2.13小数除法

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  小数除法

  

  [Figure 1]

  Lucy is baking a batch of cookies to share with her friends. She is planning to share the cookies with 6 different friends. She wants everybody (including herself!) to get 3 cookies, so she wants to make 21 cookies. Lucy also wants to make sure all her cookies are the same size. She carefully weighs her batch of cookie dough and finds that she has 39.9 ounces of dough total. How can Lucy figure out how many ounces each ball of cookie dough should weigh?

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  In this concept, you will learn how to  divide  decimals with and without  rounding .
  ::在这个概念中,你将学会如何将小数点与小数点和不四舍五入分开。

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  Dividing Decimals
  ::分裂的十进制

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  Division is useful anytime you want to split up an object or a group of objects. It is also useful when you want to reverse the multiplication done on a number.
  ::分区在您想要分割一个对象或一组对象时是有用的。当您想要倒转一个数字上的乘法时,它也是有用的。

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  Remember that the answer to a division problem is called the  quotient . The number being divided is called the  dividend  and the number you are dividing by is called the  divisor .
  ::记住, 分裂问题的答案被称为商数。 所分割的数字被称为红利, 而您所分割的数字则被称为“ divisor ” 。

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  Here is where those terms are in a division problem.
  ::这就是这些术语存在分歧问题的地方。

  $$\begin{array}{r}\text{dividend}÷\text{divisor}=\text{quotient}\end{array}$$

  $$\begin{array}{r}\stackrel{\text{quotient}}{\text{divisor}\overline{)\text{dividend }}}\end{array}$$

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  Dividing decimal numbers is very similar to dividing  whole numbers , but with a couple of extra steps. In  order  to be able to do long division like normal, you will first want to change your divisor into a whole number by moving its  decimal point . Then, in order to keep the expression equal, you will need to move the decimal point of the dividend in the same way.
  ::十进制数字与整数的除法非常相似, 但是要加几个步骤。 为了能够像正常一样进行长的除法, 您首先要移动其十进制点, 将其修改为整数 。 然后, 为了保持表达式的相等, 您需要以同样的方式移动股息的十进制点 。

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  Here are the steps for dividing decimal numbers.
  ::这是十进制数字的除法步骤 。

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  1. Move the decimal point of your divisor to the right as many places as necessary to turn it into a whole number.
     ::将您的断点的小数点移动到右边, 以便将其转换成一个完整的数字 。
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  2. Move the decimal point of your dividend the same number of places. Add zeros if necessary.
     ::将股息的十进制点移动到相同的位数。必要时添加零。
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  3. Temporarily ignore any decimal point in the dividend and divide using long division just as if you were dividing whole numbers.
     ::暂时忽略红利中的任何小数点,使用长分割法进行分割,就像将整个数字分开一样。
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  4. In your answer, insert a decimal point directly above the decimal point in your dividend.
     ::在您的答复中,在您的红利中,在小数点以上插入一个小数点。

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  Here is an example.
  ::举一个例子。

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  Divide  $\begin{array}{r}4.6÷2.3\end{array}$ .
  ::4.6和2.3分开。

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  First, note that 4.6 is your dividend and 2.3 is your divisor.
  ::首先,请注意4.6是你的红利,2.3是你的红利。

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  Now, move the decimal point of your divisor, 2.3, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 2.3 into 23.
  ::现在,将您的断点2.3的小数点向右移到右方,以便将其变成一个整数。您将把小数点1的位置向右移到右方,将2.3变成23。

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  Next, move the decimal point of your dividend, 4.6, the same number of places. Move the decimal point 1 place to the right to turn 4.6 into 46.
  ::下一步,将你的股息小数点(4.6)移动到4.6, 位数相同。将小数点1移到右边,将4.6转成46。

