1.8 采用分数操作

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  使用分数操作

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  A construction worker is cutting pipe. He originally cuts a pipe that is $\begin{align*}3 \frac{3}{4}\end{align*}$ feet long. The pipe is $\begin{align*}1 \frac{5}{6}\end{align*}$ feet too long. How long will the pipe be after the construction worker cuts the pipe to the right size?
  ::管道是156英尺长。管道在建筑工人把管道切到正确尺寸之后还要多久?

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  We will cover how to operate with fractions in this section.
  ::我们将在本节中介绍如何以分数进行操作。

  

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  Proper and Improper Fractions
  
  ::适当和不适当分分数

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  In this section, we consider fractions of the form $\begin{align*}\frac{p}{q}\end{align*}$ where p and q are integers. We have two categories of fractions highlighted in the box below.
  
  ::在本节中,我们考虑p和q为整数的pq形式的分数。下面的框中突出显示了两类分数。

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   Proper and Improper Fractions
  ::适当和不适当分分数

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  If $\begin{align*}p < q\end{align*}$ or $\begin{align*}\frac{p}{q}<1\end{align*}$ , then the fraction is called a proper fraction . 
  ::如果 p<q 或 pq < 1, 则该分数称为适当的分数 。

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  If $\begin{align*}p>q\end{align*}$ or $\begin{align*}\frac{p}{q}>1\end{align*}$ , then the fraction is called an improper fraction . 
  ::如果 p>q 或 pq>1, 则该分数被称为不适当的分数 。

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   Where to Put the Negative Sign for a Fraction
  ::将一个分数的负号放在哪里

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  We tend to put the negative sign in front of the fraction, like $\begin{align*}-\frac{p}{q}\end{align*}$ , or in the numerator , like  $\begin{align*}\frac{-p}{q}\end{align*}$ .
  ::我们倾向于在分数前面放负符号,比如-pq,或者在分子中,比如-pq。

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  If after computing the denominator is negative, you can rewrite the fraction in either of the forms above.  
  ::如果计算分母为负值,您可以重写上面两种表格中的分数。

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  Mixed Numbers
  
  ::混合数字

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  Improper fractions can be written as a whole number part and a fractional part that is less than 1.
  
  ::不正当的分数可以写成整个数字部分和小于1的分数部分。

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   Converting Between Improper Fractions and Mixed Numbers
  ::不正确分数和混合数字之间的转换

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  • To convert improper fractions into mixed numbers: divide . The quotient is the whole number part and the remainder is the numerator of the fractional part.
    For example, $$\begin{align*}\frac{17}{5} = 3 \frac{2}{5} \\ \\ \quad 3 R 2\\ 5 |\overline{17}\quad\end{align*}$$
    ::将不适当的分数转换成混合数字: 分隔。 商数是整数部分, 其余是分数部分的分子。 例如, 175=325R = 17
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  • To convert mixed numbers into fractions: multiply the denominator by the whole number part and add the numerator of the fractional part. 
    For example, $$\begin{align*}3\frac{2}{5} = \frac{5 \cdot 3 + 2}{5} = \frac{17}{5}\end{align*}$$  
    ::要将混合数字转换成分数:将分数乘以整数部分,并添加分数部分的分子。例如,325=53+25=175

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  Multiplying and Dividing Fractions
  ::乘数和分数分数

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  To multiply fractions, you multiply the numerators together and the denominators together.
  
  ::要乘以分数,您将乘以分子和分母。

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  Example 1
  ::例1

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  Find the following product:  $\begin{align*}\frac{1}{2} \cdot \frac{4}{5}\end{align*}$ .
  ::寻找以下产品:12-45。

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  Solution: Multiplying numerators with numerators and denominators with denominators, we have
  
  ::解 解 : 具有分子和分母分母的乘数分子和分母的乘数分子

  $$\begin{align*}\frac{1}{2} \cdot \frac{4}{5}=\frac{1 \times 4}{2 \times 5}=\frac{4}{10}\end{align*}$$

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  However, $\begin{align*}\frac{4}{10}\end{align*}$ is not simplified, that is, there is a common factor in the numerator and the denominator. We divide the numerator and the denominator by two.
  ::然而,410项没有简化,也就是说,分子和分母中有一个共同的系数。我们把分子和分母除以2。

  $$\begin{align*}\frac{4}{10}=\frac{4 \div 2}{10 \div 2}=\frac{2}{5}\end{align*}$$

