数和解答。

  $$\begin{array}{rl}4x& =22\times 31\\ x& =170.5\end{array}$$

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  The answer is 170.5. The baker will need 170.5 cups of flour to make 31 loaves of bread.
  ::答案是170.5。面包师需要170.5杯面粉,才能做31块面包。

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  Example 5
  ::例5

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  If the baker cut the original recipe in half, how many cups of flour would he have needed?
  ::如果面包师把原食谱切成两半 他需要多少杯面粉?

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  First, write the ratios.
  ::首先,写比率。

  $$\begin{array}{r}\frac{flour}{loaves}\frac{22}{4}\frac{x}{2}\end{array}$$

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  Next, write a proportion.
  ::下一个,写一个比例。

  $$\begin{array}{r}\frac{22}{4}=\frac{x}{2}\end{array}$$

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  Then, determine a method. It is easy to see that 4  2 = 2. Use proportional reasoning.
  ::然后,确定一种方法。很容易看到4 2 = 2. 使用比例推理。

  $$\begin{array}{r}\frac{22÷2}{4÷2}=\frac{11}{2}\end{array}$$

   The answer is 11. Making half the recipe would require 11 cups of flour.

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  Review
  ::回顾

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  Use what you have learned to solve each problem. Consider more than one strategy for solving each problem. Then choose the strategy you think will work best and use it to solve the problem.
  ::使用您所学的知识来解决每个问题。 考虑一个以上的解决每个问题的战略。 然后选择您认为最有效并用它解决问题的战略 。

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  1. A jar contains only pennies and nickels. The ratio of pennies to nickels in the jar is 2 to 7. If there are 14 nickels in the jar, how many pennies are in the jar?
     ::罐子中便便与镍之比是2比7,如果罐子中有14个镍,罐子中有多少便便?
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  2. Anya charges $40 for 5 hours of babysitting. Lionel charges $14 for 2 hours of babysitting. Which babysitter charges the cheapest rate?
     ::Anya为5小时的保姆收费40美元,Lionel为2小时的保姆收费14美元,哪个保姆收费最便宜?
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  3. On a map, Derek measured the straight-line distance between Toronto, Canada and Niagara Falls, New York to be 2 inches. The scale on the map shows that  $\begin{array}{r}\frac{1}{2}inch=11miles\end{array}$ . What is the actual straight-line distance between Toronto and Niagara Falls?
     ::在地图上,Derek测量了多伦多、加拿大和纽约尼亚加拉瀑布之间的直线距离为2英寸。地图上的尺度显示,多伦多和尼亚加拉瀑布之间的实际直线距离是1 2 i n c h = 11 m i e s。 多伦多和尼亚加拉瀑布之间的实际直线距离是多少?
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  4. A desk is 120 centimeters long. What is the length of the desk in meters? Use this unit conversion: 1 meter = 100 centimeters.
     ::办公桌长120厘米。 办公桌的长度是多少? 用这个单位转换:1米=100厘米。
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  5. On a field trip, the ratio of teachers to students is 1 : 25. If there are 5 teachers on the field trip, how many students are on the trip?
     ::在实地考察中,教师与学生的比例是1:25。 如果有5名教师在实地考察中,那么在考察中有多少学生?
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  6. Kara bought 5 pounds of Brand X roast beef for $43. Cameron bought 3 pounds of Brand Y roast beef for $27. Which brand of roast beef is the better buy?
     ::Cameron花了27美元买了3磅Brand Y烤牛肉。
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  7. If two inches on a map are equal to three miles, how many miles are represented by four inches?
     ::如果地图上两英寸等于三英里, 多少英里代表四英寸?
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  8. If eight inches on a map are equal to ten miles, how many miles are 16 inches equal to?
     ::如果地图上8英寸等于10英里 16英寸等于多少英里?
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  9. Casey drew a design for bedroom. On the picture, she used one inch to represent five feet. If her bedroom wall is ten feet long, how many inches will Casey draw on her diagram to represent this measurement?
     ::Casey画了一个卧室设计图。照片上,她用一英寸代表5英尺。如果她的卧室墙长10英尺,Casey将用多少英寸图来代表这个测量?
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  10. If two inches are equal to twelve feet, how many inches would be equal to 36 feet?
      ::如果两英寸等于12英尺, 多少英寸等于36英尺?
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  11. If four inches are equal to sixteen feet, how many feet are two inches equal to?
      ::如果四英寸等于十六英尺,那么两英寸等于多少英尺?
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  12. The carpenter chose a scale of 6” for every twelve feet. Given this measurement, how many feet would be represented by 3”?
      ::木匠选择了每12英尺6英尺的尺码。 根据这种测量,多少英尺代表3英尺? ”
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  13. If 9 inches are equal to 27 feet, how many feet are equal to three inches?
      ::如果9英寸等于27英尺,多少英尺等于3英寸?
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  14. If four inches are equal to 8 feet, how many feet are equal to two inches?
      ::如果四英寸等于八英尺, 多少英尺等于两英寸?
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  15. If six inches are equal to ten feet, how many inches are five feet equal to?
      ::如果6英寸等于10英尺,那么5英尺等于多少英寸?

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  Review (Answers)
  ::回顾(答复)

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  Click   to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

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  Resources
  ::资源