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  Now, divide as usual using long division.
  ::现在,像往常一样,用长的分裂来分裂。

  $$\begin{array}{r}\stackrel{2}{23\stackrel{¯}{)46}}\end{array}$$

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  Last, decide where the decimal point should go in your answer. Because the dividend no longer has a decimal point, your answer will not have a decimal point.
  ::最后,决定您答案中的小数点应该到哪里去。由于红利不再有小数点,您的答案就不会有小数点。

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  The answer is  $\begin{array}{r}4.6÷2.3=2\end{array}$ .
  ::答案是4.62.3=2。

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  Remember that you can always check your answer to a division problem by multiplying. You can verify that  $\begin{array}{r}2.3\times 2=4.6\end{array}$ .
  ::记住, 您可以通过乘法来检查您对分裂问题的答案 。 您可以验证2.3 × 2 = 4. 6 。

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  Here is an example where you will have to add zeros to the dividend.
  ::这里的例子就是,你必须在红利上增加零。

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  Divide  $\begin{array}{r}9÷2.25\end{array}$ .
  ::除以9+2.25。

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  First, note that 9 is your dividend and 2.25 is your divisor.
  ::首先,请注意,9是你的红利,2.25是你的顾问。

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  Now, move the decimal point of your divisor, 2.25, to the right in order to make it a whole number. You will move the decimal point 2 places to the right to turn 2.25 into 225.
  ::现在,将您的二位数的小数点 2. 25 移动到右边, 以便将其变成一个完整的数字。 您会将小数点 2 移到右边, 将 2. 25 转换为 225 。

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  Next, move the decimal point of your dividend, 9, the same number of places. Even though there is no written decimal point with the number 9, remember that 9 is the same as  $\begin{array}{r}9.000000\dots \end{array}$ . When you move the decimal point you will be adding zeros. Move the decimal point 2 places to the right to turn 9 into 900.
  ::下一步, 移动您的股息的十进位点, 9 位数相同 。 即使 9 位数没有写出十进位点 。 记住 9 与 9 千 000 相同 。 当您移动小数点时, 您将会添加零 。 将小数点 2 位数移到右边, 将 9 变成 900 。

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  Now, divide as usual using long division.
  ::现在,像往常一样,用长的分裂来分裂。

  $$\begin{array}{r}\stackrel{4}{225\stackrel{¯}{)900}}\end{array}$$

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  Last, decide where the decimal point should go in your answer. Because the dividend no longer has a decimal point, your answer will not have a decimal point.
  ::最后,决定您答案中的小数点应该到哪里去。由于红利不再有小数点,您的答案就不会有小数点。

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  The answer is  $\begin{array}{r}9÷2.25=4\end{array}$ .
  ::答案是9+2.25=4。

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  Sometimes you will want to  round  numbers before or after dividing them. Rounding is useful when you only need an approximate answer instead of an exact answer. Make sure to read the question carefully in order to figure out if you should round before or after doing the division.
  ::有时您会想要在拆分它们之前或之后整数。 舍入在只需要大致答案而不是准确答案时有用。 务必仔细阅读问题, 以便确定您在分割之前或之后是否应该绕行 。

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  Here is an example.
  ::举一个例子。

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  Round each number to the nearest tenth and then divide.
  ::每一回合到最近的第十回合,然后分成两部分。

  $\begin{array}{r}67.521÷2.243\end{array}$

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  First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point.
  ::首先,将每个数字四舍五入到最接近的十分之一。记住,十分位数是小数点右侧的第一个位数。

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  • 67.521 rounds to 67.5
    ::67.521发炮弹至67.5
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  • 2.243 rounds to 2.2
    ::2.243发至2.2发

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  Now, divide the rounded numbers. Note that 67.5 is your dividend and 2.2 is your divisor.
  ::现在,将四舍五入数字除以。请注意67.5是你的红利,2.2是你的分数。

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  Next, move the decimal point of your divisor, 2.2, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 2.2 into 22.
  ::下一步,将您的断点 2. 2 的小数点向右移动, 以便将其变成一个完整的数字。 您将把小数点 1 的位置向右移动, 将2.2 变成 22 。