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  Our final answer is $\begin{align*}\frac{2}{5}\end{align*}$ .
  ::我们的最后一个答案是25。

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  When we multiply a whole number and a fraction, we can convert the whole number into a fraction with a denominator of 1.
  ::当我们乘以一个整数和一个分数时,我们可以将整数转换成一个分数,分母为1。

  $$\begin{align*}5 \cdot \frac{1}{2}= \frac{5}{1} \cdot \frac{1}{2}=\frac{5}{2}=2 \frac{1}{2}\end{align*}$$

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  Example 2
  ::例2

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  Find $\begin{align*}3 \frac{1}{2} \cdot 2 \frac{1}{3}\end{align*}$ .
  ::找寻312-1213。

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  Solution: First, we convert each to an improper fraction.
  ::解决办法:首先,我们把每一个转换成不适当的部分。

  $$\begin{align*}3 \frac{1}{2} &= \frac{2 \cdot 3 +1}{2}= \frac{7}{2}\\ \\ 2 \frac{1}{3} &= \frac{3 \cdot 2 +1}{3} =\frac{7}{3}\end{align*}$$

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  Next, we multiply the two improper fractions.
  ::接下来,我们乘以两个不适当的部分。

  $$\begin{align*}\frac{7}{2} \cdot \frac{7}{3}=\frac{49}{6}\end{align*}$$

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  Lastly, we can convert this improper fraction to a mixed number by dividing. The numerator of the fractional part is the remainder.
  ::最后,我们可以通过除法将这一不适当的部分转换为混合数。分部分的分子是剩余部分。

  $$\begin{align*}\frac{49}{6}=8 \frac{1}{6}\end{align*}$$

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  by Mathispower4u demonstrates how to find the product and quotient of mixed numbers.
  ::Mathispower4u 展示了如何找到混合数字的产品和商数。

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  It is often helpful to simplify before multiplying. Look for any common factors between any of the numerators and the denominators to cancel.
  ::在乘法之前进行简化往往是有益的。 寻找任何一个分子和要取消的分母之间的任何共同因素 。

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  Example 3
  ::例3

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  Find $\begin{align*}\frac{2}{9} \cdot \frac{18}{30}\end{align*}$ .
  ::找到291830号

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  Solution: There are common factors between the numbers in the numerator and the numbers in the denominator. Two and 30 share a common factor of 2. We can divide both of them by 2 to simplify. Nine and 18 also have a common factor of 9. We can divide both of them by 9 to simplify.
  ::解决办法:分子数数和分母数之间有共同的因素。2和30是共同的2个因素。2和30是共同的2个因素。我们可以将两者除以2以简化。9和18也是共同的9个因素。我们可以将两者除以9以简化。

  $$\begin{align*}\frac{2}{9} \cdot \frac{18}{30}=\frac{\cancel{2}^1}{\cancel{9}^1} \cdot \frac{\cancel{18}^2}{\cancel{30}^{15}} = \frac{1}{1} \cdot \frac{2}{15} = \frac{2}{15}\end{align*}$$

   

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  To , we multiply by the reciprocal .
  ::我们乘以互惠

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  Example 4
  ::例4

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  Find $\begin{align*}\frac{6}{8} \div \frac{1}{2}\end{align*}$ .
  ::找68号 12号

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  In this example, one-half is the divisor . We need to flip the divisor so that we multiply by the reciprocal. As a multiplication problem, we have
  ::在这个例子中,一半是二分之一是二分之一。我们需要翻转二分之一,这样我们就能乘以对等乘法。作为一个乘法问题,我们有

  $$\begin{align*}\frac{6}{8} \cdot \frac{2}{1}\end{align*}$$

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  Next, we multiply across and since the numerator is larger than the denominator, we can convert to a mixed number.
  ::接下来,我们乘以一个倍数, 因为分子大于分母, 我们可以转换成混合数。

  $$\begin{align*}\frac{12}{8}=1 \frac{4}{8}\end{align*}$$

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  However, the fractional part is not simplified. There is a common factor of 2. Our last step is to simplify the fractional part of the mixed number. Our answer is $\begin{align*}1 \frac{1}{2}\end{align*}$ .
  ::然而,部分部分没有简化,共同因素是2。我们的最后一步是简化混合数字的分部分。我们的答复是112。

