2.6 理性数字除法

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  理性数字分割

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  Division of Rational Numbers 
  ::理性数字分割

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  An identity element is a number which, when combined with a mathematical operation on a number, leaves that number unchanged. For example, the identity element for addition and subtraction is zero , because adding or subtracting zero to a number doesn’t change the number. And zero is also what you get when you add together a number and its opposite, like 3 and -3.
  ::身份元素是一个数字,当与数字上的数学操作相结合时,该数字保持不变。例如,用于添加和减法的识别元素为零,因为对数字的增减零不会改变数字。当将数字和数字相加时,零也是数字,例如3和3和3。

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  Multiplicative Inverses
  ::倍倍倍倍倍倍数

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  The inverse operation of addition is subtraction—when you add a number and then subtract that same number, you end up back where you started. Also, adding a number’s opposite is the same as subtracting it—for example, $\begin{align*}4 + (-3)\end{align*}$ is the same as $\begin{align*}4 - 3\end{align*}$ .
  ::加法的反作用是减法——当您添加一个数字然后再减去同一数字时,你就会回到开始的位置。此外,加法的反向作用与减法相同——例如,4+(-3)与4-3相同。

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  Multiplication and division are also inverse operations to each other—when you multiply by a number and then divide by the same number, you end up back where you started. Multiplication and division also have an identity element: when you multiply or divide a number by one , the number doesn’t change.
  ::乘法和除法是相互反向的操作,当乘以一个数字,然后除以同一个数字时,最终会回到开始的位置。乘法和除法还有一个特性元素:当乘以一个数字或除以一个数字时,数字不会改变。

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  Just as the opposite of a number is the number you can add to it to get zero, the reciprocal of a number is the number you can multiply it by to get one. And finally, just as adding a number’s opposite is the same as subtracting the number, multiplying by a number’s reciprocal is the same as dividing by the number.
  ::数字的反面是数字可以加到数字中以获得零,数字的对等性是数字可以通过数字乘以获得一个数字。 最后,数字的对等性与减去数字相同,乘以数字的对等性与除以数字相同。

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  The reciprocal of a number $\begin{align*}x\end{align*}$ is also called the multiplicative inverse . Any number times its own multiplicative inverse equals one, and the multiplicative inverse of $\begin{align*}x\end{align*}$ is written as $\begin{align*}\frac{1}{x}\end{align*}$ .
  ::数 x 的对等也称为乘数反。它自己的乘数乘以乘以乘以乘以反等值等于一,而乘以反乘以乘以x 以一。

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  To find the multiplicative inverse of a rational number , we simply invert the fraction —that is, flip it over. In other words:
  ::为了找到一个合理数字的乘数反,我们只是颠倒了分数,即翻转它。换句话说:

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  The multiplicative inverse of $\begin{align*}\frac{a}{b}\end{align*}$ is $\begin{align*}\frac{b}{a}\end{align*}$ , as long as $\begin{align*}a \neq 0\end{align*}$ .
  ::ab的倍数反比是巴巴,直到0。

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  You’ll see why in the following exercise.
  ::你会看到为什么在接下来的练习中。

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  Finding Multiplicative Inverses 
  ::查找多种重复的逆数

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  Find the multiplicative inverse of each of the following.
  ::查找以下每一种的多倍反差。

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  a) $\begin{align*}\frac{3}{7}\end{align*}$
  :a) 37

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  When we invert the fraction $\begin{align*}\frac{3}{7}\end{align*}$ , we get $\begin{align*}\frac{7}{3}\end{align*}$ . Notice that if we multiply $\begin{align*}\frac{3}{7} \cdot \frac{7}{3}\end{align*}$ , the 3’s and the 7’s both cancel out and we end up with $\begin{align*}\frac{1}{1}\end{align*}$ , or just 1.
  ::当我们颠倒第37点时,我们得到73分, 注意如果我们乘以3773分, 3和7分都取消, 我们最后只有11分, 或只有1分。

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  b) $\begin{align*}\frac{4}{9}\end{align*}$
  :b) 49

