5.2最简形式的比率

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  最简形式的比率

  

  [Figure 1]

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  Noah's family is on a road trip across the United States in their van. They started in Buffalo, New York, and they are driving all the way to San Francisco, California. The entire trip is approximately 2,700 miles, and so far they have traveled 2,400 miles. Noah can only think of the 300 miles they have yet to go. How can you express the distance still left to go to the total number of miles as a fraction in its  simplest form ?
  ::诺亚的家人正在乘坐他们的面包车横贯美国的公路旅行。他们从纽约的布法罗出发,一路开到加利福尼亚的旧金山。整个行程大约2,700英里,迄今为止他们已经行驶了2,400英里。诺亚只能想到他们尚未行驶的300英里。你如何用最简单的形式表达距离,将仍然剩下到总英里数的距离表达成一个小数呢?

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  In this concept, you will learn to  reduce  ratios to their simplest form.
  ::在这个概念中,你将学会降低与最简单形式的比率。

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  Reducing Ratios to Simplest Form
  ::降低最简单表格比率

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  A  ratio  is the relationship, or comparison, of one  amount  to another.
  ::A比率是指一个数额与另一个数额的关系或比较。

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  To  simplify  a ratio means to reduce it to its smallest, simplest, terms. 
  ::简化比例意味着将其减少到最小、最简单、最简单的条件。

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  In  order  to reduce a ratio, or fraction, to its simplest form, it is necessary to find the greatest, or highest, common factor for both terms in the ratio or fraction.
  ::为了将比例或分数降低到最简单的形式,必须在比例或分数中找到两个方面的最大或最高共同系数。

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  The   of two numbers is the largest number that will  divide  into both terms evenly.
  ::两个数字是两个用语均匀划分的最大数字。

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  Let's look at an example:
  ::让我们举个例子:

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  Write this ratio in simplest form  $\begin{array}{r}\frac{20}{24}\end{array}$ .
  ::以最简单的形式写下这个比例 20 24 。

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  First, find the  factors  of the terms 20 and 24.
  ::首先,找出20和24这两个用语的因数。

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  The factors of 20 are 1, 2, 4, 5, 10, and 20.
  ::20的系数是1、2、4、5、10和20。

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  The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
  ::24个系数是1、2、3、4、6、8、12和24。

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  Next, find the highest common factor.
  ::接下来,找出最高的共同因素。

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  The factors that both 20 and 24 have in common are 1, 2 and 4.
  ::20和24的共同因素是1、2和4。

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  The greatest of those common factors is 4.
  ::其中最大的共同因素是4个。

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  Then, divide both terms by 4 to write the ratio in its simplest form.
  ::然后,将两个术语除以4 来以最简单的形式写出比率。

  $\begin{array}{r}\frac{20}{24}=\frac{20÷4}{24÷4}=\frac{5}{6}\end{array}$

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  The answer is   $\begin{array}{r}\frac{5}{6}\end{array}$
  ::答案是5 6

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  Simplifying ratios can be helpful in determining equivalence.
  ::简化比率可有助于确定等值。

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   are those that are equal.
  ::他们是平等的,

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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 Noah and the family cross-country road trip.
  ::早些时候,你被问及Noah和家人的 跨国公路旅行的问题。

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  Noah was dismayed as he thought about having to ride yet another 300 miles, but that was all that was left of the 2,700 mile trip. Express this distance as a fraction in its simplest terms.
  ::诺亚在想再骑三百英里时感到沮丧,但这是2,700英里行程的剩余部分。 用最简单的词语将距离表达为一小部分。

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  First, write a fraction.
  ::首先,写一个分数。

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  300 miles to go : 2,700 miles total
  ::距离300英里 总共2700英里

  $\begin{array}{r}\frac{300}{2700}\end{array}$

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  Next, recognize the two zeros at the end of each term make 100  a large factor that can be immediately removed.
  ::接下来,在每个任期结束时承认两个零,使100成为可以立即消除的巨大因素。

  $\begin{array}{r}\frac{3}{27}\end{array}$

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  Then, find the greatest common factor to further simplify and reduce both terms by division.
  ::然后找出最大的共同因素,按司进一步简化和减少这两个条件。

    $\begin{array}{r}\frac{3÷3}{27÷3}=\frac{1}{9}\end{array}$

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  The answer is  $\begin{array}{r}\frac{1}{9}\end{array}$
  ::答案是1 9

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  Noah's family has only  $\begin{array}{r}\frac{1}{9}\end{array}$   of the total distance yet to go. 
  ::诺亚的家人只有19个距离还远着呢

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  Example 2
  ::例2

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  Simplify this ratio: 14:16
  ::简化这一比率:14:16

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  First, write the ratio as a fraction.
  ::首先,将比率写成一个分数。

    $\begin{array}{r}\frac{14}{16}\end{array}$

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  Next, determine the common factors.
  ::其次,确定共同因素。

