3.13分数的除法

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  分数的除法

  

  [Figure 1]

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  Robert is making cookies to sell at the bake sale at school. He wants to make as many cookies as possible. As he pulls the ingredients out of the pantry he notices that he doesn't have that much sugar. He measures his sugar and sees that he has  $\begin{array}{r}4\frac{1}{2}\end{array}$  cups of sugar. To make one batch of cookies he will need  $\begin{array}{r}\frac{3}{4}\end{array}$  cup of sugar. How many batches of cookies can he make?
  ::罗伯特在学校的烤面包销售会上做饼干出售。 他想做尽可能多的饼干。 当他从储藏室抽取原料时, 他注意到自己没有那么多糖。 他量了糖, 发现自己有412杯糖。 要做一批饼干, 他需要3 4杯糖。 他能做多少批饼干?

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  In this concept, you will learn how to  divide  fractions and mixed numbers.
  ::在这个概念中,你会学会如何分化分数和混合数字。

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  Dividing Fractions and Mixed Numbers
  ::分数分数和混合数字

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  Dividing fractions is very similar to multiplying fractions. In fact, to complete a fraction division problem you will rewrite it as a fraction multiplication problem and then solve. In  order  to turn a fraction division problem into a fraction multiplication problem you will need to find the  reciprocal  of one of the fractions. The  reciprocal  of a number is one divided by the number. When working with fractions, the reciprocal of a fraction is just the fraction flipped.
  ::分数分数与分数的乘数分数非常相似。 事实上, 要完成一个分数分数问题, 您将重写为分数乘法问题, 然后解决。 为了将分数分数问题转换成分数乘法问题, 您需要找到一个分数分数的对等性。 一个数字的对等性除以数字。 使用分数时, 一个分数的对等性只是翻转的分数。

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  Here are some examples.
  ::以下是一些例子。

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  • The reciprocal of  $\begin{array}{r}\frac{2}{5}\end{array}$  is  $\begin{array}{r}\frac{5}{2}\end{array}$ .
    ::两五对等是五二
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  • The reciprocal of  $\begin{array}{r}\frac{1}{3}\end{array}$  is  $\begin{array}{r}\frac{3}{1}\end{array}$ .
    ::13对等是3对1对1。
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  • The reciprocal of  $\begin{array}{r}\frac{7}{6}\end{array}$  is  $\begin{array}{r}\frac{6}{7}\end{array}$ .
    ::7 6是6 7。

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  To turn a fraction division problem into a fraction multiplication problem, take the reciprocal of the  divisor . 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 .
  ::将分数分解问题转换成分数乘法问题, 请使用差数的对等法。 记住, 分数问题的答案叫做商数 。 所分割的数字叫做红利, 您所分割的数字则叫做 divisor 。

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

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  Here are the steps for dividing fractions.
  ::这是分分数的阶梯。

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  1. Rewrite the division problem as a multiplication problem by finding the reciprocal of the divisor.
     ::将分裂问题重写为乘法问题,
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  2. Multiply the fractions.
     ::乘以分数 。
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  3. Simplify  your answer. Rewrite as a  mixed number  if necessary.
     ::简化您的答复。 如有必要, 重写为混合数 。

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  Here is an example.
  ::举一个例子。

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  Divide  $\begin{array}{r}\frac{6}{8}÷\frac{1}{2}\end{array}$ .
  ::除以 6 8 1 2 。

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  First, find the reciprocal of the divisor,  $\begin{array}{r}\frac{1}{2}\end{array}$ .
  ::首先,找到对等的食堂,12。

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  The reciprocal of  $\begin{array}{r}\frac{1}{2}\end{array}$  is  $\begin{array}{r}\frac{2}{1}\end{array}$ .
  ::12对等为2对1。

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  Next, rewrite the division problem as a multiplication problem by multiplying the dividend by the reciprocal of the divisor.
  ::其次,将分割问题改写为乘法问题,将红利乘以分股权的对等。

  $$\begin{array}{r}\frac{6}{8}\cdot \frac{2}{1}\end{array}$$

   

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  Now, multiply.
  ::现在,乘数。

  $$\begin{array}{r}\frac{6}{8}\cdot \frac{2}{1}=\frac{12}{8}\end{array}$$

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  Finally, simplify.
  ::最后,简化。

  $$\begin{array}{r}\frac{12}{8}=\frac{3}{2}=1\frac{1}{2}\end{array}$$

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  The answer is  $\begin{array}{r}\frac{6}{8}÷\frac{1}{2}=1\frac{1}{2}\end{array}$ .
  ::答案是6 8 1 2 = 1 2 。