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  Similarly, the inverse of $\begin{align*}\frac{4}{9}\end{align*}$ is $\begin{align*}\frac{9}{4}\end{align*}$ ; if we multiply those two fractions together, the 4’s and the 9’s cancel out and we’re left with 1. That’s why the rule “invert the fraction to find the multiplicative inverse” works: the numerator and the denominator always end up canceling out, leaving 1.
  ::类似地,49的反比是94;如果我们将这两个部分相乘,4个和9个取消,我们只剩下1个,这就是为什么规则 " 颠倒部分以找到多倍反向 " 起作用:分子和分母总是最后取消,留下1个。

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  c) $\begin{align*}3\frac{1}{2}\end{align*}$
  :c) 312

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  To find the multiplicative inverse of $\begin{align*}3\frac{1}{2}\end{align*}$ we first need to convert it to an improper fraction . Three wholes is six halves, so $\begin{align*}3\frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}\end{align*}$ . That means the inverse is $\begin{align*}\frac{2}{7}\end{align*}$ .
  ::要找到312的倍数反差, 我们首先需要将其转换为不适当的分数。 三个整数是六分之一, 所以 312=62+12=72。 这意味着反数是27 。

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  d) $\begin{align*}-\frac{x}{y}\end{align*}$
  :d)-xy

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  Don’t let the negative sign confuse you. The multiplicative inverse of a negative number is also negative! Just ignore the negative sign and flip the fraction as usual. 
  ::不要让负面迹象混淆你。 负数的倍数反比也是负数! 只要忽略负数迹象, 照常翻转部分, 就可以了 。

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  The multiplicative inverse of $\begin{align*}-\frac{x}{y}\end{align*}$ is $\begin{align*}-\frac{y}{x}\end{align*}$ .
  ::-xy 的倍增效应为 - yx 。

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  e) $\begin{align*}\frac{1}{11}\end{align*}$
  ::e) 111

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  The multiplicative inverse of $\begin{align*}\frac{1}{11}\end{align*}$ is $\begin{align*}\frac{11}{1}\end{align*}$ , or simply 11.
  ::111的乘数反比为111,或仅11。

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  Look again at the last example. When we took the multiplicative inverse of $\begin{align*}\frac{1}{11}\end{align*}$ we got a whole number, 11. That’s because we can treat that whole number like a fraction with a denominator of 1. Any number, even a non-rational one, can be treated this way, so we can always find a number’s multiplicative inverse using the same method.
  ::再看看最后一个例子。 当我们以111的倍数反转时,我们得到了一个完整的数字,11。 这是因为我们可以把整个数字当作一个分数,分母为1。 任何数字,即使是非理性数字,都可以这样处理,这样我们就可以用同样的方法发现一个数字的倍数反转。

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  Divide Rational Numbers
  ::分裂性理性数字

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  Earlier, we mentioned that multiplying by a number’s reciprocal is the same as dividing by the number. That’s how we can divide rational numbers; to divide by a rational number, just multiply by that number’s reciprocal. In more formal terms:
  ::早些时候,我们曾提到,乘以一个数字的对等乘法与除以数字是相同的。 这就是我们如何区分合理数字;用一个合理数字来除以一个合理数字,只是乘以这个数字的对等乘法。 用更正式的术语说:

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  $$\begin{align*}\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.\end{align*}$$
  ::ab-cd=abxdc。

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  Divide the following , giving your answer in the simplest form .
  ::除以下方,以最简单的形式给出答案。

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  a) $\begin{align*}\frac{1}{2} \div \frac{1}{4}\end{align*}$
  :a) 1214

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  Replace $\begin{align*}\frac{1}{4}\end{align*}$ with $\begin{align*}\frac{4}{1}\end{align*}$ and multiply: $\begin{align*}\frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2\end{align*}$ .
  ::以41取代14,乘以12x41=42=2。

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  b) $\begin{align*}\frac{7}{3} \div \frac{2}{3}\end{align*}$
  :b) 7323

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  Replace $\begin{align*}\frac{2}{3}\end{align*}$ with $\begin{align*}\frac{3}{2}\end{align*}$ and multiply: $\begin{align*}\frac{7}{3} \times \frac{3}{2} = \frac{7 \cdot 3}{3 \cdot 2} = \frac{7}{2}\end{align*}$ .
  ::将23改为32,乘以:73x32=733=72。

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  c) $\begin{align*}\frac{x}{2} \div \frac{1}{4y}\end{align*}$
  ::c) x214y