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  Example 5
  ::例5

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  Find $\begin{align*}4 \frac{1}{3} \div 2 \frac{1}{6}\end{align*}$ .
  ::找413216

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  Solution: First, we convert both of these mixed numbers to improper fractions. Let’s rewrite the problem with these numbers.
  ::解决方案:首先,我们把这些混合数字转换为不适当的分数。 让我们用这些数字重写问题。

  $$\begin{align*}4 &\frac{1}{3}=\frac{3 \cdot 4 + 1}{3}=\frac{13}{3}\\ \\ 2 &\frac{1}{6}=\frac{6 \cdot 2 +1}{6}=\frac{13}{6} \\ \\ 4 &\frac{1}{3} \div 2 \frac{1}{6} =\frac{13}{3} \div \frac{13}{6}\end{align*}$$

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  Now we change this to a multiplication problem by multiplying by the reciprocal.
  ::现在我们把它变成一个乘以乘以对等的乘法的乘数问题。

  $$\begin{align*}\frac{13}{3} \cdot \frac{6}{13}\end{align*}$$

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  Next, we can simplify by canceling out the 13's and the common factor of 3 between 3 and 6.
  ::其次,我们可以简化,取消13岁和3岁和6岁之间的共同系数3。

  $$\begin{align*}\frac{\cancel{13}^1}{\cancel{3}^1} \cdot \frac{\cancel{6}^2}{\cancel{13}^1} = \frac{1}{1} \cdot \frac{2}{1} = 2\end{align*}$$

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  by Mathispower4u demonstrates how to divide fractions and explains what is happening with division involving fractions. 
  ::由 Mathispower4u 演示如何分割分数并解释分数的分数。

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  Adding and Subtracting Fractions
  ::添加和减减分数

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  To add or subtract fractions, we need to first express the fractions with a common denominator . The common denominator allows us to add or subtract pieces that are equal in size. We cannot add fractions that are different in size. For example, $\begin{align*}\frac{1}{2} + \frac{1}{3} \ne \frac{2}{5}\end{align*}$ or less than one-half of say a cake as you can see in the image below.
  
  ::要添加或减减分数, 我们需要首先表达带有共同分母的分数。 共同分母允许我们增加或减大小相等的分数。 我们不能增加大小不同的分数。 例如, 12+13+25 或小于一半的蛋糕, 您可以在下面的图像中看到 。

   

  

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  Instead, we need to first create , fractions that are equal to the original fraction but represented with different sized pieces. For example, $\begin{align*}\frac{1}{2}\end{align*}$  and $\begin{align*}\frac{2}{4}\end{align*}$  are equivalent fractions. They are equal but for the former the whole is divided into two pieces and for the latter the whole is divided into four pieces. Creating equivalent fractions will allow us to combine fractions of the same size.  
  ::相反,我们首先需要创建一个分数,这些分数等于原始分数,但以不同大小的分数表示。例如,12和24是等效的分数。它们相等,但前者则全部分为两块,后者则全部分为四块。创建等值的分数将允许我们混合相同大小的分数。

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  To create equivalent fractions, we f irst determine the least common multiple of the denominators in our original fractions. This is called the least common denominator . In this example,  the least common denominator is 6. Now, ask yourself what do I need to multiply the original denominator by to obtain the least common denominator. Then, multiply both the denominator and numerator by that number. Remember that any number divided by itself is equal to one and one is the multiplicative identity element . 
  ::为了创建等值分数, 我们首先确定我们原始分数中的分母中最小常见的多个。 这被称为最小的分母。 在此示例中, 最小的分母是 6 。 现在, 请问您自己, 要将原始分母乘以最小的分母, 我还需要什么来乘以最小的分母 。 然后, 将分母和分子乘以这个数字。 记住, 任何数字本身除以等于一个, 一个是多倍化的特性元素 。