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  You can also  . Just like when multiplying mixed numbers, when dividing mixed numbers you will first rewrite each mixed number as an  improper fraction .
  ::您也可以。 就像乘以混合数字一样, 当除以混合数字时, 您将首先将每个混合数字重写为不适当的分数 。

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  Here are the steps for dividing mixed numbers.
  ::这是混合数字的分化步骤。

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  1. Convert the mixed numbers to improper fractions.
     ::将混合数字转换成不适当的分数。
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  2. Rewrite the division problem as a multiplication problem by finding the reciprocal of the divisor.
     ::将分裂问题重写为乘法问题,
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  3. Multiply the fractions.
     ::乘以分数 。
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  4. Simplify your answer. Rewrite as a mixed number if necessary.
     ::简化您的答复。 如有必要, 重写为混合数 。

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  Here is an example.
  ::举一个例子。

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  Divide  $\begin{array}{r}4\frac{1}{3}÷2\frac{1}{6}\end{array}$ .
  ::除以4 1 3 + 2 1 6 。

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  First, rewrite both mixed numbers as improper fractions.
  ::首先,改写两个混合数字,作为不适当的分数。

  • $\begin{array}{r}4\frac{1}{3}=\frac{(3\cdot 4)+1}{3}=\frac{13}{3}\end{array}$
  • $\begin{array}{r}2\frac{1}{6}=\frac{(6\cdot 2)+1}{6}=\frac{13}{6}\end{array}$

  $$\begin{array}{r}\frac{13}{3}÷\frac{13}{6}\end{array}$$

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  Now, find the reciprocal of the divisor,  $\begin{array}{r}\frac{13}{6}\end{array}$ .
  ::现在,找出13 6对等的食堂

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  The reciprocal of  $\begin{array}{r}\frac{13}{6}\end{array}$  is  $\begin{array}{r}\frac{6}{13}\end{array}$ .
  ::13 6是6 13。

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  Next, rewrite the division problem as a multiplication problem by multiplying the dividend by the reciprocal of the divisor.
  ::其次,将分割问题改写为乘法问题,将红利乘以分股权的对等。

  $$\begin{array}{r}\frac{13}{3}\cdot \frac{6}{13}\end{array}$$

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  Now, multiply.
  ::现在,乘数。

  $$\begin{array}{r}\frac{13}{3}\cdot \frac{6}{13}=\frac{78}{39}\end{array}$$

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  Finally, simplify.
  ::最后,简化。

  $$\begin{array}{r}\frac{78}{39}=2\end{array}$$

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  The answer is  $\begin{array}{r}4\frac{1}{3}÷2\frac{1}{6}=2\end{array}$ .
  ::答案是4 1 3 + 2 1 6 = 2 。

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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 Robert and his cookies.
  ::之前,你得到一个问题 关于罗伯特和他的饼干。

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  Robert is making cookies for a bake sale. He needs  $\begin{array}{r}\frac{3}{4}\end{array}$  cup of sugar for each batch of cookies. He has  $\begin{array}{r}4\frac{1}{2}\end{array}$  cups of sugar in his pantry. Robert needs to figure out how many batches of cookies he can make.
  ::罗伯特正在做饼干做面包销售。 他每批饼干需要34杯糖。 他有412杯糖。 罗伯特需要弄清楚他能做多少批饼干。

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  In order to determine how many batches of cookies he can make, Robert needs to divide  $\begin{array}{r}4\frac{1}{2}\end{array}$  by  $\begin{array}{r}\frac{3}{4}\end{array}$ .
  ::为了确定他能做多少批饼干 罗伯特需要将412除以34

  $$\begin{array}{r}4\frac{1}{2}÷\frac{3}{4}\end{array}$$

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  First, Robert should rewrite the mixed number as an improper fraction.
  ::首先,罗伯特应该改写混合数字,作为一个不适当的部分。

  • $\begin{array}{r}4\frac{1}{2}=\frac{(2\cdot 4)+1}{2}=\frac{9}{2}\end{array}$

  $$\begin{array}{r}\frac{9}{2}÷\frac{3}{4}\end{array}$$

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  Now, he should find the reciprocal of the divisor,  $\begin{array}{r}\frac{3}{4}\end{array}$ .
  ::现在,他应该 找到对等的 法官,3 4 。

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  The reciprocal of  $\begin{array}{r}\frac{3}{4}\end{array}$  is  $\begin{array}{r}\frac{4}{3}\end{array}$ .
  ::3 4的对等是4 3。

