7.4三分数积

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  三分数积

  

  [Figure 1]

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  Roberto is working on his math homework. He needs to find the volume of a rectangular prism. The formula is for the volume of a rectangular prism is:
  ::Roberto正在做数学作业。 他需要找到长方形棱柱的体积。 公式是长方形棱柱的体积 :

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  $\begin{array}{r}V=\frac{3}{5}\times \frac{1}{4}\times \frac{5}{6}\end{array}$
  ::V = 3 5 5 × 1 4 × 5 6

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  The given dimensions are length =  $\begin{array}{r}\frac{3}{5}\end{array}$ , width =  $\begin{array}{r}\frac{1}{4}\end{array}$ , and height  $\begin{array}{r}\frac{5}{6}\end{array}$ . The units are in inches. What is the volume of the  prism?
  ::给定的尺寸是长度 = 3 5,宽 = 14,身高 5 6。单位在英寸内。棱镜的体积是多少?

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  In this concept, you will learn how to multiply three fractions.
  ::在这个概念中,你会学会如何乘以三个分数。

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  Multiplying Three Fractions
  ::乘法三分数

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  Multiplying three fractions only sounds more complicated than multiplying two fractions. The procedure is the same. Multiply the numerators and the denominators to find the product. But before multiplying,  simplify  the fractions if you can. This way, you won’t end up with a large fraction that is challenging to simplify at the end.
  ::乘法三分数只听起来比乘法两分数要复杂。 程序是一样的。 乘法乘法乘法乘法乘法和分母以寻找产品。 但在乘法之前,如果可以的话,请简化分数。 这样,你就不会有最后难以简化的大分数。

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  There are two ways to simplify before multiplying:
  ::在乘法之前有两种简化方式:

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  1. Simplify any whole fraction. 
     ::简化任何整分数 。

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  Example:  $\begin{array}{r}\frac{3}{6}=\frac{1}{2}\\ \frac{3}{6}\times \frac{1}{9}& =\frac{1}{2}\times \frac{1}{9}\end{array}$  
  ::示例: 3 6 6 = 1 2 3 6 × 1 9 = 1 2 × 1 9

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  2. Cross simplify using the   (GCF).
     ::使用(GCF)进行交叉简化。

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  Example: The GCF of 3 and 9 is 3.  $\begin{array}{r}\frac{\cancel{3}}{6}\times \frac{1}{\cancel{9}}=\frac{1}{6}\times \frac{1}{3}\end{array}$
  ::例如:全球合作框架3和9是3、3、6x1、9=1、6x1、3

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

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  Multiply the following fractions. 
  ::乘以下列分数。

  $$\begin{array}{r}\frac{1}{4}\times \frac{2}{6}\times \frac{4}{5}=\underset{\_}{}\end{array}$$

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  First, simplify the fractions. Check to see if any of the individual fractions can be simplified. The fraction  $\begin{array}{r}\frac{2}{6}\end{array}$  can be simplified.  
  ::首先,简化分数。检查是否可以简化单个分数。分数 2 6 可以简化。

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

  $$\begin{array}{r}\frac{1}{4}\times \frac{1}{3}\times \frac{4}{5}\end{array}$$

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  Also check the diagonals. The two 4s can be cross simplified with the GCF of 4. Each one simplifies to 1.
  ::检查对角线。两个4可以与全球合作框架的4相互简化,每个简化为1。

  $$\begin{array}{r}\frac{1}{{\cancel{4}}^1}\times \frac{1}{3}\times \frac{{\cancel{4}}^1}{5}=\frac{1}{1}\times \frac{1}{3}\times \frac{1}{5}\end{array}$$

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  Then, multiply the fractions. Find the product of the numerators over the product of the denominators.
  ::然后,乘以分数。在分母的产物上找到分子的产物。

  $$\begin{array}{r}\frac{1\times 1\times 1}{1\times 3\times 5}=\frac{1}{15}\end{array}$$

