1.9 使用十进制操作

章节大纲

• 选择节常规

  折叠 展开

  常规

  全部折叠 展开全部

  • 选择活动新闻通告

    

    新闻通告 讨论区
• 选择节使用十进数操作

  折叠 展开

  使用十进数操作

  播放段落

  Jake weighs 197.5 pounds and he wants to lose a few pounds over the next month. In the first week, he loses 1.4 pounds; in the second week, he loses 0.3 pounds; in the third week, he loses 1.1 pounds; and in the final week, he loses 0.4 pounds. How can Jake determine how much he weighs now?
  ::Jake的体重为197.5磅,他想在下一个月减掉几磅。 在第一周,他减了1.4磅;第二周,他减了0.3磅;第三周,他减了1.1磅;最后一周,他减了0.4磅。 Jake如何确定他现在的体重?

  播放段落

  In this section, we discuss how to operate with decimals.
  ::在本节中,我们讨论如何用小数点来操作。

  

  播放段落

  Adding and Subtracting Decimals
  ::添加和减减十进数

  播放段落

  To , line the place values up with each other and compute as if the decimal point was not there.
  
  ::要将位置排列成线,请将位置的数值对齐,然后进行计算,仿佛小数点不在那里一样。

  播放段落

  Example 1
  
  ::例1

  播放段落

  Find  $\begin{array}{r}5+3.45+0.56\end{array}$ .
  ::查找 5+3.45+0.56。

  播放段落

  Solution: First, we stack the numbers vertically. We particularly focus on lining up the decimal points of the numbers to make sure that we add numbers that are in the same place value . It is also helpful to add zeros as placeholders in spots where there are not other numbers.  
  ::解决方案 : 首先, 我们垂直堆叠数字 。 我们特别侧重于排列数字的十进制点, 以确保添加相同位置值的数字 。 在没有其它数字的点添加零作为占位符也是有益的 。

  $$\begin{array}{r}5.00\\ 3.45\\ \underset{\_}{+0.56}\end{array}$$

  播放段落

  Notice how each number in the problem has the same number of digits and how all of the decimals are lined up. Finally, we add the numbers in each column.
  ::注意问题中每个数字的位数和小数组的行数相同。 最后, 在每列中添加数字 。

  $$\begin{array}{r}5.00\\ 3.45\\ \underset{\_}{+0.56}\\ 9.01\end{array}$$

  播放段落

  Example 2
  ::例2

  播放段落

  Jake weighs 197.5 pounds and he wants to lose a few pounds over the next month. In the first week, he loses 1.4 pounds; in the second week, he loses 0.3 pounds; in the third week, he loses 1.1 pounds; and in the final week, he loses 0.4 pounds. How can Jake determine how much he weighs now?
  ::Jake的体重为197.5磅,他想在下一个月减掉几磅。 在第一周,他减了1.4磅;第二周,他减了0.3磅;第三周,他减了1.1磅;最后一周,他减了0.4磅。 Jake如何确定他现在的体重?

  播放段落

  Solution: Let's first add up the total amount of weight Jake has lost.
  
  ::解决方案:让我们首先把Jake损失的总重量加起来。

  $$\begin{array}{r}1.4\\ 0.3\\ 1.1\\ \underset{\_}{+0.4}\\ 3.2\end{array}$$

  播放段落

  Now, we can subtract the total weight lost from Jake's original weight. 
  
  ::现在,我们可以从Jake原来的体重中 减去损失的总重量

  $$\begin{array}{r}197.5\\ \underset{\_}{-3.2}\\ 194.3\end{array}$$

  播放段落

  Jake's current weight is 194.3 pounds.  
  ::Jake现在的体重是194.3磅

  播放段落

  Multiplying Decimals
  
  ::乘进数十进数

  播放段落

  To multiply decimals, there are two things you need to keep in mind. First, you multiply the numbers as if there were no decimal points. Second, you need to add up the total number of decimal places in the problem and reflect that in your final solution.
  
  ::要将小数点乘以十进制,需要记住两件事。首先,你将数字乘以,好像没有小数点。第二,你需要将问题中小数点的总数加在一起,并在最终解决方案中反映这一点。

  播放段落

  Example 3
  
  ::例3

  播放段落

  Micah is going on a climbing trip. Micah wants to buy 195.5 feet of rope that costs $1.59 per foot. How much will the rope cost?
  ::Micah要买195.5英尺的绳子,每英尺1.59美元。绳子要花多少钱?

