2.15解十进制方程的除法

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• 选择节解十进制方程的除法

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  解十进制方程的除法

  

  [Figure 1]

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  Peter has decided to open up a special savings account at the bank. He wants to earn interest on the money he has saved. He has $178 saved up that he is planning to deposit into his new account. He learns that his account will earn 3.2%   each year. The amount of interest,  $\begin{array}{r}I\end{array}$ , he will earn in  $\begin{array}{r}t\end{array}$  years is given by the following  formula :
  ::Peter决定在该银行开设一个特别储蓄账户。 他想从他储蓄的钱上赚取利息。 他存了178美元,计划存入他的新账户。 他了解到他的账户每年将赚取3.2%。 利息的数额,我,他将在两年内赚到的金额由以下公式提供:

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  $\begin{array}{r}I=5.696t\end{array}$  
  ::I=5.696吨

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  Peter's goal is to get his account to $200. How can Peter determine approximately how many years that will take?
  ::Peter的目标是让他的帐户达到200美元 Peter怎么确定要花多少年呢?

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  In this concept, you will learn how to apply the  inverse property of multiplication  to solve decimal equations.
  ::在此概念中,您将学会如何应用乘法的反向属性来解析小数方程式。

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  Using Division to Solve Decimal Equations
  ::使用分解来解析十进制等量

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  Recall that  inverse   operations  are operations that reverse one another.  Addition  and  subtraction  are inverse operations.  Multiplication  and  division  are also inverse operations. For example, if you take any number and multiply it by 3 and then divide the result by 3, you will be back to the original number. The division reversed the multiplication.
  ::回顾反向操作是互相反向的操作。增减是反向操作。乘法和除法也是反向操作。例如,如果将任何数字乘以3再乘以3,结果再除以3,结果就会回到原来的数字。除法将乘数反向。

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  The  Inverse Property of Multiplication  states that the product of any non-zero number and its  reciprocal  is one. In symbols, it says that for any non-zero number  $\begin{array}{r}a\end{array}$ :
  ::反乘法属性表示,任何非零数的产物及其对等是一。在符号中,它表示,对于任何非零数 a :

  $$\begin{array}{r}a\cdot \frac{1}{a}=1\end{array}$$

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  The  multiplicative inverse  of a number is another word for the reciprocal of a number. The multiplicative inverse of  $\begin{array}{r}a\end{array}$  is  $\begin{array}{r}\frac{1}{a}\end{array}$  as long as  $\begin{array}{r}a\end{array}$  is  not equal to  zero.
  ::乘数的乘数反差是数字对等的另一个词。乘数反差为 1 等于 1 等于 1 等于 等于 0 。

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  You can use the inverse property of multiplication to help you to solve equations that would be difficult to solve using mental math. Remember that when you are solving an  equation  your goal is to  isolate  the  variable , which means you want to get the variable by itself on one side of the equation.
  ::您可以使用乘法的反向属性来帮助您解析使用心理数学难以解析的方程式。 记住, 当您正在解析方程式时, 您的目标是分离变量, 这意味着您想要将变量自己放在方程式的一边 。

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  To use the inverse property of multiplication, if your variable is being multiplied by a number, you can divide both sides by that number to  isolate the variable .
  ::要使用乘法的反向属性,如果变量乘以一个数字,您可以将两边除以该数字以孤立变量。

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  Here is an example.
  ::举一个例子。

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  Solve the following equation for  $\begin{array}{r}x\end{array}$ .
  ::解决 x 的以下方程式 。

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  $\begin{array}{r}2.7x=3.78\end{array}$
  ::2.7 x = 3.78

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  First, notice that  $\begin{array}{r}x\end{array}$  is being multiplied by 2.7 on one side of the equation. Divide both sides by 2.7 to isolate  $\begin{array}{r}x\end{array}$ . This is the same as multiplying both sides by  $\begin{array}{r}\frac{1}{2.7}\end{array}$  .
  ::首先,请注意 x 在方程的一边乘以 2.7 。 将两边除以 2. 7 以孤立 x 。 这与将两边乘以 1. 2. 7 相同 。

  $$\begin{array}{r}\frac{2.7x}{2.7}=\frac{3.78}{2.7}\end{array}$$

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  Next, simplify the left side of the equation.  $\begin{array}{r}\frac{2.7}{2.7}\end{array}$  is equal to 1, which leaves you with  $\begin{array}{r}1x\end{array}$  or  $\begin{array}{r}x\end{array}$  on the left side of the equation.
  ::接下来,简化方程的左侧。 2.7 2.7 等于 1, 使您在方程的左侧留下 1 x 或 x 。

  $$\begin{array}{r}\begin{array}{rcl}1x& =& \frac{3.78}{2.7}\\ x& =& \frac{3.78}{2.7}\end{array}\end{array}$$