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  The final answer is  $\begin{array}{r}\frac{1}{15}\end{array}$ .
  ::最后一个答案是11点15分

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  Here is another example.
  ::下面是另一个例子。

  $$\begin{array}{r}\frac{5}{9}\times \frac{7}{14}\times \frac{3}{5}=\underset{\_}{}\end{array}$$

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  First, simplify the fraction. The 5s simplify with the GCF of 5. The 3 and 9 also simplify with the GCF of 3.
  ::第一,简化分数,5个简化,5个简化,5个简化,5个简化,3个简化,9个简化,3个简化,3个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,3个简化,9个简化,3个简化,9个简化,3个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,5个简化,3个全球合作框架,3个简化,9个简化,3个简化,3个简化,9个简化,3个简化。

  $$\begin{array}{r}\frac{{}^1\cancel{5}}{{}^3\cancel{9}}\times \frac{7}{14}\times \frac{{}^1\cancel{3}}{{}^1\cancel{5}}=\frac{1}{3}\times \frac{7}{14}\times \frac{1}{1}\end{array}$$

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  The fraction  $\begin{array}{r}\frac{7}{14}\end{array}$  can also be simplified.
  ::分数7-14也可以简化。

  $$\begin{array}{rl}\frac{1}{3}\times \frac{{\cancel{7}}^1}{{\cancel{14}}^2}\times \frac{1}{1}& =\frac{1}{3}\times \frac{1}{2}\times \frac{1}{1}\end{array}$$

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  Then, multiply the fractions. Find the product of the numerators over the product of the denominators.
  ::然后,乘以分数。在分母的产物上找到分子的产物。

   

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

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  The final answer is  $\begin{array}{r}\frac{1}{6}\end{array}$ .
  ::最后一个答案是16。

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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 Roberto's math question.
  ::之前,你对罗伯托的数学问题有疑问

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  Roberto needs to find the volume of a rectangular prism with the dimensions of length =  $\begin{array}{r}\frac{3}{5}\end{array}$ , width =  $\begin{array}{r}\frac{1}{4}\end{array}$ , and height =  $\begin{array}{r}\frac{5}{6}\end{array}$ . Use the formula for the volume of a rectangular prism to find the volume.
  ::Roberto需要找到长度 = 3 5,宽 = 1 4,高度 = 5 6的矩形棱晶体体积。使用矩形棱晶体体积的公式来找到体积。

  $$\begin{array}{r}V=l\times w\times h\end{array}$$

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  First, substitute the known values.
  ::首先,取代已知价值。

  $$\begin{array}{r}V=\frac{3}{5}\times \frac{1}{4}\times \frac{5}{6}\end{array}$$

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  Then, simplify the fractions. 
  ::然后,简化分数。

  $$\begin{array}{r}V=\frac{{\cancel{3}}^1}{{\cancel{5}}^1}\times \frac{1}{4}\times \frac{{\cancel{5}}^1}{{\cancel{6}}^2}=\frac{1}{1}\times \frac{1}{4}\times \frac{1}{2}\end{array}$$

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  Next, multiply the fractions. Remember to include the units. 
  ::下一步, 乘以分数。 记住要包含单位 。

  $$\begin{array}{r}V=\frac{1\times 1\times 1}{1\times 4\times 2}=\frac{1}{8}{\text{inches}}^3\end{array}$$

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  The volume of the rectangular prism is  $\begin{array}{r}\frac{1}{8}{\text{inches}}^3\end{array}$ .
  ::矩形棱柱的体积是18英寸3英寸

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  Example 2
  ::例2

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  Multiply the following fractions. Answer in  simplest form .
  ::乘以下列分数。回答以最简单的形式。

  $$\begin{array}{r}\frac{5}{6}\times \frac{2}{4}\times \frac{6}{9}\end{array}$$

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  First, simplify the fractions. Simplify the second and third fractions before multiplying.
  ::首先,简化分数。在乘法前简化第二和第三分数。

