1.4整数除法

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  整数除法

  

  [Figure 1]

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  Jessica won a bet with her friends and received a lunchbox full of mints as her reward. Jessica counts 286 mints in total. She wants this supply to last until the end of the semester, which is 5 weeks away. If she only has mints on school days, how many can Jessica eat  per  day in  order  to make her supply last?
  ::杰西卡和朋友打赌赢了赌,并收到了一个满满薄荷糖的午餐盒作为她的奖赏。 杰西卡总共点了286薄荷糖。 她希望这一供应持续到学期结束,也就是5周后。 如果她在上课日里只有薄荷糖,那么杰西卡每天能吃多少饭才能让她的供给持续到最后?

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  In this concept, you will learn how to  divide   whole numbers .
  ::在这个概念中,你将学会如何分配整个数字。

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  Dividing Whole Numbers
  ::Divid 全部数字

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  The opposite operation of multiplication is division. To multiply means to add groups of matching things together, to divide means to split up into matching groups.
  ::乘法的相反操作是分法。 乘法的乘法可以将相匹配的事物组合起来, 将相匹配的事物分割成相匹配的组合 。

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

  $$\begin{array}{r}72÷9=\underset{\_}{}\end{array}$$

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  In this problem, 72 is the  dividend  - it is the number being divided. The  divisor  is the number of parts that the dividend is being split into, in this case, 9. The answer to a division problem is called the  quotient .  One way to complete this problem and find the quotient is to recall multiplication facts and work backwards.
  ::在这个问题中,72是股息 — — 正在分割的数字。 分数是股息分成多少部分。 分数是股息分成多少部分,在这个例子中是9个部分。 分数问题的答案叫做商数。 解决这个问题和找到商数的方法之一是回顾乘数事实和倒行逆施。

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  To divide 72 by 9, start be asking "What number multiplied by 9 equals 72?"
  ::将72除以9 开始问“乘以多少乘以9等于72?”

  $$\begin{array}{r}9\times 8=72\end{array}$$

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  If 8 groups of 9 equal 72, then of course 72 can be split into 8 groups of 9.
  ::如果8组9人等于72人,那么当然72人可以分为8组9人。

  $$\begin{array}{r}72÷9=8\end{array}$$

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  The quotient is 8.
  ::商数是8。

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

  $$\begin{array}{r}15÷2=\underset{\_}{}\end{array}$$

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  This is tricky because 15 is not an even number. This means it won't divide evenly. When this happens and you are using only whole numbers, will be a  remainder .
  ::这是棘手的, 因为 15 不是一个偶数数字。 这意味着它不会平均分配 。 当这种情况发生时, 您只使用整个数字时, 剩下的就是 。

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  Start by asking "What number mutliplied by 2 comes closest to 15, without going over?"
  ::开始问,“2比15最接近2的变形数字是多少,没有过头吗?”

  $$\begin{array}{r}2\times 7=14\end{array}$$

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  So, 7 groups of 2 comes closest to 15, with 1 left over. That is the remainder. Use " r"  to show that there is a remainder. 
  ::所以, 7组 2 最接近 15 组, 剩下 1 组。 这是剩下的部分。 使用“ r” 来显示剩余部分 。

  $$\begin{array}{r}15÷2=7r1\end{array}$$

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  The answer is 7 r1.
  ::答案是7 r1。

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  When dividing larger numbers, it may be easier to keep things organized with a division box.
  ::当除去较大数字时,可能比较容易用一个分隔框来安排。

  $$\begin{array}{r}8\overline{)825}\end{array}$$

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  Here there is a one digit divisor, 8, and a three digit dividend, 825. You need to figure out how many times 8 goes into 825. To do this, divide the divisor (8) into each digit of the dividend. 
  ::这里有一个位数位数, 8, 3位数红利, 825。 您需要弄清楚8乘以多少乘以 825 。 要做到这一点, 将 (8) 分成每个位数 。

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  First, divide 8 into 8. Of course the answer is 1. Put the 1 on top of the division box above the 8. Then multiply the 1 by 8 and subtract the result (8), from the dividend (8 - 8 = 0), and bring down the next number in the dividend (2), as shown by the arrow:
  ::第一,将8分为8。当然答案是1. 将1放在8之上的分数框上方,然后乘以1乘8,从红利中减去结果(8)(8-8=0),并按箭头所示,将红利(2)中的下一个数字降下来:

  $$\begin{array}{rl}& 1\\ & 8\overline{)825}\\ & \underset{\_}{-8}\downarrow \\ & 02\end{array}$$