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  Then, move the decimal point of your dividend, 67.5, the same number of places. Move the decimal point 1 place to the right to turn 67.5 into 675.
  ::然后将你的股息的小数点( 67.5) 移动到67.5 。 位数相同。 将小数点 1 移到右侧, 将67.5 转换为675 。

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  Now, divide as usual using long division. The decimal point in your answer will go directly above the decimal point in the dividend.
  ::现在, 使用长除法按常态分隔。 您的回答中的小数点将直接超过红利中的小数点 。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{30.68181\dots }{22\overline{)675.000000}}\\ & & \underset{\_}{-66}\\ & & 150\\ & & \underset{\_}{-132}\\ & & 180\\ & & \underset{\_}{-176}\\ & & 40\\ & & \underset{\_}{-22}\\ & & 180\\ & & \underset{\_}{-176}\\ & & 40\\ & & \underset{\_}{-22}\\ & & 180\end{array}\end{array}$$

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  Notice that this time there is a repeating pattern in the long division. This means the answer is a  repeating decimal . The  $\begin{array}{r}818181\dots \end{array}$  pattern will repeat over and over forever. In this case, you have two options.
  ::请注意, 这一次在长分隔线中存在重复模式。 这意味着答案是重复小数小数。 81881... 模式将永远重复。 在此情况下, 您有两个选项 。

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  1. You can use a bar above the repeating portion of the quotient to indicate that it repeats:  $\begin{array}{r}30.6\overline{81}\end{array}$ .
     ::您可以在重覆的商数部分上方使用条表示重复: 30.6 81 。
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  2. You can round your answer: 30.682.
     ::你可以绕过你的答案: 30.682。

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  In this case, because the original dividend and divisor were already rounded and your answer is already only an approximate answer, it makes sense to round.
  ::在这种情况下,因为最初的红利和股息已经四舍五入,而你的答案已经只是一个大致答案,圆起来是有道理的。

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  The answer is  $\begin{array}{r}67.521÷2.243\end{array}$  is approximately 30.682.
  ::答案是67.521 2.243 约为 30.682。

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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 Lucy and her cookies.
  ::早些时候,你得到一个问题 关于露西和她的饼干。

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  She's baking up 21 cookies to share with her friends and she has 39.9 ounces of dough. She wants to figure out how much each ball of cookie dough should weigh so that she will have 21 equally sized cookies.
  ::她正在烤21个饼干,与朋友分享,她有39.9盎司的面团。她想弄清楚每个球的饼干应该有多重,这样她就可以有21个同等大小的饼干。

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  To figure this out, Lucy needs to divide  $\begin{array}{r}39.9÷21\end{array}$ .
  ::露西需要分出39.9 21

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  First, she should note that 39.9 is her dividend and 21 is her divisor.
  ::首先,她应当指出,39.9是她的红利,21是她的顾问。

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  Her divisor is already a whole number, so she will not need to move any decimal points.
  ::她的腹部已经是一个完整的数字, 所以她不需要移动任何小数点 。

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  Next, she should divide as usual using long division. The decimal point in her answer will go directly above the decimal point in the dividend.
  ::其次,她应该像往常一样使用长分割法分割。她的答复中的小数点将直接超过红利中的小数点。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{1.9}{21\overline{)39.9}}\\ & & \underset{\_}{-21}\\ & & 189\\ & & \underset{\_}{-189}\\ & & 0\end{array}\end{array}$$

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  The answer is  $\begin{array}{r}39.9÷21=1.9\end{array}$ . Lucy should make each ball of cookie dough 1.9 ounces.
  ::答案是39.9 21 = 1.9. 露西应该做每个球 饼干面团1.9盎司.

