8.6使用交叉积的比例

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  使用交叉积的比例

  

  [Figure 1]

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  Tony is training for a marathon. A marathon is about 26 miles. Last week, it took him 120 minutes to  run  13 miles. If Tony runs at the same  speed , how long will it take him to run a full marathon?
  ::托尼正在训练马拉松,马拉松大约26英里,上周他花了120分钟跑13英里。如果托尼以同样的速度跑,他要跑多久才能跑完马拉松?

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  In this concept, you will learn to use the cross product to identify proportions.
  ::在这个概念中,你将学会使用交叉产品来确定比例。

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  Using Cross Products to Identify Proportions
  ::使用交叉产品确定比例

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  A  proportion  is two equal ratios. Proportions can be written as equations. A  ratio  compares two quantities. One way to check if two ratios are equivalent is to compare the simplest form of each ratio.
  ::比例是两个等值比率。比例可以以等式形式写成。一个比例比较两个数量。检查两种比率是否相等的一种方法是比较每种比率的最简单的形式。

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  Compare the ratios 1:4 and 5:20. Write the ratios as fractions and simplify to compare.
  ::比较1:4和5:20的比率。将比率写成分数,并简化比较。

  $$\begin{array}{r}\frac{1}{4}\text{and}\frac{5}{20}\end{array}$$

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  $\begin{array}{r}\frac{1}{4}\end{array}$  is in simplest form.  $\begin{array}{r}\frac{5}{20}\end{array}$  can be simplified by dividing the numerator and denominator by the  greatest common factor , 5.
  ::1 4 以最简单的形式出现。 5 20 可以通过将分子和分母除以最大共同系数5 来简化。

  $$\begin{array}{r}\frac{1}{4}\text{and}\frac{5÷5}{20÷5}=\frac{1}{4}\end{array}$$

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  The ratios 1:4 and 5:20 are  equivalent ratios .
  ::1:4和5:20是等同比率。

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  Another way to check if two ratios are equivalent is to cross multiply and compare the cross products. A  cross product  is the result of multiplying the numerator of one ratio with the denominator of another. If the cross products are equal, then the ratios are proportional.
  ::另一种检查两种比率是否相等的方法是交叉乘数和比较交叉产品。交叉产品是将一个比率的分子与另一个比率的分母相乘的结果。如果交叉产品相等,则比例是成比例的。

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  For ratios  $\begin{array}{r}\frac{a}{b}\end{array}$  and  $\begin{array}{r}\frac{c}{d}\end{array}$ , if  $\begin{array}{r}ad=bc\end{array}$ , then  $\begin{array}{r}\frac{a}{b}\end{array}$  and  $\begin{array}{r}\frac{c}{d}\end{array}$  are proportional and can be written as  $\begin{array}{r}\frac{a}{b}=\frac{c}{d}\end{array}$ .
  ::对于比率a b 和 c d ,如果 d = b c,那么a b 和 c d 是比例的,可以写成 b = c d 。

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  Let’s use cross products to compare the ratios  $\begin{array}{r}\frac{2}{3}\end{array}$  and  $\begin{array}{r}\frac{4}{6}\end{array}$  to determine if they are proportional.
  ::让我们用交叉产品来比较比率2、3和4、6, 以确定它们是否成比例。

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  First, write an  equation  with the ratios.
  ::首先,用比率来写一个方程。

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

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  Next, cross multiply to find the cross products.
  ::接下来,交叉倍增以找到交叉产品。

  $$\begin{array}{r}2\times 6=3\times 4\end{array}$$

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  Then, simplify both sides of the equation by multiplying and check if they are equal.
  ::然后,通过乘法简化方程式的两边, 并检查它们是否相等 。

  $$\begin{array}{r}12=12\end{array}$$

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  Both sides are equal. The ratios  $\begin{array}{r}\frac{2}{3}\end{array}$  and  $\begin{array}{r}\frac{4}{6}\end{array}$  are proportional.
  ::双方是平等的,比例是2、3和4、6,比例是相称的。