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  Now, simplify the right side of the equation. Use what you have learned about decimal division. Move the decimal points 1 place to the right to change 2.7 into 27 and 3.78 into 37.8. Then, divide using long division.
  ::现在, 简化方程的右侧。 使用您所学到的关于小数分数的知识。 将小数点 1 移到右侧, 将2. 7 改为 27 和 3. 78 改为 37. 8 。 然后, 用长分数分隔 。

  $$\begin{array}{r}\begin{array}{rc}& \stackrel{1.4}{27\overline{)37.8}}\\ & \underset{\_}{-27}\\ & 108\\ & \underset{\_}{-108}\\ & 0\end{array}\end{array}$$

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  $\begin{array}{r}x=1.4\end{array}$
  ::x = 1. 4 = 1. 4

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  The answer is  $\begin{array}{r}x=1.4\end{array}$ .
  ::答案是 x = 1. 4 。

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  You can check the solution by substituting that value for  $\begin{array}{r}x\end{array}$  back into the original equation and verifying that it makes both sides equal. Use what you have learned about decimal multiplication.
  ::您可以将 x 的值替换为原方程, 并验证它是否使两边相等, 以此检查解决方案。 使用您所学的关于十进制乘法的知识 。

  $$\begin{array}{r}\begin{array}{rcl}2.7x& =& 3.78\\ 2.7(1.4)& =& 3.78\\ 3.78& =& 3.78\end{array}\end{array}$$

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  The answer is correct.
  ::答案是正确的。

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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 Peter and his new savings account.
  ::早些时候,你得到一个问题 彼得和他的新储蓄账户。

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  He is depositing $178 into the account. The amount of interest,  $\begin{array}{r}I\end{array}$ , he will earn in  $\begin{array}{r}t\end{array}$  years is given by the formula  $\begin{array}{r}I=5.696t\end{array}$ . Peter wants to know how many years it will take before he has $200 in his account.
  ::他在账户中存入178美元。利息的数额,我,他将在两年内赚到的利息由公式I = 5.696 t给付。Peter想知道,在他有200美元之前,要花多少年的时间。

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  First, Peter needs to figure out how much interest he would need to earn in  order  to have $200 total. If he starts with $178, he would need  $\begin{array}{r}\$200-\$178=\$22\end{array}$  in interest in order to have $200 total.
  ::首先,Peter需要弄清楚他需要赚取多少利息才能获得200美元的总和。 如果他从178美元开始,他需要200美元 - 178美元 - 22美元利息才能获得200美元的总和。

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  Now, Peter should  substitute  $22 for  $\begin{array}{r}I\end{array}$  in the formula.
  ::现在,彼得应该用22美元代替我的配方

  $$\begin{array}{r}22=5.696t\end{array}$$

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  Next, Peter should solve the equation for  $\begin{array}{r}t\end{array}$ . Notice that  $\begin{array}{r}t\end{array}$  is being multiplied by 5.696 on one side of the equation. He needs to divide both sides by 5.696 to isolate  $\begin{array}{r}t\end{array}$ .
  ::接下来,彼得应该解决 t. t. 的方程式。 请注意, t 在方程式的一边正乘以5. 696 。 他需要将两边除以 5. 696 来隔离 t 。

  $$\begin{array}{r}\frac{22}{5.696}=\frac{5.696t}{5.696}\end{array}$$

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  Now, Peter should simplify the right side of the equation.  $\begin{array}{r}\frac{5.696}{5.696}\end{array}$  is equal to 1, which leaves him with  $\begin{array}{r}t\end{array}$  on the right side of the equation.
  ::现在,彼得应该简化方程的右侧。 5.696 5.696等于1, 使他在方程的右侧留下 t。