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  Next, Robert should rewrite the division problem as a multiplication problem by multiplying the dividend by the reciprocal of the divisor.
  ::接下来,罗伯特应该把分裂问题改写成一个乘法问题,将股息乘以分股权的对等。

  $$\begin{array}{r}\frac{9}{2}\cdot \frac{4}{3}\end{array}$$

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  Now, he should multiply.
  ::现在,他应该成倍增长。

  $$\begin{array}{r}\frac{9}{2}\cdot \frac{4}{3}=\frac{36}{6}\end{array}$$

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  Finally, he should simplify.
  ::最后,他应当简化。

  $$\begin{array}{r}\frac{36}{6}=6\end{array}$$

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  The answer is Robert can make 6 batches of cookies with the sugar he has.
  ::答案是罗伯特可以用他的糖做6批饼干

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  Example 2
  ::例2

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  Trina is building a sailboat. She needs 8 planks that are each  $\begin{array}{r}\frac{3}{4}\end{array}$  foot in length. She has a piece of wood that is  $\begin{array}{r}6\frac{1}{2}\end{array}$  feet long. How many planks can she cut from this board? Does she have enough to make her sailboat?
  ::Trina正在建造一艘帆船,她需要8个木板,每条34英尺长。她有一块木板,长612英尺。她可以从这块板上切出多少木板?她有足够的木板做帆船吗?

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  You will need to divide  $\begin{array}{r}6\frac{1}{2}\end{array}$  by  $\begin{array}{r}\frac{3}{4}\end{array}$  to determine how many planks Trina can cut from this board.
  ::你需要将612除以34 以确定Trina能从这块板上切掉多少木板。

  $$\begin{array}{r}6\frac{1}{2}÷\frac{3}{4}\end{array}$$

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  First, rewrite the mixed number as an improper fraction.
  ::首先,重写混合数字,作为不适当的部分。

  • $\begin{array}{r}6\frac{1}{2}=\frac{(2\cdot 6)+1}{2}=\frac{13}{2}\end{array}$

  $$\begin{array}{r}\frac{13}{2}÷\frac{3}{4}\end{array}$$

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  Now, find the reciprocal of the divisor,  $\begin{array}{r}\frac{3}{4}\end{array}$ .
  ::现在,找到对等的食堂,3 -4。

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  The reciprocal of  $\begin{array}{r}\frac{3}{4}\end{array}$  is  $\begin{array}{r}\frac{4}{3}\end{array}$ .
  ::3 4的对等是4 3。

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  Next, rewrite the division problem as a multiplication problem by multiplying the dividend by the reciprocal of the divisor.
  ::其次,将分割问题改写为乘法问题,将红利乘以分股权的对等。

  $$\begin{array}{r}\frac{13}{2}\cdot \frac{4}{3}\end{array}$$

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  Now, multiply.
  ::现在,乘数。

  $$\begin{array}{r}\frac{13}{2}\cdot \frac{4}{3}=\frac{52}{6}\end{array}$$

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  Finally, simplify.
  ::最后,简化。

  $$\begin{array}{r}\frac{52}{6}=\frac{26}{3}=8\frac{2}{3}\end{array}$$

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  The answer is Trina can cut  $\begin{array}{r}8\frac{2}{3}\end{array}$  planks from the piece of wood that she has. She has enough to make her sailboat.
  ::答案是Trina可以从她的木头上剪掉823木板 她有足够的钱做帆船

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  Example 3
  ::例3

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  Divide  $\begin{array}{r}\frac{5}{10}÷\frac{1}{2}\end{array}$ .
  ::除以5 10+1 2。

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  First, find the reciprocal of the divisor,  $\begin{array}{r}\frac{1}{2}\end{array}$ .
  ::首先,找到对等的食堂,12。

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  The reciprocal of  $\begin{array}{r}\frac{1}{2}\end{array}$  is  $\begin{array}{r}\frac{2}{1}\end{array}$ .
  ::12对等为2对1。

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  Next, rewrite the division problem as a multiplication problem by multiplying the dividend by the reciprocal of the divisor.
  ::其次,将分割问题改写为乘法问题,将红利乘以分股权的对等。

  $$\begin{array}{r}\frac{5}{10}\cdot \frac{2}{1}\end{array}$$

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  Now, multiply.
  ::现在,乘数。

  $$\begin{array}{r}\frac{5}{10}\cdot \frac{2}{1}=\frac{10}{10}\end{array}$$

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  Finally, simplify.
  ::最后,简化。

  $$\begin{array}{r}\frac{10}{10}=1\end{array}$$

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  The answer is  $\begin{array}{r}\frac{5}{10}÷\frac{1}{2}=1\end{array}$ .
  ::答案是5 10 + 1 2 = 1 。

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  Example 4
  ::例4

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  Divide  $\begin{array}{r}\frac{6}{8}÷\frac{1}{4}\end{array}$ .
  ::除以 6 8 1 4 。