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  $\begin{align*}\frac{x}{2} \div \frac{1}{4y} = \frac{x}{2} \times \frac{4y}{1} = \frac{4xy}{2} = \frac{2xy}{1} = 2xy\end{align*}$
  ::x214y=x2×4y1=4xy2=2xy1=2xy1=2xy

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  d) $\begin{align*}\frac{11}{2x} \div \left ( -\frac{x}{y} \right )\end{align*}$
  :d) 112x(-xy)

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  $\begin{align*}\frac{11}{2x} \div \left ( -\frac{x}{y} \right ) = \frac{11}{2x} \times \left ( -\frac{y}{x} \right ) = -\frac{11y}{2x^2} \end{align*}$
  ::112x(- xy) = 112xxxx(- yx) 11y2x2

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  Solve Real-World Problems Using Division
  ::利用司解决现实世界问题

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  Speed, Distance and Time
  ::速度、距离和时间

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  An object moving at a certain speed will cover a fixed distance in a set time . The quantities speed, distance and time are related through the equation $\begin{align*}\text{Speed} = \frac{\text{Distance}}{\text{Time}}\end{align*}$ .
  ::以一定速度移动的物体将在固定时间内覆盖固定距离。数量速度、距离和时间通过方程式“速度”=“距离时间”。

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  Anne runs a mile and a half in a quarter hour. What is her speed in miles per hour?
  ::Anne在四分之一小时里跑1.5英里 她的速度是每小时1英里多快?

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  We already have the distance and time in the correct units (miles and hours), so we just need to write them as fractions and plug them into the equation.
  ::我们已经有了正确的单位(英里和小时)的距离和时间, 所以我们只需要把它们写成分数, 并把它们插进方程中。

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  $$\begin{align*}\text{Speed} = \frac{1\frac{1}{2}}{\frac{1}{4}} = \frac{3}{2} \div \frac{1}{4} = \frac{3}{2} \times \frac{4}{1} = \frac{3 \cdot 4}{2 \cdot 1} = \frac{12}{2} = 6\end{align*}$$
  ::速度=11214=3214=32×41=3421=122=6

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  Anne runs at 6 miles per hour.
  ::安妮每小时跑6英里

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  Examples
  ::实例

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  Divide the following rational numbers, giving your answer in the simplest form .
  ::将下列合理数字分开,以最简单的形式回答。

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  Example 1
  ::例1

  $\begin{align*}\frac{3}{10} \div \frac{7}{5}\end{align*}$

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  Replace $\begin{align*}\frac{7}{5}\end{align*}$ with $\begin{align*}\frac{5}{7}\end{align*}$ and multiply: $\begin{align*}\frac{3}{10} \times \frac{5}{7} = \frac{15}{70} =\frac{3}{14}\end{align*}$ .
  ::将75改为57,乘以:310x57=1570=314。

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  Example 2
  ::例2

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  $\begin{align*}\frac{9x}{5} \div \frac{9}{5}\end{align*}$
  ::9x595

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  Replace $\begin{align*}\frac{9}{5}\end{align*}$ with $\begin{align*}\frac{5}{9}\end{align*}$ and multiply: $\begin{align*}\frac{9x}{5} \times \frac{5}{9} = \frac{45x}{45} = x\end{align*}$ .
  ::将95改为59,乘以:9x5x59=45x45=x。

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  Review 
  ::回顾

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  For 1-5, find the multiplicative inverse of each of the following.
  ::1-5, 找到以下每一种的多倍反差 。

  1. 100
  2. $\begin{align*}\frac{2}{8}\end{align*}$
  3. $\begin{align*}-\frac{19}{21}\end{align*}$
  4. 7
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  5. $\begin{align*}-\frac{z^3}{2xy^2}\end{align*}$
     ::- 兹32xy2

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  For 6-10, divide the following rational numbers. Write your answer in the simplest form.
  ::6 - 10, 将下列合理数字除以。 请以最简单的形式写下您的答复 。

  6. $\begin{align*}\frac{5}{2} \div \frac{1}{4}\end{align*}$
  7. $\begin{align*}\frac{1}{2} \div \frac{7}{9}\end{align*}$
  8. $\begin{align*}\frac{5}{11} \div \frac{6}{7}\end{align*}$
  9. $\begin{align*}\frac{1}{2} \div \frac{1}{2}\end{align*}$
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  10. $\begin{align*}-\frac{x}{2} \div \frac{5}{7}\end{align*}$
      ::- x257

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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 。