  $$\begin{array}{r}\frac{22}{5.696}=t\end{array}$$

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  Next, Peter should simplify the left side of the equation. He should use what he has learned about decimal division. Move the decimal points 3 places to the right to change 5.696 into 5696 and 22 into 22000. Then, he can divide using long division.
  ::接下来, 彼得应该简化方程的左侧。 他应该使用他所学的小数分数。 将小数点三移到5. 696 改为 5696 和 22 改为 2000 。 然后, 他可以使用长分数进行分割 。

  $$\begin{array}{r}\begin{array}{rc}& \stackrel{\mathrm{3.8...}}{5696\overline{)22000.00}}\\ & \underset{\_}{-17088}\\ & 49120\\ & \underset{\_}{-45568}\\ & 35520\end{array}\end{array}$$

  
  Because Peter only wanted to know approximately how many years it will take, he can stop his division here.  $\begin{array}{r}t\end{array}$  is approximately equal to 3.8, which means it will take about 4 years.

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  The answer is it will take approximately 4 years for Peter to have $200 in his account.
  ::答案是,彼得要花大约4年的时间 才能有200美元入帐。

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  Example 2
  ::例2

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  Solve the following equation for  $\begin{array}{r}x\end{array}$ .
  ::解决 x 的以下方程式 。

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  $\begin{array}{r}4.5x=12.6\end{array}$
  ::4.5x=12.6

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  First, notice that  $\begin{array}{r}x\end{array}$  is being multiplied by 4.5 on one side of the equation. Divide both sides by 4.5 to isolate  $\begin{array}{r}x\end{array}$ .
  ::首先,请注意 x 乘以方程一面4.5。 将两面除以4.5 以孤立 x 。

  $$\begin{array}{r}\frac{4.5x}{4.5}=\frac{12.6}{4.5}\end{array}$$

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  Next, simplify the left side of the equation.  $\begin{array}{r}\frac{4.5}{4.5}\end{array}$  is equal to 1, which leaves you with  $\begin{array}{r}1x\end{array}$  or  $\begin{array}{r}x\end{array}$  on the left side of the equation.
  ::接下来,简化方程的左侧。 4.5 4.5 等于 1, 使您在方程的左侧留下 1 x 或 x 。

  $$\begin{array}{r}x=\frac{12.6}{4.5}\end{array}$$

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  Now, simplify the right side of the equation. Use what you have learned about decimal division. Move the decimal points 1 place to the right to change 4.5 into 45 and 12.6 into 126. Then, divide using long division.
  ::现在, 简化方程的右侧。 使用您所学到的关于小数分数的知识。 将小数点 1 移到将4.5 改为 45 和 12. 6 改为 126. 的权利上, 然后用长分数分隔 。

  $$\begin{array}{r}\begin{array}{rc}& \stackrel{2.8}{45\overline{)126.0}}\\ & \underset{\_}{-90}\\ & 360\\ & \underset{\_}{-360}\\ & 0\end{array}\end{array}$$

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  $\begin{array}{r}x=2.8\end{array}$
  ::x = 2.8

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  The answer is  $\begin{array}{r}x=2.8\end{array}$ .
  ::答案是x=2.8。

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  Now, check the answer.
  ::现在,检查答案。

  $$\begin{array}{r}\begin{array}{rcl}4.5x& =& 12.6\\ 4.5(2.8)& =& 12.6\\ 12.6& =& 12.6\end{array}\end{array}$$

   

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  The answer is correct.
  ::答案是正确的。

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  Example 3
  ::例3

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  Solve the following equation for  $\begin{array}{r}x\end{array}$ .
  ::解决 x 的以下方程式 。

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  $\begin{array}{r}2.3x=5.06\end{array}$
  ::2.3 x = 5.06

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  First, notice that  $\begin{array}{r}x\end{array}$  is being multiplied by 2.3 on one side of the equation. Divide both sides by 2.3 to isolate  $\begin{array}{r}x\end{array}$ .
  ::首先,请注意 x 乘以方程一面2.3。 将两面除以2.3 以孤立 x 。

  $$\begin{array}{r}\frac{2.3x}{2.3}=\frac{5.06}{2.3}\end{array}$$

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  Next, simplify the left side of the equation.  $\begin{array}{r}\frac{2.3}{2.3}\end{array}$  is equal to 1, which leaves you with  $\begin{array}{r}1x\end{array}$  or  $\begin{array}{r}x\end{array}$  on the left side of the equation.
  ::接下来,简化方程的左侧。 2.3 2.3 等于 1, 使您在方程的左侧留下 1 x 或 x 。