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  Example 2
  ::例2

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  The Landry’s living room is a rectangle with an area of 97.92 square meters and a length of 13.6 meters. What is the width of the living room?
  ::Landry的客厅长13.6米,面积97.92平方米。 客厅宽度是多少?

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  First, remember that the area of a rectangle is given by length times width. This means that if you have the area and divide by the length, you will get the width.
  ::首先, 记住矩形区域是用长度乘以宽度的。 这意味着如果您有区域, 并用长度除以, 您将得到宽度 。

  $$\begin{array}{r}\begin{array}{rcl}w& =& A÷l\\ w& =& 97.92÷13.6\end{array}\end{array}$$

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  Note that for this problem, 97.92 is your dividend and 13.6 is your divisor.
  ::请注意,对于这个问题,97.92是你的红利,13.6是你的红利。

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  Now, move the decimal point of your divisor, 13.6, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 13.6 into 136.
  ::现在,将您的断点的小数点13.6向右移动, 以便将其变成一个完整的数字。 您将把小数点 1 移到右边, 将13.6 转换为136 。

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  Next, move the decimal point of your dividend, 97.92, the same number of places. Move the decimal point 1 place to the right to turn 97.92 into 979.2.
  ::下一步, 移动您的股息的十进位点, 97.92, 与位数相同。 将小数点 1 移到右边, 将97.92 移到979. 2 。

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  Now, divide as usual using long division. The decimal point in your answer will go directly above the decimal point in the dividend.
  ::现在, 使用长除法按常态分隔。 您的回答中的小数点将直接超过红利中的小数点 。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{7.2}{136\overline{)979.2}}\\ & & \underset{\_}{-952}\\ & & 272\\ & & \underset{\_}{-272}\\ & & 0\end{array}\end{array}$$

  The answer is the width is 7.2 meters.

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  Example 3
  ::例3

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  Divide  $\begin{array}{r}3.96÷1.2\end{array}$ .
  ::3.96=1.2除以3.96=1.2。

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  First, note that 3.96 is your dividend and 1.2 is your divisor.
  ::首先,请注意3.96是你的红利,1.2是你的红利。

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  Now, move the decimal point of your divisor, 1.2, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 1.2 into 12.
  ::现在,将您的断点1.2的小数点向右移动, 以便将其整数化为一个整数。 您将把小数点1 移到右边, 将 1.2 转换为 12 。

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  Next, move the decimal point of your dividend, 3.96, the same number of places. Move the decimal point 1 place to the right to turn 3.96 into 39.6.
  ::下一步, 移动股息的小数点为3. 96, 位数相同。 将小数点 1 移到右边, 将3. 96 移到39.6 。

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  Now, divide as usual using long division. The decimal point in your answer will go directly above the decimal point in the dividend.
  ::现在, 使用长除法按常态分隔。 您的回答中的小数点将直接超过红利中的小数点 。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{3.3}{12\overline{)39.6}}\\ & & \underset{\_}{-36}\\ & & 36\\ & & \underset{\_}{-36}\\ & & 0\end{array}\end{array}$$

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  The answer is  $\begin{array}{r}3.96÷1.2=3.3\end{array}$ .
  ::答案是3.961.2=3.3。

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  Example 4
  ::例4

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  Find the quotient of  $\begin{array}{r}24.288÷9.6\end{array}$ .
  ::查找24.288的商数 9.6

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  First, note that 24.288 is your dividend and 9.6 is your divisor.
  ::首先,请注意24 288是你的红利,9.6是你的红利。

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  Now, move the decimal point of your divisor, 9.6, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 9.6 into 96.
  ::现在,将您的断点的小数点( 9. 6) 移动到右边, 以便将其变成一个完整的数字。 您将把小数点 1 移到右边, 将9. 6 转换为96 。

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  Next, move the decimal point of your dividend, 24.288, the same number of places. Move the decimal point 1 place to the right to turn 24.288 into 242.88.
  ::下一步, 移动您的股息的小数点( 24. 288) , 即相同的位数。 将小数点 1 移到右边, 将24 288 移到 242. 88 。