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  $$\begin{align*}&\frac{1}{2} \cdot \frac{3}{3} = \frac{3}{6} && \qquad \qquad {\color{gray}2 \cdot 3 = 6 \ \ \text{and} \ \ \frac{3}{3}=1} \\ \\ &\frac{1}{3} \cdot \frac{2}{2} = \frac{2}{6} && \qquad \qquad {\color{gray}3 \cdot 2 = 6 \ \ \text{and} \ \ \frac{2}{2}=1} \\\\\end{align*}$$
  ::1233=3623=6和33=11322=2632=6和22=1

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  Now that we are looking at the same division of the cake, that is, into 6 pieces, we can combine our fractions to determine how much cake we have. 
  ::现在我们看到的是蛋糕的同一部分, 也就是6个部分, 我们可以把分数结合起来, 来决定我们有多少蛋糕。

  $$\begin{align*}\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\end{align*}$$
  

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  Let's review our steps.
  
  ::让我们回顾一下我们的步骤。

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    To Add or Subtract Fractions
  ::添加或减减分数

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  1. Find the least common denominator.
  ::1. 找出最小的共同标准。

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  2. Create equivalent fractions for the fractions that do not have the least common denominator as their denominators.
  ::2. 为分母不具有最小共同分母的分母创建等值分数分数。

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  3. Replace the original fractions in the problem with their new equivalent forms.
  ::3. 将问题中的原有分数改为新的等同形式。

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  4. Add or subtract the numerators.
  ::4. 增加或减去分子。

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  5. Simplify the fraction by dividing the numerator and denominator by any common factors.
  ::5. 通过将分子和分母除以任何共同因素来简化分数。

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  Example 6
  
  ::例6

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  Subtract $\begin{align*}\frac{2}{3}-\frac{2}{9}\end{align*}$ .
  ::减号23-29。

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  Solution: First, find a common denominator. You are looking for a number that is a multiple of both 3 and 9. 9 is the least common multiple of 3 and 9.
  ::解决方案: 首先, 找到一个共同的分母。 您正在寻找一个数字, 这个数字是 3 和 9 的倍数, 9 9 是 3 和 9 中最不常见的 3 和 9 的 倍数 。

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  Now, rewrite each fraction as an equivalent fraction with a denominator of 9. $\begin{align*}\frac{2}{9}\end{align*}$  already has a denominator of 9 so it will not change.
  ::现在,将每一分数重写为等效分数,分母为9.29,分母为9,所以不会改变。

  $$\begin{align*} \frac{2}{3} &= \frac{2}{3} \cdot \frac{3}{3}=\frac{6}{9} \\ \\ \frac{2}{9} &= \frac{2}{9} \end{align*}$$

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  Next, subtract the equivalent fractions. You will subtract the numerators and keep the denominator the same.
  ::下一步,减去相等的分数。您将减去分子数,并将分母保持原样。

  $$\begin{align*}\frac{6}{9}-\frac{2}{9}=\frac{4}{9}\end{align*}$$

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  Finally, check to make sure your answer is in simplest form . $\begin{align*}\frac{4}{9}\end{align*}$  is in simplest form because 4 and 9 do not have any common factors besides 1.
  ::最后,检查以确保你的答复最简单。 49 最简单,因为4和9除了1之外没有任何共同因素。

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  The answer is $\begin{align*} \frac{2}{3}-\frac{2}{9}=\frac{4}{9}\end{align*}$ .
  ::答案是23-29=49。

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  by Mathispower4u demonstrates how to add and subtract fractions.
  ::Mathispower4u 演示如何增减分数。

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  Example  7
  ::例7

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  A construction worker is cutting pipe. He originally cuts a pipe that is $\begin{align*}3 \frac{3}{4}\end{align*}$ feet long. The pipe is $\begin{align*}1 \frac{5}{6}\end{align*}$ feet too long. How long will the pipe be after the construction worker cuts the pipe to the right size?
  ::管道是156英尺长。管道在建筑工人把管道切到正确尺寸之后还要多久?

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  Solution: We can approach this problem different ways. In this solution, we will convert the mixed numbers to improper fractions and then follow the steps above.
  