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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 Tony training for the marathon.
  ::之前,你遇到一个 托尼在马拉松上训练的问题

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  Tony took 120 minutes to run 13 miles. Use cross products to find how long it will take Tony to run 26 miles if he continues at the same speed. (Note: Write the corresponding units in the same location. The minutes are in the numerator and miles are in the denominator.)
  ::Tony花了120分钟跑13英里。 使用交叉产品寻找 Tony如果继续以同样的速度跑26英里需要多长时间。 (注: 在同一地点写相应的单位。 分钟在分子中, 里程在分母中 )

  $$\begin{array}{r}\frac{\text{minutes}}{\text{miles}}=\frac{\text{minutes}}{\text{miles}}\end{array}$$

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  First, write an equation with the ratios. Use  $\begin{array}{r}x\end{array}$  to represent the unknown quantity.
  ::首先,根据比率写一个方程式。使用 x 表示未知的数量 。

  $$\begin{array}{r}\frac{120}{13}=\frac{x}{26}\end{array}$$

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  Next, cross multiply to find the cross products.
  ::接下来,交叉倍增以找到交叉产品。

  $$\begin{array}{r}120\times 26=13x\end{array}$$

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  Then, simplify by multiplying 120 and 26.
  ::然后简化 乘以120和26

  $$\begin{array}{r}3120=13x\end{array}$$

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  Divide both sides by 13 to find the unknown quantity.
  ::双方除以13 以找到未知的数量

  $$\begin{array}{r}\begin{array}{rcl}\frac{3120}{13}& =& \frac{13x}{13}\\ 240& =& x\end{array}\end{array}$$

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  It will take Tony about 240 minutes or 4 hours to run the marathon.
  ::托尼要花240分钟或4个小时跑马拉松

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  Example 2
  ::例2

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  Use the cross products to determine if the ratios  $\begin{array}{r}\frac{6}{9}\end{array}$  and  $\begin{array}{r}\frac{3}{4.5}\end{array}$  are proportional.
  ::利用交叉产品确定比率6 9和3 4.5是否相称。

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  First, write an equation with the ratios.
  ::首先,用比率来写一个方程。

  $$\begin{array}{r}\frac{6}{9}=\frac{3}{4.5}\end{array}$$

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  Next, cross multiply to find the cross products.
  ::接下来,交叉倍增以找到交叉产品。

  $$\begin{array}{r}6\times 4.5=9\times 3\end{array}$$

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  Then, simplify both sides of the equation by multiplying and check if they are equal.
  ::然后,通过乘法简化方程式的两边, 并检查它们是否相等 。

  $$\begin{array}{r}27=27\end{array}$$

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  Both sides are equal. The ratios  $\begin{array}{r}\frac{6}{9}\end{array}$  and  $\begin{array}{r}\frac{3}{4.5}\end{array}$  are proportional.
  ::双方平等,比例为6-9和3-4.5。

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  Example 3
  ::例3

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  Use the cross products to determine if the ratios  $\begin{array}{r}\frac{2}{5}\text{and}\frac{5}{9}\end{array}$  are proportional.
  ::使用交叉产品来确定比率2 5和5 9是否相称。

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  First, write an equation with the ratios.
  ::首先,用比率来写一个方程。

  $$\begin{array}{r}\frac{2}{5}=\frac{5}{9}\end{array}$$

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  Next, cross multiply to find the cross products.
  ::接下来,交叉倍增以找到交叉产品。

  $$\begin{array}{r}2\times 9=5\times 5\end{array}$$

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  Then, simplify both sides of the equation by multiplying and check if they are equal.
  ::然后,通过乘法简化方程式的两边, 并检查它们是否相等 。

  $$\begin{array}{r}18\ne 25\end{array}$$

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  18 is  not equal to  25.  $\begin{array}{r}\frac{2}{5}\end{array}$  and  $\begin{array}{r}\frac{5}{9}\end{array}$  are not proportional.
  ::18不等于25,2,5,5,9不相称。