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  Now, look at the next digit in the dividend. There are no 8's in 2, so put a 0 in the answer, next to the 1, and bring down the next number (5):
  ::现在,看看股息的下一个数位数。2分里没有8分,所以在答案中放一个0分,紧靠1分,然后把下一个数字(5分)放下来:

  $$\begin{array}{rl}& 10\\ & 8\overline{)825}\\ & \underset{\_}{-8}\phantom{\mathrm{}}\downarrow \\ & 025\end{array}$$

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  Next, see how many times 8 goes into the new number, 25. 8 multiplied by 4 is 24, so put 24 under 25 and subtract. Since there are no more numbers to bring down, there is a remainder of 1. Add this into the answer.
  ::接下来,请看新数字乘以多少乘以8, 25, 8乘以4乘以24, 24乘以24, 所以将24置于25之下, 减去。 因为没有更多的数字可以调低, 剩下的是 1 。 在答案中加上这个 。

  $$\begin{array}{rl}& 103r1\\ & 8\overline{)825}\\ & \underset{\_}{-8}\\ & 025\\ & \phantom{\mathrm{}}\underset{\_}{-24}\\ & \phantom{\mathrm{}}1\end{array}$$

   The answer is 103 r1.

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  You can check your work by multiplying the answer by the divisor.
  ::您可以通过将答案乘以 divisor 来检查您的工作 。

  $$\begin{array}{rl}& 103\\ & \underset{\_}{\times 8}\\ & 824+r\text{of}1=825\end{array}$$

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  The answer checks out.
  ::答案检查出来。

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  You can apply these same steps to any division problem even if the divisor has two or three digits. Work through each value of the dividend with the value of the divisor. Then check your work by multiplying your answer by the divisor.
  ::您可以将这些相同的步骤应用到任何分裂问题, 即使 divisor 有两三个位数 。 以 divisor 值来检查红利的每一个值 。 然后通过 divisor 乘以 divisor 来检查您的工作 。

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  Practice on your own!
  ::自己练习!

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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 Jessica and her minty mother lode.
  
  ::早些时候,你得到一个问题 杰西卡和她的 薄荷妈妈的树叶。

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  Jessica wants to make 286 mints last 5 school weeks.
  ::杰西卡想让286薄荷糖 持续5个学周

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  First, figure out how many days Jessica needs to consider in her calculation, since she is only going to eat mints 5 days per week (on school days).
  ::首先,找出Jessica在计算时需要考虑的天数,因为她每周(在上课日)只吃5天薄荷糖。

  $$\begin{array}{r}5\times 5=25\end{array}$$

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  Next, Jessica needs to divide the large number of items by the small number of days.
  ::接下来,杰西卡需要将大量物品除以少量天数。

  $$\begin{array}{r}286\text{ mints}÷25\text{ days}\end{array}$$

   
  

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  Perhaps it will be easier to use a division box
  ::也许使用一个分划框会比较容易些

  $$\begin{array}{r}25\overline{)286}\end{array}$$

   

  $$\begin{array}{rl}{}_{011}& {}_{\text{r11}}\\ 25\overline{)286}& \text{First, how many times does 25 go into 2? Put a 0 above the answer line.}\\ 28& \text{Next, bring down the 8.}\\ \underset{\_}{-25}& \text{Next, how many times does 25 go into 28? Put a 1 above the answer line.}\\ 36& 28-25=3,\text{Then, bring down the 6.}\\ \underset{\_}{-25}& \text{Finally, how many times does 25 go into 36? Put a 1 above the answer line.}\\ 11& 36-25=11.\text{This is the remainder.}\end{array}$$

  
  

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  The quotient is 11 remainder 11.
  ::商数为11,其余11。

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  Jessica can have 11 mints per day and she will have 11 left at the end of the semester.
  ::杰西卡每天可以有11个薄荷糖 学期结束时她还有11个

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  Example 2
  ::例2

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  Let’s look at a problem with a two-digit divisor.
  ::让我们看看两位数的二位数断字器的问题。

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  The quotient is 204.
  ::商数是204

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  You can check your work by multiplying:  $\begin{array}{r}204\times 12\end{array}$ .
  ::您可以通过乘法检查您的工作: 204 x 12 。

  $$\begin{array}{rl}204& \\ \underset{\_}{\times 12}& \\ 408& \\ \underset{\_}{+2040}& \\ 2448& \end{array}$$