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  First, find the reciprocal of the divisor,  $\begin{array}{r}\frac{1}{4}\end{array}$ .
  ::首先,找到对等的食堂,14。

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  The reciprocal of  $\begin{array}{r}\frac{1}{4}\end{array}$  is  $\begin{array}{r}\frac{4}{1}\end{array}$ .
  ::14对等是4对1。

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  Next, rewrite the division problem as a multiplication problem by multiplying the dividend by the reciprocal of the divisor.
  ::其次,将分割问题改写为乘法问题,将红利乘以分股权的对等。

  $$\begin{array}{r}\frac{6}{8}\cdot \frac{4}{1}\end{array}$$

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  Now, multiply.
  ::现在,乘数。

  $$\begin{array}{r}\frac{6}{8}\cdot \frac{4}{1}=\frac{24}{8}\end{array}$$

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  Finally, simplify.
  ::最后,简化。

  $$\begin{array}{r}\frac{24}{8}=3\end{array}$$

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  The answer is  $\begin{array}{r}\frac{6}{8}÷\frac{1}{4}=3\end{array}$ .
  ::答案是6 8 + 1 4 = 3 。

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  Example 5
  ::例5

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  Divide  $\begin{array}{r}6\frac{1}{4}÷1\frac{1}{2}\end{array}$ .
  ::除以6 1 4 + 1 1 2 。

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  First, rewrite both mixed numbers as improper fractions.
  ::首先,改写两个混合数字,作为不适当的分数。

  • $\begin{array}{r}6\frac{1}{4}=\frac{(4\cdot 6)+1}{4}=\frac{25}{4}\end{array}$
  • $\begin{array}{r}1\frac{1}{2}=\frac{(2\cdot 1)+1}{2}=\frac{3}{2}\end{array}$

  $$\begin{array}{r}\frac{25}{4}÷\frac{3}{2}\end{array}$$

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  Now, find the reciprocal of the divisor,  $\begin{array}{r}\frac{3}{2}\end{array}$ .
  ::现在,找到三二对等的食堂

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  The reciprocal of  $\begin{array}{r}\frac{3}{2}\end{array}$  is  $\begin{array}{r}\frac{2}{3}\end{array}$ .
  ::3 2 的对等为 2 3 。

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  Next, rewrite the division problem as a multiplication problem by multiplying the dividend by the reciprocal of the divisor.
  ::其次,将分割问题改写为乘法问题,将红利乘以分股权的对等。

  $$\begin{array}{r}\frac{25}{4}\cdot \frac{2}{3}\end{array}$$

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  Now, multiply.
  ::现在,乘数。

  $$\begin{array}{r}\frac{25}{4}\cdot \frac{2}{3}=\frac{50}{12}\end{array}$$

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  Finally, simplify.
  ::最后,简化。

  $$\begin{array}{r}\frac{50}{12}=\frac{25}{6}=4\frac{1}{6}\end{array}$$

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  The answer is   $\begin{array}{r}6\frac{1}{4}÷1\frac{1}{2}=4\frac{1}{6}\end{array}$ .
  ::答案是6 1 4 1 2 = 4 1 6 。

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  Review
  ::回顾

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  Divide.
  ::分开。

  1. $\begin{array}{r}\frac{1}{10}÷\frac{3}{4}\end{array}$
  2. $\begin{array}{r}\frac{5}{9}÷\frac{1}{6}\end{array}$
  3. $\begin{array}{r}\frac{1}{5}÷\frac{5}{8}\end{array}$  
  4. $\begin{array}{r}\frac{2}{3}÷\frac{1}{4}\end{array}$
  5. $\begin{array}{r}2÷\frac{1}{2}\end{array}$
  6. $\begin{array}{r}1\frac{1}{3}÷3\frac{1}{8}\end{array}$
  7. $\begin{array}{r}5\frac{4}{5}÷1\frac{1}{5}\end{array}$
  8. $\begin{array}{r}2\frac{1}{4}÷\frac{1}{7}\end{array}$
  9. $\begin{array}{r}4\frac{5}{8}÷2\end{array}$
  10. $\begin{array}{r}\frac{1}{7}÷\frac{1}{6}\end{array}$
  11. $\begin{array}{r}3\frac{5}{6}÷1\frac{2}{3}\end{array}$
  12. $\begin{array}{r}6\frac{1}{5}÷\frac{7}{12}\end{array}$
  13. $\begin{array}{r}\frac{1}{5}÷\frac{7}{12}\end{array}$
  14. $\begin{array}{r}9\frac{1}{5}÷\frac{3}{12}\end{array}$
  15.   $\begin{array}{r}4\frac{1}{5}÷\frac{8}{9}\end{array}$

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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