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  Now, divide as usual using long division. The decimal point in your answer will go directly above the decimal point in the dividend.
  ::现在, 使用长除法按常态分隔。 您的回答中的小数点将直接超过红利中的小数点 。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{2.53}{96\overline{)242.88}}\\ & & \underset{\_}{-192}\\ & & 508\\ & & \underset{\_}{-480}\\ & & 288\\ & & \underset{\_}{-288}\\ & & 0\end{array}\end{array}$$

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  The answer is  $\begin{array}{r}24.288÷9.6=2.53\end{array}$ .
  ::答案是24.2889.6=2.53。

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  Example 5
  ::例5

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  Round to the nearest tenth and then divide.
  ::回合到最近的第十回合,然后分成两部分。

  $\begin{array}{r}4.721÷2.465\end{array}$

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  First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point.
  ::首先,将每个数字四舍五入到最接近的十分之一。记住,十分位数是小数点右侧的第一个位数。

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  • 4.721 rounds to 4.7
    ::4.721发至4.7
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  • 2.465 rounds to 2.5
    ::2.465发至2.5发

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  Now, divide the rounded numbers. Note that 4.7 is your dividend and 2.5 is your divisor.
  ::现在,将四舍五入的数字除以。请注意,4.7是你的红利,2.5是你的红利。

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  Next, move the decimal point of your divisor, 2.5, to the right in order to make it a whole number. You will move the decimal point 1 place to the right to turn 2.5 into 25.
  ::下一页,将您的断点的小数点2.5移到右侧,以便将其变成一个完整的数字。您将把小数点1移到右侧,将2.5转换为25。

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  Then, move the decimal point of your dividend, 4.7, the same number of places. Move the decimal point 1 place to the right to turn 4.7 into 47.
  ::然后将你的股息的小数点, 4.7 移动到相同的位数。将小数点 1 移到右边,将4.7 转换为47。

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  Now, divide as usual using long division. The decimal point in your answer will go directly above the decimal point in the dividend.
  ::现在, 使用长除法按常态分隔。 您的回答中的小数点将直接超过红利中的小数点 。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{1.88}{25\overline{)47.00}}\\ & & \underset{\_}{-25}\\ & & 220\\ & & \underset{\_}{-200}\\ & & 200\\ & & \underset{\_}{-200}\\ & & 0\end{array}\end{array}$$

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  The answer is  $\begin{array}{r}4.721÷2.465\end{array}$  is approximately 1.88.
  ::答案是4.721 2.465 约1.88。

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  Review
  ::回顾

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  Find the quotient.
  ::找到商数。

  1. $\begin{array}{r}5.4÷4.5\end{array}$  
  2. $\begin{array}{r}8.71÷6.7\end{array}$  
  3. $\begin{array}{r}3.375÷2.25\end{array}$  
  4. $\begin{array}{r}11.2÷5.6\end{array}$  
  5. $\begin{array}{r}19.11÷1.3\end{array}$  
  6. $\begin{array}{r}28.992÷18.12\end{array}$  
  7. $\begin{array}{r}113.52÷12.9\end{array}$  
  8. $\begin{array}{r}31.93÷3.1\end{array}$  
  9. $\begin{array}{r}46.125÷6.15\end{array}$  
  10. $\begin{array}{r}84.28÷17.2\end{array}$  

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  Find the quotient and then round to the nearest hundredth.
  ::找到商数,然后绕到最近的一百个。

  11. $\begin{array}{r}113.409÷25.21\end{array}$  
  12. $\begin{array}{r}81.862÷6.4\end{array}$  
  13. $\begin{array}{r}377.15÷1.54\end{array}$  
  14. $\begin{array}{r}9.17÷4.5\end{array}$  
  15. $\begin{array}{r}7.56÷2.12\end{array}$  

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