2.14检查十进制除法的估计

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  检查十进制除法的估计

  

  [Figure 1]

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  Steve’s dad is looking to buy a new car. He is considering a car that would cost $21,045.68 if he paid for it over 48 months. Before deciding on the car, Steve’s dad wants to figure out what his monthly payments would be. He doesn’t want a car payment of more than $400 a month. How can Steve help his dad to determine whether or not he should buy this car?
  ::史蒂夫的爸爸正在寻找买一辆新车。 他正在考虑一辆汽车,如果他支付48个月以上,车价将达21,045.68美元。 在决定该车之前,史蒂夫的爸爸想弄清楚他每月要支付多少钱。 他不希望每月支付400美元以上的汽车费用。 史蒂夫如何帮助他父亲决定是否应该买这辆车?

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  In this concept, you will learn to estimate decimal quotients by dividing leading digits.
  ::在此概念中,您将学会通过将位数除以位数来估计小数位数。

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  Estimating Decimal Quotients
  ::估计十进数引数

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  Recall that  estimation  is a process by which you find an approximate solution. When you are working with decimal numbers with many digits, one way to estimate their  quotient  is to  divide  only their two leading digits.
  ::回顾这一估算是一个过程,您可以通过它找到一个大致的解决方案。当您使用多位数的十进制数字工作时,一种估计数字的方法就是只将其两位位数除以两个位数。

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  Here are the steps for dividing decimal numbers using leading digits.
  ::这是使用前位数字来除以十进制数字的步骤 。

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  1. Identify the two leading (left-most) digits of each number, preserving the location of the  decimal point . Note that if 0 is the only digit to the left of a decimal point, it does not count as one of the leading digits.
     ::识别每个数字的两个前两位(最左)位数,保留小数点的位置。请注意,如果小数点左侧只有0位数,那么它不会算作前两位数之一。
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  2. Divide the leading digits using your steps for  .
     ::使用您的步骤除以前位数字 。

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  Here is an example.
  ::举一个例子。

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  Estimate the quotient of  $\begin{array}{r}4.819÷1.245\end{array}$ .
  ::估计商数为4.819 1.245。

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  First, identify the two leading digits of each number.
  ::首先,确定每个数字的两位前两位数。

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  • The leading digits of 4.819 are 4.8
    ::4.819的前两位数是4.8
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  • The leading digits of 1.245 are 1.2
    ::1.245名前数是1.245名前数是1.2名

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  Now, note that 4.8 is your  dividend  and 1.2 is your  divisor .
  ::现在,请注意4.8是你的红利,1.2是你的红利。

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  Next, move the decimal points. You will move the decimal point of your divisor, 1.2, 1 place to the right to turn it into 12. Move the decimal point of your dividend, 4.8, 1 place as well to turn it into 48.
  ::下一步,移动小数点。您将把您的断点的小数点, 1.2, 1 位移到右侧, 将其转换为12 。 移动您的红利的小数点, 4.8, 1 位移到右侧, 并将其转换为48 。

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  Now, divide as usual using long division.
  ::现在,像往常一样,用长的分裂来分裂。

  $$\begin{array}{r}\stackrel{4}{12\stackrel{¯}{)48}}\end{array}$$

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  The answer is  $\begin{array}{r}4.819÷1.245\end{array}$  is approximately 4.
  ::答案是4.8191.245约4。

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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 Steve's dad, who is looking to buy a new car.
  ::之前,你被问及史蒂夫的爸爸, 史蒂夫的爸爸,他正在寻找 买一辆新车。

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  The car he is looking at would cost $21,045.68 and he would have 48 months to pay for it. Steve wants to figure out what his dad’s monthly payments would be to make sure they are not more than $400.
  ::他正在看的汽车将花费21,045.68美元,他有48个月的时间来支付。 Steve想弄清楚他父亲的月工资是多少才能保证不超过400美元。

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  Steve needs to divide  $\begin{array}{r}\$21,045.68÷48\end{array}$ . Because Steve only wants to see if the monthly payment is above or below $400, he can estimate the answer by dividing the leading digits.
  ::史蒂夫需要分配21美元,045.68美元, 48。 因为史蒂夫只想知道每月的付款是否超过或低于400美元, 他可以通过将位数除以来估计答案。

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  First, he should identify the two leading digits of each number.
  ::首先,他应确定每个数字的两个前两位数。

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  • The leading digits of 21,045.68 are 21,000 (remember that you have to maintain the placement of the decimal point!)
    ::21 045.68的前两位数是21 000(记住您必须保持小数点的位置 !)
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  • The leading digits of 48 are 48
    ::48个前位数为48个前位数

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  Now, note that 21,000 is your dividend and 48 is your divisor. Because your divisor is already a whole number, you do not need to move any decimal points.
  ::现在请注意, 21,000 是你的红利, 48 是你的红利。 您的 divisor 已经是一个完整的数字, 您不需要移动任何小数点 。

