2.10小数乘法

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  小数乘法

  

  [Figure 1]

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  Peaches are Jackson’s favorite fruit. Each year he looks forward to peach season when he can go out and pick peaches at a local farm. He learns that this year, the peaches at the farm will cost $2.50  per  pound. Jackson and his mom pick peaches until they fill up one large box. Their box filled with peaches weighs 15.8 pounds. How can Jackson calculate how much their box of peaches will cost?
  ::桃子是杰克逊最喜欢的果子。 每年,他都期待桃子季节到来,在那里的农场摘桃子。 他知道今年农场的桃子每磅将花费2.50美元。 杰克逊和他的母亲在桃子堆满一个大盒子之前都摘桃子。 他们装满桃子的盒子重15.8磅。 杰克逊如何计算他们的桃子要花多少钱?

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  In this concept, you will learn how to multiply decimals with and without  rounding .
  ::在这个概念中,你将学会如何在不四舍五入的情况下乘以十进制。

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  Multiplying Decimals
  ::乘进数十进数

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  Recall that in a decimal number, the  decimal point  divides the whole part of the number from the fractional part of the number.
  ::回顾在小数点数中,小数点将数字的整个部分与数字的分数部分分开。

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  Multiplying decimal numbers is very similar to multiplying  whole numbers , but you will have to watch out for the decimal points. Remember that the answer to a multiplication problem is called the  product .
  ::乘以小数数与乘以整数非常相似, 但是您必须注意小数点。 记住乘法问题的答案叫做产品 。

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  To multiply decimal numbers:
  ::乘以小数数字:

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  1. Arrange the numbers vertically so that the right-most digits for each number are lined up. Do not line up the decimal points, just the right-most digits.
     ::垂直排列数字,使每个数字的最右位数字排成一行。不要在小数点上排成一行,只需在最右位数字上排成一行。
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  2. Temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.
     ::暂时忽略小数点,然后像乘整数一样乘以乘以整数。
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  3. Count the total number of digits after the decimal points in the original numbers.
     ::计算在原数小数点之后的数字总数。
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  4. In your answer, count the same number of places from right to left and insert a decimal point.
     ::在您的答复中,从右向左计算相同数目的位置,并插入小数点。

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  Here is an example.
  ::举一个例子。

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  Find the product of 3.25 and 1.2.
  ::查找3.25和1.2的产物。

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  First, arrange the numbers vertically so that their right-most digits line up.
  ::首先,垂直排列数字,以便他们的最右位数排成一排。

  $$\begin{array}{r}\begin{array}{rcl}& & 3.25\\ & & \underset{\_}{\times 1.2}\end{array}\end{array}$$

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  Next, temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.
  ::下一步,暂时忽略小数点,然后像乘整数一样乘以乘以整数。

  $$\begin{array}{r}\begin{array}{rcl}& & 3.25\\ & & \underset{\_}{\times 1.2}\\ & & 650\\ & & \underset{\_}{+3250}\\ & & 3900\end{array}\end{array}$$

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  Now, count the total number of places after the decimal points in the original two numbers.
  ::现在,在最初的两个数字中,计算小数点之后的位数总数。

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  • 3.25 has 2 digits after the decimal point.
    ::3.25小数点后有两个位数。
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  • 1.2 has 1 digit after the decimal point.
    ::1.2 在小数点后为1位数。

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  $\begin{array}{r}2+1=3\end{array}$ , so there are 3 digits total after the decimal points in the original numbers.
  ::2 + 1 = 3, 所以在原数小数点之后, 总共有三个位数。

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  Finally, count 3 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 3 and the 9.
  ::最后,在答案中从右到左倒数三位数,然后插入小数点。您将把小数点放在三点和九点之间。

  $$\begin{array}{r}3.900\end{array}$$

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  The answer is  $\begin{array}{r}3.25\times 1.2=3.9\end{array}$ . Notice that you don’t have to write the zeros at the end of the answer after the decimal point if you don’t want to. 3.900 is the same as 3.9.
  ::答案是 3. 25 × 1.2 = 3. 9 。 请注意, 如果您不想, 无需在小数点后填写答案结尾处的零。 3. 900 和 3. 9 相同 。