  播放段落

  Solution: Multiply 195.5 by $1.59 to find the total cost of the rope.
  ::解决办法:乘以195.5乘以1.59,以找出绳索的总成本。

  播放段落

  First, ignore the decimal point and multiply the numbers as if they were two whole numbers .
  ::首先,忽略小数点,将数字乘以两个整数。

  $$\begin{array}{r}\begin{array}{rcl}& & 195.5\\ & & \underset{\_}{\times 1.59}\\ & & 17595\\ & & 9775\\ & & \underset{\_}{+1995}\\ & & 310845\end{array}\end{array}$$

  播放段落

  Then, place the decimal point in the product. There are a total of 3 decimal places in the numbers being multiplied. Move the decimal point 3 places to the left.
  ::然后在产品中放置小数点。在乘以的数字中,总小数点为3位。将小数点3位移到左边。

  $$\begin{array}{r}310.\underset{\leftarrow }{845}\end{array}$$

   

  播放段落

  The cost of the rope should be $310.845, but remember that dollars are only calculated to the hundredths place. Round the decimal to the nearest hundredths place. (The symbol  $\begin{array}{r}\approx \end{array}$  means approximately.)
  ::绳索的成本应该是310.845美元, 但记住美元只计算到第100位。 将小数点舍入到最近的第100位 。 (符号 意指约 。 )

  $$\begin{array}{r}\begin{array}{rcl}& & 310.8\underset{\_}{4}5\approx 310.85\\ & & \uparrow \end{array}\end{array}$$

  播放段落

  Micah will spend $310.85 on climbing rope.
  ::Micah会花310.85美元用于攀爬绳子

  播放段落

  by Mathispower4u demonstrates how to multiply decimals.
  ::通过 Mathispower4u 演示如何乘以十进制 。

  播放段落

  Dividing Decimals
  
  ::分裂的十进制

  播放段落

  We do not have a method to divide decimals, but we do have a method to into decimals. If we can convert the divisor , the number you divide by, into a whole number, then we can use long division.
  
  ::我们没有方法将小数点除以,但我们有方法将小数点转换为小数点。如果我们能将小数点转换成小数点,那么我们就可以使用长的分隔。

  播放段落

  Let's consider an example:
  
  ::举个例子:

  $$\begin{array}{r}10.4÷2.6\end{array}$$

  播放段落

  The divisor is 2.6. To change a decimal number to a whole number, multiply the decimal number by a power of ten to move the decimal point. Multiply 2.6 by 10 to move the decimal point one space to the right.
  ::divisor 是 2. 6。 要将小数点数改为整数, 将小数点数乘以十乘以十, 以移动小数点。 乘以 2. 6 乘以 10 将小数点一空格向右移动 。

  $$\begin{array}{r}2.6\times 10=26\end{array}$$

  播放段落

  Dividing by 26 is easier than .6. However, dividing 10.4 by 26 is not the same as dividing 10.4 by 2.6. If you change the divisor, you must also change the dividend , the number you divide into.
  ::然而,将10.4除以26并不等于将10.4除以2.6。如果改变分数,你还必须改变红利,即你所划分的红利。

  播放段落

  Remember that a division problem can also be written as a fraction. The dividend is placed in the numerator and the divisor is placed in the denominator . In this problem, 10.4 is the dividend and is placed in the numerator. 2.6 is the divisor and is placed in the denominator.
  ::记住分裂问题也可以作为一个分数来写。 红利放在分子中, 分数放在分母中。 在这个问题中, 10.4 是红利, 放在分子中。 2. 6 是分数, 放在分数中 。

  $$\begin{array}{r}10.4÷2.6=\frac{10.4}{2.6}\end{array}$$

  播放段落

  If you change the denominator, you must also change the numerator in the same way in order for the fraction to have the same value. Multiply the numerator and denominator by 10.
  ::如果您更改分母, 您也必须以同样的方式更改分子数, 以使分数的值相同。 乘以分子数和分母乘以 10 。

  $$\begin{array}{r}10.4÷2.6=\frac{10.4}{2.6}\times \frac{10}{10}=\frac{104}{26}\end{array}$$

  播放段落

  Now write the new fraction as a division problem and divide to find the quotient .
  ::现在将新分数写成分裂问题, 并进行分割以找到商数 。

  播放段落

  $$\begin{array}{r}104÷26\text{ or }\stackrel{}{26\overline{)104}}\end{array}$$

  
  ::10426或26104

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{4}{26\stackrel{¯}{)104}}\\ & & \underset{\_}{-104}\\ & & 0\end{array}\end{array}$$

  播放段落

  $\begin{array}{r}10.4÷2.6\end{array}$  is the same as $\begin{array}{r}104÷26\end{array}$ . The quotient of 10.4 divided by 2.6 is 4.
  ::10.4.4.除以2.6除以4的商数为4。

  播放段落

  Here is another division problem. This time the divisor has two decimal places.
  ::还有一个分裂问题。 这次, divisor 有两个小数点位数 。

  播放段落

  Example 4
  
  ::例4

  $$\begin{array}{r}\stackrel{}{0.45\overline{)1.44}}\end{array}$$

  播放段落

  Solution: First, change the divisor to a whole number by multiplying it by a power of ten. Multiply 0.45 by 100. Then, do the same thing to the dividend.
  ::解决方案: 首先, 将二分位数乘以十乘以整数。 乘以0. 45乘以100, 然后对红利做同样的事情 。

  $$\begin{array}{r}\begin{array}{rcl}0.45\times 100& =& 45\\ 1.44\times 100& =& 144\end{array}\end{array}$$