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  Now, divide as usual using long division.
  ::现在,像往常一样,用长的分裂来分裂。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{437.5}{48\overline{)21000.0}}\\ & & \underset{\_}{-192}\\ & & 180\\ & & \underset{\_}{-144}\\ & & 360\\ & & \underset{\_}{-336}\\ & & 240\\ & & \underset{\_}{-240}\\ & & 0\end{array}\end{array}$$

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  The answer is that the monthly payments will be approximately $437.50. Steve’s dad shouldn’t buy this car because the payments will be more than $400 a month.
  ::答案是每月的付款约为437.50美元。 史蒂夫的父亲不应该买这辆车,因为每月的付款将超过400美元。

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  Example 2
  ::例2

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  Estimate the quotient by dividing the leading digits. If your answer is a  repeating decimal ,  round  it to the nearest tenth.
  ::将前位位数除以估计商数。如果回答是重复的十进制数字,则将它四舍五入到最近的十进制数字。

  $\begin{array}{r}6.4256÷2.2453\end{array}$

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  First, identify the two leading digits of each number.
  ::首先,确定每个数字的两位前两位数。

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  • The leading digits of 6.4256 are 6.4
    ::6.4256的前位数字为6.4。
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  • The leading digits of 2.2453 are 2.2
    ::2.2453的前两位数为2.2。

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  Now, note that 6.4 is your dividend and 2.2 is your divisor.
  ::现在,请注意6.4是你的红利,2.2是你的红利。

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  Next, move the decimal points. You will move the decimal point of your divisor, 2.2, 1 place to the right to turn it into 22. Move the decimal point of your dividend, 6.4, 1 place as well to turn it into 64.
  ::下一步,移动小数点。您将把您的断点的小数点, 2. 2, 1 位移到右侧将其转换为22 。移动您的红利的小数点, 6.4, 1 位移到右侧,并将其转换为64 。

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  Now, divide as usual using long division.
  ::现在,像往常一样,用长的分裂来分裂。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{2.9090\dots }{22\overline{)64.0000}}\\ & & \underset{\_}{-44}\\ & & 200\\ & & \underset{\_}{-198}\\ & & 20\\ & & \underset{\_}{-0}\\ & & 200\\ & & \underset{\_}{-198}\\ & & 20\end{array}\end{array}$$

  
  This time your answer is a repeating decimal. The  $\begin{array}{r}9090\dots \end{array}$  pattern will repeat over and over. You can round your answer to the nearest tenth.

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  The answer is  $\begin{array}{r}6.4256÷2.2453\end{array}$  is approximately 2.9.
  ::答案是6.4256 2.2453 约2.9。

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  Example 3
  ::例3

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  Estimate the quotient of  $\begin{array}{r}15.934÷2.57\end{array}$  using leading digits.
  ::使用前位数估算的商数为15.934 2.57。

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  First, identify the two leading digits of each number.
  ::首先,确定每个数字的两位前两位数。

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  • The leading digits of 15.934 are 15
    ::15 934名前数为15名
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  • The leading digits of 2.57 are 2.5
    ::2.57的前两位数为2.5。

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  Now, note that 15 is your dividend and 2.5 is your divisor.
  ::现在,请注意15是你的红利,2.5是你的红利。

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  Next, move the decimal points. You will move the decimal point of your divisor, 2.5, 1 place to the right to turn it into 25. Move the decimal point of your dividend, 15, 1 place as well to turn it into 150.
  ::下一步,移动小数点。您将把您的断点的小数点, 2.5, 1 位移到右侧, 将其转换为 25 。 移动您的红利的小数点, 15, 1 位移到右侧, 并将其转换为 150 。

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  Now, divide as usual using long division.
  ::现在,像往常一样,用长的分裂来分裂。

  $$\begin{array}{r}\stackrel{6}{25\stackrel{¯}{)150}}\end{array}$$

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  The answer is  $\begin{array}{r}15.934÷2.57\end{array}$  is approximately 6.
  ::答案是15.934 2.57 大约是 6。

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  Example 4
  ::例4

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  Estimate the quotient of  $\begin{array}{r}4.368÷3.12\end{array}$  using leading digits. Round your answer to the nearest hundredth.
  ::使用前导数字估计4.368 3. 12 的商数。 将您的答复四舍五入到最近的第一百位 。

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  First, identify the two leading digits of each number.
  ::首先,确定每个数字的两位前两位数。

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  • The leading digits of 4.368 are 4.3
    ::4.368的前位数字为4.3
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  • The leading digits of 3.12 are 3.1
    ::3.12的前位数字为3.1