  $$\begin{array}{r}x=\frac{5.06}{2.3}\end{array}$$

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  Now, simplify the right side of the equation. Use what you have learned about decimal division. Move the decimal points 1 place to the right to change 2.3 into 23 and 5.06 into 50.6. Then, divide using long division.
  ::现在, 简化方程的右侧。 使用您所学到的关于小数分数的知识。 将小数点 1 移到右侧, 将2.3 改为 23 和 5. 06 改为 50.6 。 然后, 用长分数分隔 。

  $$\begin{array}{r}\begin{array}{rc}& \stackrel{2.2}{23\overline{)50.6}}\\ & \underset{\_}{-46}\\ & 46\\ & \underset{\_}{-46}\\ & 0\end{array}\end{array}$$

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  $\begin{array}{r}x=2.2\end{array}$
  ::x = 2.2

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  The answer is  $\begin{array}{r}x=2.2\end{array}$ .
  ::答案是x=2.2。

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  Now, check the answer.
  ::现在,检查答案。

  $$\begin{array}{r}\begin{array}{rcl}2.3x& =& 5.06\\ 2.3(2.2)& =& 5.06\\ 5.06& =& 5.06\end{array}\end{array}$$

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  The answer is correct.
  ::答案是正确的。

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  Example 4
  ::例4

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  Solve the following equation for  $\begin{array}{r}x\end{array}$ .
  ::解决 x 的以下方程式 。

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  $\begin{array}{r}1.6x=5.76\end{array}$  
  ::1.6x=5.76

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  First, notice that  $\begin{array}{r}x\end{array}$  is being multiplied by 1.6 on one side of the equation. Divide both sides by 1.6 to isolate  $\begin{array}{r}x\end{array}$ .
  ::首先,请注意 x 乘以1.6 方程的一面。 将两面除以1.6 以孤立 x 。

  $$\begin{array}{r}\frac{1.6x}{1.6}=\frac{5.76}{1.6}\end{array}$$

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  Next, simplify the left side of the equation.  $\begin{array}{r}\frac{1.6}{1.6}\end{array}$  is equal to 1, which leaves you with  $\begin{array}{r}1x\end{array}$  or  $\begin{array}{r}x\end{array}$  on the left side of the equation.
  ::接下来,简化方程的左侧。 1. 6 1. 6 等于 1, 使您在方程的左侧留下 1 x 或 x 。

  $$\begin{array}{r}x=\frac{5.76}{1.6}\end{array}$$

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  Now, simplify the right side of the equation. Use what you have learned about decimal division. Move the decimal points 1 place to the right to change 1.6 into 16 and 5.76 into 57.6. Then, divide using long division.
  ::现在, 简化方程的右侧。 使用您所学的有关小数分数的知识。 将小数点 1 移动到将1.6 改为 16 和 5. 76 改为 57. 6 的权利上。 然后, 用长分数分隔 。

  $$\begin{array}{r}\begin{array}{rc}& \stackrel{3.6}{16\overline{)57.6}}\\ & \underset{\_}{-48}\\ & 96\\ & \underset{\_}{-96}\\ & 0\end{array}\end{array}$$

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  $\begin{array}{r}x=3.6\end{array}$
  ::x = 3.6

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  The answer is  $\begin{array}{r}x=3.6\end{array}$ .
  ::答案是x=3.6。

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  Now, check the answer.
  ::现在,检查答案。

  $$\begin{array}{r}\begin{array}{rcl}1.6x& =& 5.76\\ 1.6(3.6)& =& 5.76\\ 5.76& =& 5.76\end{array}\end{array}$$

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  The answer is correct.
  ::答案是正确的。

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  Example 5
  ::例5

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  Solve the following equation for  $\begin{array}{r}x\end{array}$ .
  ::解决 x 的以下方程式 。

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  $\begin{array}{r}4.7x=10.81\end{array}$
  ::4.7 x = 10.81