  播放段落

  Here is the transformed division problem. Divide to find the quotient.
  ::这就是改变的分裂问题。为了找到商数而分裂。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{3}{45\stackrel{¯}{)144}}\\ & & \underset{\_}{-135}\\ & & 9\end{array}\end{array}$$

  播放段落

  Use zero as a placeholder to continue dividing.
  ::使用零作为占位符继续分隔 。

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{3.2}{45\overline{)144.0}}\\ & & \underset{\_}{-135}\\ & & 90\\ & & \underset{\_}{-90}\\ & & 0\end{array}\end{array}$$

  播放段落

  The quotient of 1.44 divided by 0.45 is 3.2.
  ::1.44除以0.45的商数为3.2。

  播放段落

  by Mathispower4u demonstrates how to divide decimals.
  ::用 Mathispower4u 演示如何除以十进制 。

   

  播放段落

  Summary
  
  ::摘要

  播放段落
  • To add or subtract decimals, stack the numbers so that their place values are aligned and then add or subtract as if the decimal was not there. Remember to bring the decimal point down between the ones and the tenths place. 
    ::要添加或减去小数点,请堆叠数字,使其位置值对齐,然后加或减以小数点不在那里。记住将小数点从小数点移到十分点之间。
  播放段落
  • To multiply decimals, multiply the numbers as you would if a decimal point was not present. Then, count the number of decimal places. That is the number of decimal places that should be in your result.
    ::要乘以小数,如果小数点没有出现,则按小数点乘以数字。然后,计算小数点位数。这就是结果中应该包含的小数点位数。
  播放段落
  • To divide decimals, multiply the dividend and the divisor by powers of 10 to make sure the divisor is not a decimal. Then, divide as you would with whole numbers. Remember, to bring the decimal point up between the ones and the tenths places.
    ::要将小数点除以十位数,则将红利和分数乘以十位数,以确保分数不是小数点数。然后,以整位数进行除法。记住,将小数点在十位数和十位数之间拉高。

  播放段落

  Review
  
  ::回顾

  播放段落

  Calculate the following.
  ::计算如下。

  1. $\begin{array}{r}17.32+0.365\end{array}$  
  2. $\begin{array}{r}2.147+-1.24\end{array}$  
  3. $\begin{array}{r}-39.1+-4.986\end{array}$  
  4. $\begin{array}{r}-2.93-3.162\end{array}$  
  5. $\begin{array}{r}5.841-2.53\end{array}$  
  6. $\begin{array}{r}-24.79-(-8.371)\end{array}$  
  7. $\begin{array}{r}2.72\times 3.1\end{array}$  
  8. $\begin{array}{r}-81.5\times 4.2\end{array}$  
  9. $\begin{array}{r}12.36÷0.04\end{array}$  
  10. $\begin{array}{r}81.46÷1.16\end{array}$  

  播放段落

  Explore More
  ::探索更多

  播放段落

  1. Use the order of operations to solve. Round your answer to the nearest millionth (if necessary). 
  ::1. 使用操作顺序解答。将您的答复转至最接近的百万位数( 必要时) 。

  播放段落

   a.  $\begin{array}{r}-1.2-3.45(-42.39+1.76)÷1.5\end{array}$
  ::a.-1.2-3.45(-4239+1.76)

  播放段落

   b.  $\begin{array}{r}(7.39-15.89)÷-4.5\times 3.93\end{array}$
  ::b. (7.39-15.89) 4.5x3.93

  播放段落

  2. The current cost of gas is $3.28. Liz’s vehicle needs 12.35 gallons of gas. How much will it cost Liz to fill her gas tank?
  ::2. 目前的天然气成本为3.28美元,Liz的车辆需要12.35加仑的天然气,Liz填充其油箱需要多少钱?

  播放段落

  3. Liza is doing her Christmas shopping at a local department store. She buys her mother a bracelet which costs $72.33. She purchases a pair of hiking boots for her father which cost $49.99. She selects a pair of pajamas for her sister which cost $27.49. She also returns a dress that cost $21.95 and receives store credit for the item. How much will Liza’s total bill be before tax?
  ::3. Liza在当地百货商店买圣诞用品,给母亲买手镯,费用72.33美元,给父亲买双徒步靴,费用49.99美元,给姐姐买睡衣,费用27.49美元,还退回一件价值21.95美元的礼服,并获得商店信贷,Liza的税前帐单总额有多少?

  播放段落

  4. Jason wants to distribute a barrel of milk into 8 bottles. Each bottle can hold 1.2 liters and there is 0.5 liters of milk left. How much milk was in the barrel? ___ liters
  ::4. Jason想将一桶牛奶分到8瓶,每瓶可持有1.2升牛奶,还有0.5升牛奶,桶里有多少牛奶?

  播放段落

  5. Jack ran for 39 minutes at a speed of 315.5 yards per minute. If he has 195.5 yards left in his run, what is the total distance of Jack's run? ___ yards
  ::5. 杰克以每分钟315.5码的速度跑了39分钟,如果他还有195.5码,杰克跑的距离是多少?

  播放段落

  Answers to Review and Explore More Problems
  
  ::对审查和探讨更多问题的答复

  播放段落

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