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  Now, note that 4.3 is your dividend and 3.1 is your divisor.
  ::请注意,4.3是你的红利,3.1是你的红利。

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  Next, move the decimal points. You will move the decimal point of your divisor, 3.1, 1 place to the right to turn it into 31. Move the decimal point of your dividend, 4.3, 1 place as well to turn it into 43.
  ::下一步,移动小数点。您将把您的断点的小数点, 3.1, 1 移动到右侧, 将其转换为31 。 移动您的红利的小数点, 4.3, 1 移动到右侧, 并把它转换为43 。

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  Now, divide as usual using long division.
  ::现在,像往常一样,用长的分裂来分裂。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{1.387\dots }{31\overline{)43.000}}\\ & & \underset{\_}{-31}\\ & & 120\\ & & \underset{\_}{-93}\\ & & 270\\ & & \underset{\_}{-248}\\ & & 220\\ & & \underset{\_}{-217}\\ & & 30\end{array}\end{array}$$

  Because the question said to round your answer to the nearest hundredth, you can stop the long division once you’ve reached the thousandths place.  $\begin{array}{r}1.387\dots \end{array}$  rounds to 1.39.

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  The answer is  $\begin{array}{r}4.368÷3.12\end{array}$  is approximately 1.39.
  ::答案是4.368 3.12 约1.39。

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  Example 5
  ::例5

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  Estimate the quotient of  $\begin{array}{r}6.16÷1.12\end{array}$  using leading digits. Round your answer to the nearest hundredth.
  ::使用前列位数来估计 6.16 = 1. 12 的商数。 将您的答复四舍五入到最近的一百位 。

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  First, identify the two leading digits of each number.
  ::首先,确定每个数字的两位前两位数。

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  • The leading digits of 6.16 are 6.1
    ::6.16的前位数字为6.16,前位数字为6.1
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  • The leading digits of 1.12 are 1.1
    ::1.12的前数为1.1。

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  Now, note that 6.1 is your dividend and 1.1 is your divisor.
  ::现在,请注意,6.1是你的红利,1.1是你的顾问。

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  Next, move the decimal points. You will move the decimal point of your divisor, 1.1, 1 place to the right to turn it into 11. Move the decimal point of your dividend, 6.1, 1 place as well to turn it into 61.
  ::下一步,移动小数点。您将把您的断点的小数点, 1. 1, 1, 1 移到右边, 将其转换为 11 。 移动您的红利的小数点, 6.1, 1 并把它转换为 61 。

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  Now, divide as usual using long division.
  ::现在,像往常一样,用长的分裂来分裂。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{5.545\dots }{11\overline{)61.0000}}\\ & & \underset{\_}{-55}\\ & & 60\\ & & \underset{\_}{-55}\\ & & 50\\ & & \underset{\_}{-44}\\ & & 60\\ & & \underset{\_}{-55}\\ & & 50\end{array}\end{array}$$

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  Because the question said to round your answer to the nearest hundredth, you can stop the long division once you’ve reached the thousandths place.  $\begin{array}{r}5.545\dots \end{array}$  rounds to 5.55.
  ::问题在于你对最近的一百多岁的答复, 一旦你到达千分之一的地方, 你就可以停止长期的分裂。 5. 545 ... 回合到5. 555 回合。

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  The answer is  $\begin{array}{r}6.16÷1.12\end{array}$  is approximately 5.55.
  ::答案是6.16 1.12 约5.55。

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  Review
  ::回顾

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  Estimate the quotients by dividing the two leading digits. If your answer is a repeating decimal, round it to the nearest tenth.
  ::将两个位数除以估计商数。 如果您的回答是重复的十进制, 则将四舍五入到最近的十进制 。

  1. $\begin{array}{r}4.992÷.07123\end{array}$  
  2. $\begin{array}{r}1.8921÷6.0341\end{array}$
  3. $\begin{array}{r}26.2129÷1.3612\end{array}$
  4. $\begin{array}{r}1.00765÷.25\end{array}$
  5. $\begin{array}{r}36.2129÷2.5612\end{array}$
  6. $\begin{array}{r}.42129÷.15612\end{array}$
  7. $\begin{array}{r}6.4129÷2.2612\end{array}$
  8. $\begin{array}{r}26.2129÷13.5612\end{array}$
  9. $\begin{array}{r}8.2129÷2.2612\end{array}$
  10. $\begin{array}{r}42.2129÷8.2612\end{array}$
  11. $\begin{array}{r}16.2129÷4.1612\end{array}$
  12. $\begin{array}{r}19.0029÷3.599\end{array}$
  13. $\begin{array}{r}.45632÷.09123\end{array}$
  14. $\begin{array}{r}8.765÷1.098\end{array}$
  15. $\begin{array}{r}.145632÷.701023\end{array}$

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  Review (Answers)
  ::回顾(答复)