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  First, notice that  $\begin{array}{r}x\end{array}$  is being multiplied by 4.7 on one side of the equation. Divide both sides by 4.7 to isolate  $\begin{array}{r}x\end{array}$ .
  ::首先,注意方程式一侧x乘以4.7。

  $$\begin{array}{r}\frac{4.7x}{4.7}=\frac{10.81}{4.7}\end{array}$$

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  Next, simplify the left side of the equation.  $\begin{array}{r}\frac{4.7}{4.7}\end{array}$  is equal to 1, which leaves you with  $\begin{array}{r}1x\end{array}$  or  $\begin{array}{r}x\end{array}$  on the left side of the equation.
  ::接下来,简化方程的左侧。 4.7 4.7 等于 1, 使您在方程的左侧留下 1 x 或 x 。

  $$\begin{array}{r}x=\frac{10.81}{4.7}\end{array}$$

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  Now, simplify the right side of the equation. Use what you have learned about decimal division. Move the decimal points 1 place to the right to change 4.7 into 47 and 10.81 into 108.1. Then, divide using long division.
  ::现在, 简化方程的右侧。 使用您所学到的关于小数分数的知识。 将小数点 1 移动到右侧, 将4. 7 改为 47 和 10. 81 改为108.1 。 然后, 用长分数分隔 。

  $$\begin{array}{r}\begin{array}{rc}& \stackrel{2.3}{47\overline{)108.1}}\\ & \underset{\_}{-94}\\ & 141\\ & \underset{\_}{-141}\\ & 0\end{array}\end{array}$$

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  $\begin{array}{r}x=2.3\end{array}$
  ::x=2.3 = 2.3

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  The answer is  $\begin{array}{r}x=2.3\end{array}$ .
  ::答案是x=2.3。

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  Now, check the answer.
  ::现在,检查答案。

  $$\begin{array}{r}\begin{array}{rcl}4.7x& =& 10.81\\ 4.7(2.3)& =& 10.81\\ 10.81& =& 10.81\end{array}\end{array}$$

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  The answer is correct.
  ::答案是正确的。

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  Review
  ::回顾

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  Solve the following equations for  $\begin{array}{r}x\end{array}$ .
  ::为 x 解决以下方程式。

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  1.   $\begin{array}{r}3.7x=7.77\end{array}$  
     ::3.7 x = 7.77
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  2. $\begin{array}{r}3.1x=10.23\end{array}$
     ::3.1xx=10.23
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  3. $\begin{array}{r}7.2x=29.52\end{array}$
     ::7.2 x = 29.52
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  4. $\begin{array}{r}2.7x=11.34\end{array}$
     ::2.7 x = 11.34
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  5. $\begin{array}{r}1.2x=6.72\end{array}$
     ::1.2 x = 6.72
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  6. $\begin{array}{r}11x=27.5\end{array}$
     ::11x=27.5
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  7. $\begin{array}{r}6.7x=42.21\end{array}$
     ::6.7 x = 42.21
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  8. $\begin{array}{r}8.2x=51.66\end{array}$
     ::8.2 x = 51.66
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  9. $\begin{array}{r}1.9x=12.92\end{array}$
     ::1.9 x = 12.92
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  10. $\begin{array}{r}5.7x=54.72\end{array}$
      ::5.7 x = 54.72
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  11. $\begin{array}{r}0.55x=.31955\end{array}$
      ::0.55 x = 31955
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  12. $\begin{array}{r}9.8x=114.66\end{array}$
      ::9.8 x = 114.66
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  13. In a week of track practice, Rose ran 3.12 times more than Jamie. If Rose ran 17.16 kilometers, how many kilometers did Jamie run? Write an equation and solve.
      ::在一周的赛道练习中,玫瑰跑的比Jamie跑的比Jamie多3.12倍。如果玫瑰跑的是17.16公里,Jamie跑的是多少公里?写一个方程和解答。
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  14. Ling’s flower bed has an area of  $\begin{array}{r}23.12m^2\end{array}$  and a width of 3.4 meters. What is the length of Ling’s flower bed? Write an equation and solve.
      ::玲的花床面积为23.12米2,宽度为3.4米。 玲的花床长度是多少? 写一个方程和解答。
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  15. A jet airplane travels 6.5 times faster than a car. If the jet travels at 627.51 kilometers per hour, how fast is the car? Write an equation and solve.
      ::喷气式飞机的飞行速度比汽车快6.5倍,如果喷气式飞机的飞行速度为每小时627.51公里,车速有多快?

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