  ::解决方案:我们可以以不同的方式解决这个问题。在这个解决方案中,我们将把混合数字转换成不适当的部分,然后采取上述步骤。

  $$\begin{align*}3\frac{3}{4} = \frac{4 \cdot 3 + 3}{4} = \frac{15}{4} \\ \\ 1\frac{5}{6} = \frac{6 \cdot 1 +5}{6} = \frac{11}{6}\end{align*}$$

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  Now, that we have improper fractions to subtract them we need a common denominator.
  
  ::现在,我们有不适当的分数来减,我们需要一个共同的分母。

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  $$\begin{align*}4 = 2 \cdot 2 \\ && \text{LCD} = 2 \cdot 2 \cdot 3 = 12\\ 6 = 2 \cdot 3\\\end{align*}$$  
  
  ::4=22LCD=223=126=23

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  With the common denominator, we can create equivalent fractions.
  
  ::有了共同的分母,我们就能创造出等量的分数。

  $$\begin{align*}\frac{15}{4} \cdot \frac{3}{3} = \frac{45}{12} \\ \\ \frac{11}{6} \cdot \frac{2}{2} = \frac{22}{12} \\\end{align*}$$
  

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  Last, we perform the operation.
  
  ::最后,我们执行行动

  $$\begin{align*}3\frac{3}{4} - 1\frac{5}{6} = \frac{15}{4} - \frac{11}{6} &= \frac{45}{12} - \frac{22}{12} = \frac{45-22}{12} = \frac{23}{12} =1\frac{11}{12} \\ \end{align*}$$

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  To convert back to a mixed number, divide 23 by 12. The whole number is the quotient. The numerator of the fraction is the remainder and the denominator of the fraction is the divisor.  
  
  ::要转换回混合数字,将23除以12。整个数字是商数。分数的分子是剩余数,分数的分母是分数。

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    WARNING
  ::警告

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  Be careful to avoid the following errors.
  ::注意避免下列错误。

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  • $\begin{align*}\frac{1}{a} + \frac{1}{b} \ne \frac{1}{a+b}\end{align*}$
    ::1a+1b+1b+1a+b
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  •   $\begin{align*}\frac{\cancel{a}+b}{\cancel{a}} \ne b\end{align*}$  
    ::a+bab

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  Feature: Pieces of Eight—Fractions on the Stock Market
  
  ::特色:股票市场八分之一的一小块

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  by Jen Kershaw
  
  ::由Jen Kershaw著

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  Did you know that the stock market originally relied on fractions instead of decimals? Read on to find out how.
  ::您知道股市最初依赖于分数而不是小数点吗? 继续阅读以找出方法 。

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  Happening History
  ::发生历史

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  During the beginnings of the American stock market in the 18th century, the United States wanted people to be able to invest in businesses by purchasing or trading stocks , or stakes in a company. The U.S. dollar had been based on Spanish currency at the time, the real , which was also known as peso de ocho , or “piece of eight.” The real was actually a silver coin that could be physically cut into eighths, quarters, or halves. Different fractions of the silver coin were worth different values—hence the name “piece of eight.”
  ::在18世纪美国股市之初,美国希望人们能够通过购买或交易股票或公司股本对企业进行投资。 美元当时以西班牙货币为基础,真实货币,也称为比索德奥乔(peso de ocho ) , 或“八分之一 ” 。 真实货币实际上是一枚银币,可以切成八分之八、四分之一或一半。 银币的不同部分价值不同 — — 即“八分之八”的名称。

  

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  Since the U.S. had based its own currency on the Spanish system, American stocks were assigned values in one-eighths of a dollar. Up until 2001, stock exchanges, such as the New York Stock Exchange, still reported prices and stock values in fractions. Because our monetary system came to revolve around decimals, each fraction of stock value had to be converted into decimals before funds could be awarded. Finally, in 2001, all of the fractions of the stock market were converted to decimals in order to make the arithmetic and money easier to manage.
  ::由于美国将自己的货币以西班牙体系为基础,美国股票的定值为美元八分之一。 直到2001年,股票交易所,如纽约证券交易所,仍然以小数报告价格和股票价值。 由于我们的货币体系以小数点左右,在获得资金之前,股票价值的每一部分都必须转换成小数。 最后,在2001年,股票市场的所有部分都转换成小数,以便更容易地管理算术和货币。

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  Take a look at the origins of the stock market.
  ::看看股市的起源。