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  The answer checks out.
  ::答案检查出来。

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  Example 3
  ::例3

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  Find the quotient:  $\begin{array}{r}4\overline{)469}\end{array}$
  ::找到商数 : 4 ) 469

   

  $$\begin{array}{rl}{}_1{}_1{}_7& {}_{\text{r1}}\\ 4\overline{)469}& \\ \underset{\_}{-4}& \text{First, how many times does 4 go into 4? Put a 1 above the answer line.}\\ 06& \text{Next, bring down the 6.}\\ \underset{\_}{-4}& \text{How many times does 4 go into 6? Put a 1 above the answer line.}\\ 2& 4-6=2\\ 29& \text{Then bring down the 9. How many times does 4 go into 29? Put a 7 above the answer line.}\\ \underset{\_}{-28}& \\ 1& \text{The one left over is the remainder.}\end{array}$$

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  The answer is 117 remainder 1.
  ::答案是117 余下1。

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  Example 4
  ::例4

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  Find the quotient:  $\begin{array}{r}18\overline{)3678}\end{array}$
  ::找到商数: 18) 3678

  $$\begin{array}{rl}{}_{0204}& {}_{r2}\\ 18\overline{)3678}& \text{First, how many times does 18 go into 3? Put a 0 above the answer line.}\\ 36& \text{Next, bring down the 6.}\\ \underset{\_}{-36}& \text{Next, how many times does 18 go into 36? Put a 2 above the answer line.}\\ 07& \text{Then, bring down the 7.}\\ & \text{Then, how many times does 18 go into 7? Put a 0 above the answer line.}\\ 78& \text{Then, bring down the 8.}\\ \underset{\_}{-76}& \text{Finally, how many times does 18 go into 78? Put a 4 above the answer line.}\\ 2& 78-76=2.\text{This is the remainder.}\end{array}$$

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  The answer is 204 remainder 2.
  ::答案是204余下2。

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  Example 5
  ::例5

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  Find the quotient:  $\begin{array}{r}20\overline{)5020}\end{array}$
  ::找到商数: 20) 5020

  
  

  $$\begin{array}{rl}{}_{0251}& \\ 20\overline{)5020}& \text{First, how many times does 20 go into 5? Put a 0 above the answer line.}\\ 50& \text{Next, bring down the 0.}\\ \underset{\_}{-40}& \text{Next, how many times does 20 go into 50? Put a 2 above the answer line.}\\ 102& 50-40=10\text{Bring down the 2.}\\ \underset{\_}{-100}& \text{Next, how many times does 20 go into 102? Put a 5 above the answer line.}\\ 20& 100-102=2.\text{Then, bring down the 0.}\\ \underset{\_}{-20}& 20\text{ goes into 20 once.}\\ 0& \text{There is no remainder.}\end{array}$$

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  The answer is 251.
  ::答案是251个

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  Here are some more challenging practice questions. How many can you solve?
  ::这里有一些更具挑战性的实践问题。你能解决多少问题?

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  Review
  ::回顾

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  Find the quotient.
  ::找到商数。

  1. $\begin{array}{r}12÷6=\underset{\_}{}\end{array}$  
  2. $\begin{array}{r}13÷4=\underset{\_}{}\end{array}$  
  3. $\begin{array}{r}132÷7=\underset{\_}{}\end{array}$  
  4. $\begin{array}{r}124÷4=\underset{\_}{}\end{array}$  
  5. $\begin{array}{r}130÷5=\underset{\_}{}\end{array}$  
  6. $\begin{array}{r}216÷6=\underset{\_}{}\end{array}$  
  7. $\begin{array}{r}1,161÷43=\underset{\_}{}\end{array}$  
  8. $\begin{array}{r}400÷16=\underset{\_}{}\end{array}$  
  9. $\begin{array}{r}1,827÷21=\underset{\_}{}\end{array}$  
  10. $\begin{array}{r}1,244÷40=\underset{\_}{}\end{array}$  
  11. $\begin{array}{r}248÷18=\underset{\_}{}\end{array}$  
  12. $\begin{array}{r}3,264÷16=\underset{\_}{}\end{array}$  
  13. $\begin{array}{r}4,440÷20=\underset{\_}{}\end{array}$  
  14. $\begin{array}{r}7,380÷123=\underset{\_}{}\end{array}$  
  15. $\begin{array}{r}102,000÷200=\underset{\_}{}\end{array}$  
  16.   $\begin{array}{r}10,976÷98=\underset{\_}{}\end{array}$