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

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

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  Next, rewrite the problem.
  ::下一个,重写问题。

  $$\begin{array}{r}\frac{5}{6}\times \frac{1}{2}\times \frac{2}{3}\end{array}$$

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  Also, simplify the twos.
  ::另外,简化两个。

  $$\begin{array}{r}\frac{5}{6}\times \frac{1}{{\cancel{2}}^1}\times \frac{{\cancel{2}}^1}{3}=\frac{5}{6}\times \frac{1}{1}\times \frac{1}{3}\end{array}$$

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  Then, multiply the fractions.
  ::然后,乘以分数。

   

  $$\begin{array}{r}\frac{5\times 1\times 1}{6\times 1\times 3}=\frac{5}{18}\end{array}$$

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  The product is  $\begin{array}{r}\frac{5}{18}\end{array}$ .
  ::产品为5 18。

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  Example 3
  ::例3

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  Multiply the following fractions:  $\begin{array}{r}\frac{1}{5}\times \frac{5}{6}\times \frac{1}{2}=\underset{\_}{}\end{array}$ . Answer in simplest form.
  ::乘以下列分数: 1 5 × 5 6 × 1 2 = _. 回答最简单。

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  First, simplify the fractions. 
  ::首先,简化分数。

  $$\begin{array}{r}\frac{1}{{\cancel{5}}^1}\times \frac{{\cancel{5}}^1}{6}\times \frac{1}{2}=\frac{1}{1}\times \frac{1}{6}\times \frac{1}{2}\end{array}$$

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  Then, multiply.
  ::然后,乘数。

   

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

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  The product is  $\begin{array}{r}\frac{1}{12}\end{array}$ .
  ::产品是1112。

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  Example 4
  ::例4

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  Multiply the following fractions:  $\begin{array}{r}\frac{8}{9}\times \frac{3}{4}\times \frac{1}{3}=\underset{\_}{}\end{array}$ . Answer in simplest form.
  ::乘以下列分数: 8 9 × 3 4 × 1 3 = _. 回答最简单。

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  First, simplify the fractions. 
  ::首先,简化分数。

  $$\begin{array}{r}\frac{{\cancel{8}}^2}{9}\times \frac{{\cancel{3}}^1}{{\cancel{4}}^1}\times \frac{1}{{\cancel{3}}^1}=\frac{2}{9}\times \frac{1}{1}\times \frac{1}{1}\end{array}$$

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  Then, multiply. Note that  $\begin{array}{r}\frac{1}{1}\end{array}$  is also equal to 1.  
  ::那么,乘数。请注意, 1 1 也等于 1 。

  $$\begin{array}{r}\frac{2}{9}\times 1\times 1=\frac{2}{9}\end{array}$$

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  The product is  $\begin{array}{r}\frac{2}{9}\end{array}$ .
  ::产品为2 9。

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  Example 5
  ::例5

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  Multiply the following fractions:  $\begin{array}{r}\frac{6}{7}\times \frac{7}{14}\times \frac{2}{10}=\underset{\_}{}\end{array}$ . Answer in simplest form.
  ::乘以下列分数: 6 7× 7 14× 2 10 = _。 回答最简单。

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  First, simplify the fractions. 
  ::首先,简化分数。

  $$\begin{array}{r}\frac{{\cancel{6}}^3}{{\cancel{7}}^1}\times \frac{{\cancel{7}}^1}{{\cancel{14}}^7}\times \frac{{\cancel{2}}^1}{{\cancel{10}}^5}=\frac{3}{1}\times \frac{1}{7}\times \frac{1}{5}\end{array}$$

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  Then, multiply.
  ::然后,乘数。

  $$\begin{array}{r}\frac{3\times 1\times 1}{1\times 7\times 5}=\frac{3}{35}\end{array}$$

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  The product is  $\begin{array}{r}\frac{3}{35}\end{array}$ .
  
  ::产品是335

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  Review
  ::回顾

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  Multiply the following fractions. Answer in simplest form.
  ::乘以下列分数。回答以最简单的形式。

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

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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

   

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