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  by themoneysideoflife provides an overview of the history of the stock market.
  ::以生命的金钱来提供股票市场历史的概览。

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  Summary
  
  ::摘要

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  • To add or subtract fractions, you need to first express the fractions with a common denominator. Then, you can combine the numerators.
    ::要添加或减减分数,您需要首先用一个共同分母来表示分数。然后,您可以将分子组合在一起。
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  • To multiply fractions, multiply the numerators and multiply the denominators.
    ::乘以分数,乘以分子,乘以分母。
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  • To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
    ::要分割分数,第一个分数乘以第二个分数的对等数。

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  Review
  
  ::回顾

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  Evaluate the following expressions:
  ::评价以下表述:

  1. $\begin{align*}-\frac{4}{9} + -\frac{13}{9}\end{align*}$  
  2. $\begin{align*}\frac{4}{3} + \frac{7}{6}\end{align*}$  
  3. $\begin{align*}\frac{2}{5} + \frac{4}{7} + \frac{1}{2}\end{align*}$  
  4. $\begin{align*}\frac{11}{4} - \frac{3}{4}\end{align*}$  
  5. $\begin{align*}-\frac{5}{6}-\frac{7}{8}\end{align*}$  
  6. $\begin{align*}\frac{9}{5} - \frac{3}{8} + \frac{4}{5}\end{align*}$  
  7. $\begin{align*}\frac{12}{7} \cdot \frac{8}{9}\end{align*}$  
  8. $\begin{align*}-\frac{2}{3} \cdot \frac{3}{11}\end{align*}$  
  9. $\begin{align*}\frac{4}{13} \div \frac{2}{3} \end{align*}$  
  10. $\begin{align*}\frac{8}{15} \div -\frac{3}{5}\end{align*}$  

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  Explore More
  ::探索更多

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  1. Use the order of operations to solve.
  ::1. 使用操作顺序解决。

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  a. $\begin{align*}\frac{1}{5} + -\frac{2}{3} \div \frac{3}{7} \cdot \frac{5}{7} - \frac{1}{2} \end{align*}$  
  
  ::a. 15_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  b. $\begin{align*}-\frac{2}{3} \cdot -\frac{4}{5} \div \frac{2}{7} + \frac{1}{2}\end{align*}$  
  
  ::b. - 234527+12

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  2. Last Month, Jane spent $\begin{align*}\frac{1}{3}\end{align*}$   of her monthly salary on food, $\begin{align*}\frac{2}{5}\end{align*}$   of her monthly salary on her tuition fees. Jane spent  $\begin{align*}\frac{3}{4}\end{align*}$  of the remaining amount (after deducting her food and tuition expenses) on transportation. If she decided to save the rest, what fraction of her salary did she save?
  ::2. 上个月,Jane将其月薪的13份花在食品上,25份月薪花在学费上,其余34份(扣除她的食品和学费后)花在交通上,如果她决定节省其余部分,她节省了多少工资?

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  3. A recipe for 6 muffins calls for 1 $\begin{align*}\frac{3}{4}\end{align*}$  cups of sugar. How much sugar should be used for 8 muffins?
  ::3. 6个松饼的配方需要134杯糖,8个松饼应使用多少糖?

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  4. In one week a sneaker company’s stock rose from  $\begin{align*}56\frac{3}{4}\end{align*}$   to  $\begin{align*}57 \frac{1}{3}\end{align*}$ . How much did the stock increase?
  ::4. 一周内,一个运动鞋公司的股票从5634增加到5713,股票增加多少?

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  5. Dan is cutting plywood for his science fair project.  He cuts a board that is 314 feet long. After he cuts it, he realizes that he really needs a piece about $\begin{align*}\frac{2}{3}\end{align*}$  of this length. How long will the new piece of wood that Dan cuts be?
  ::5. 丹正在为其科学博览会项目切割胶合板,他切了314英尺长的板子,在切开板子后,他意识到他真的需要这一长度的23英尺左右的一块。

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  Answers for Review and Explore More Problems
  ::回顾和探讨更多问题的答复

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  Please see the Appendix.
  
  ::请参看附录。

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  PLIX
  ::PLIX

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  Try these interactives that reinforce the concepts explored in this section:
  ::尝试这些强化本节所探讨概念的交互作用 :