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  The factors of 14 are 1, 2, 7, and 14.
  ::14个系数为1、2、7和14。

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  The factors of 16 are 1, 2, 4, 8, and 16.
  ::16个系数为1、2、4、8和16。

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  Then, determine the greatest common factor.
  ::然后,决定最大的共同因素。

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  Common factors are 1 and 2.
  ::共同因素是1和2。

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  2 is the greatest common factor.
  ::2是最大的共同因素。

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  Next, divide both terms in the fraction by the greatest common factor.
  ::其次,将两个术语的分数除以最大的共同系数。

  $\begin{array}{r}\frac{14÷2}{16÷2}=\frac{7}{8}\end{array}$

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  The answer is  $\begin{array}{r}\frac{7}{8}\end{array}$  or 7:8
  ::答案是7 8或7: 8

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  Example 3
  ::例3

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  Simplify this ratio: 3 to 18
  ::简化这一比率:3至18

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  First, write the ratio as a fraction.
  ::首先,将比率写成一个分数。

  $\begin{array}{r}\frac{3}{18}\end{array}$

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  Next, determine the factors of each term.
  ::接下来,决定每个术语的因数。

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  The factors of 3 are 1 and 3.
  ::3的系数为1和3。

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  The factors of 18 are 1, 2, 3, 6, 9, and 18.
  ::18个系数是1、2、3、4、6、9和18。

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  Then, divide both terms by the highest common factor.
  ::然后,将两个条件除以最高共同系数。

  $\begin{array}{r}\frac{3÷3}{18÷3}=\frac{1}{6}\end{array}$

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  The answer is  $\begin{array}{r}\frac{1}{6}\end{array}$  or 1 to 6
  ::答案是16或1至6

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  Example 4
  ::例4

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  Write an equivalent ratio for 16 to 20 in its lowest terms:
  ::以最低值计算,16至20的对应比率为16至20:

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  First, write the ratio as a fraction.
  ::首先,将比率写成一个分数。

  $\begin{array}{r}\frac{16}{20}\end{array}$

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  Then, find the factors for each term.
  ::然后找出每个学期的因数。

  1, 2, 4, 8, 16

  1, 2, 4, 5, 10, 20

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  Next, divide by the greatest common factor.
  ::其次,除以最大的共同因素。

    $\begin{array}{r}\frac{16÷4}{20÷4}=\frac{4}{5}\end{array}$

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  The answer is  $\begin{array}{r}\frac{4}{5}\end{array}$ or 4 to 5.
  ::答案是 4 5 或 4 至 5 或 4 至 5 。

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  Example 5
  ::例5

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  Beatrice has her own dog-walking service. Each day, she walks 2 dogs in the morning and 4 in the evening. In simplest terms, express the ratio of dogs that Beatrice walks in the morning.
  ::比阿特丽斯有自己的狗行走服务。她每天早上走两只狗,晚上走四只狗。用最简单的话说,表达比阿特丽斯早上走的狗比例。

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  First, remember that a ratio expresses a part to a whole and realize that the total number of dogs is:
  ::首先,记住比例表示整体的一部分,并认识到狗的总数是:

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  2 in the morning + 4 in the evening = 6 dogs total
  ::上午2点+晚上4点=总共6只狗

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  Then, write the ratio (2 dogs in the morning, 6 dogs total)  2 to 6 as a fraction.
  ::然后,写下这个比率(早上2只狗,总共6只狗) 2比6作为一个分数。

    $\begin{array}{r}\frac{2}{6}\end{array}$

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  Next, find the factors.
  ::接下来,找出因素。

  1, 2

  1, 2, 6

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  Then, divide both terms by the greatest common factor.
  ::然后,将两个条件除以最大的共同因素。

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

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  The answer is  $\begin{array}{r}\frac{1}{3}\end{array}$
  ::答案是1 3

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  Beatrice walks one third of the total number of dogs in the morning.
  ::比阿特丽斯早上走的狗 占了所有狗的三分之一

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  Review
  ::回顾

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  Simplify each ratio. Write the simplified version in the same form of the ratio.
  ::简化每个比例。 以相同比例的形式写入简化版本 。

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  1. 3 to 6
     ::3至6 3至6
  2. 5:20
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  3. 18 to 22
     ::18 至 22
  4. $\begin{array}{r}\frac{18}{20}\end{array}$
  5. $\begin{array}{r}\frac{25}{55}\end{array}$
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  6. 6 to 42
     ::6至42岁
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  7. 18 to 10
     ::18 至 10
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  8. 12 to 4
     ::12至4 12至4
  9. 16:8
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  10. 24 to 16
      ::24至16岁
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  11. 18 to 36
      ::18至36 18至36
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  12. 54 to 9
      ::54至9
  13. 81:27
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  14. 72 to 9
      ::72至9
  15. $\begin{array}{r}\frac{18}{12}\end{array}$

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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