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  Example 4
  ::例4

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  Use the cross products to determine if the ratios  $\begin{array}{r}\frac{3}{6}\text{and}\frac{5}{10}\end{array}$  are proportional.
  ::使用交叉产品来确定比例3 6和5 10是否相称。

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  First, write an equation with the ratios.
  ::首先,用比率来写一个方程。

  $$\begin{array}{r}\frac{3}{6}=\frac{5}{10}\end{array}$$

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  Next, cross multiply to find the cross products.
  ::接下来,交叉倍增以找到交叉产品。

  $$\begin{array}{r}3\times 10=6\times 5\end{array}$$

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  Then, simplify both sides of the equation by multiplying and check if they are equal.
  ::然后,通过乘法简化方程式的两边, 并检查它们是否相等 。

  $$\begin{array}{r}30=30\end{array}$$

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  Both sides are equal.  $\begin{array}{r}\frac{3}{6}\end{array}$  and  $\begin{array}{r}\frac{5}{10}\end{array}$  are proportional.
  ::双方平等。 3,6和5,10是比例的。

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  Example 5
  ::例5

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  Use the cross products to determine if the ratios  $\begin{array}{r}\frac{4}{7}\text{and}\frac{12}{28}\end{array}$  are proportional.
  ::利用交叉产品确定比率4 7和12 28是否相称。

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  First, write an equation with the ratios.
  ::首先,用比率来写一个方程。

  $$\begin{array}{r}\frac{4}{7}=\frac{12}{28}\end{array}$$

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  Next, cross multiply to find the cross products.
  ::接下来,交叉倍增以找到交叉产品。

  $$\begin{array}{r}4\times 28=7\times 12\end{array}$$

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  Then, simplify both sides of the equation by multiplying and check if they are equal.
  ::然后,通过乘法简化方程式的两边, 并检查它们是否相等 。

  $$\begin{array}{r}112\ne 84\end{array}$$

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  112 is not equal to 84.  $\begin{array}{r}\frac{4}{7}\end{array}$  and  $\begin{array}{r}\frac{12}{28}\end{array}$  are not proportional.
  ::112不等于84,4,7,12,28不相称。

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  Review
  ::回顾

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  Use the cross products to determine if the following ratios are proportional.
  ::利用交叉产品确定以下比例是否相称。

  1. $\begin{array}{r}\frac{1}{2}=\frac{6}{12}\end{array}$
  2. $\begin{array}{r}\frac{1}{3}=\frac{4}{12}\end{array}$
  3. $\begin{array}{r}\frac{1}{4}=\frac{3}{15}\end{array}$
  4. $\begin{array}{r}\frac{5}{6}=\frac{10}{12}\end{array}$
  5. $\begin{array}{r}\frac{3}{4}=\frac{6}{10}\end{array}$
  6. $\begin{array}{r}\frac{2}{5}=\frac{6}{15}\end{array}$
  7. $\begin{array}{r}\frac{2}{7}=\frac{4}{21}\end{array}$
  8. $\begin{array}{r}\frac{4}{7}=\frac{12}{21}\end{array}$  
  9. $\begin{array}{r}\frac{7}{8}=\frac{14}{16}\end{array}$
  10. $\begin{array}{r}\frac{25}{75}=\frac{1}{3}\end{array}$
  11. $\begin{array}{r}\frac{11}{33}=\frac{1}{3}\end{array}$
  12. $\begin{array}{r}\frac{15}{33}=\frac{2}{3}\end{array}$
  13. $\begin{array}{r}\frac{18}{30}=\frac{36}{60}\end{array}$
  14. $\begin{array}{r}\frac{1}{3}=\frac{6}{12}\end{array}$
  15. $\begin{array}{r}\frac{85}{100}=\frac{43.5}{50}\end{array}$

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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