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  Sometimes you will want to  round  numbers before multiplying them. Rounding is useful when you only need an approximate answer instead of an exact answer.
  ::有时您会想要在乘以数字之前圆圆数字。 当您只需要大致答案而不是精确答案时, 圆圆是有用的 。

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  Here is an example.
  ::举一个例子。

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  Round the numbers to the nearest hundredth and then find the product.
  ::将数字四舍五入到最近的第一百位,然后找到产品。

  $\begin{array}{r}3.748\times 8.095\end{array}$

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  First, round each number to the nearest hundredth. Remember that the hundredths place is the second digit to the right of the decimal point.
  ::首先, 将每个数字轮到最近的一百位。 记住, 第一百位位是小数点右侧的第二位数 。

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  • 3.748 rounds to 3.75
    ::3.748发至3.75子弹
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  • 8.095 rounds to 8.10
    ::8.095发子弹至8.10

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  Now, multiply the rounded numbers. Start by arranging the numbers vertically so that their right-most digits line up.
  ::现在,乘以四舍五入的数字。从垂直排列数字开始,这样他们最右侧的数字就会排成一行。

  $$\begin{array}{r}\begin{array}{rcl}& & 3.75\\ & & \underset{\_}{\times 8.10}\end{array}\end{array}$$

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  Next, temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.
  ::下一步,暂时忽略小数点,然后像乘整数一样乘以乘以整数。

  $$\begin{array}{r}\begin{array}{rcl}& & 3.75\\ & & \underset{\_}{\times 8.10}\\ & & 0\\ & & 3750\\ & & \underset{\_}{+300000}\\ & & 303750\end{array}\end{array}$$

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  Now, count the total number of places after the decimal points in the original two numbers. There are 4 digits total after the decimal points in the original numbers.
  ::现在,在原两个数字中计算小数点之后的位数总数。在原数字中,小数点之后的总位数为4位数。

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  Finally, count 4 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 0 and the 3.
  ::最后,在答案中从右到左倒数四个位数,然后插入小数点。您将把小数点位于 0 和 3 之间。

  $$\begin{array}{r}30.3750\end{array}$$

   

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  The answer is  $\begin{array}{r}3.75\times 8.10=30.375\end{array}$ .
  ::答案是3.75 × 8.10 = 30.375。

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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 Jackson and his peaches.
  ::早些时候,你被问及 杰克逊和他的桃子的问题

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  Jackson and his mom picked 15.8 pounds of peaches and the peaches cost $2.50 per pound. Jackson needs to calculate how much their peaches will cost.
  ::杰克逊和他的妈妈挑选了15.8磅的桃子,桃子每磅花费2.50美元。 杰克逊需要计算他们的桃子的成本。

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  To figure out the total cost, Jackson needs to multiply 2.50 by 15.8.
  ::要弄清楚总成本 杰克逊需要乘以2.50乘以15.8

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  First, Jackson should arrange the numbers vertically so that their right-most digits line up.
  ::首先,杰克逊应该垂直排列数字 以便他们的最右位数排成一排

  $$\begin{array}{r}\begin{array}{rcl}& & 2.50\\ & & \underset{\_}{\times 15.8}\end{array}\end{array}$$

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  Next, he should temporarily ignore the decimal points and multiply just as if he were multiplying whole numbers.
  ::接下来,他应该暂时忽略小数点,然后像乘整数一样乘以乘法。

  $$\begin{array}{r}\begin{array}{rcl}& & 2.50\\ & & \underset{\_}{\times 15.8}\\ & & 2000\\ & & 12500\\ & & \underset{\_}{+25000}\\ & & 39500\end{array}\end{array}$$

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  Now, he should count the total number of places after the decimal points in the original two numbers. There are 3 digits total after the decimal points in the original numbers.
  ::现在, 他应该计算两位数中小数点后的总位数。 在两位数中, 小数点后共有三位数。 在两位数中, 小数点后共有三位数。

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  Finally, he should count 3 places from right to left in the answer and insert a decimal point. He will place the decimal point between the 9 and the 5.
  ::最后, 他应该在答案中从右到左点数三个位数, 并插入小数点 。 他将把小数点放在 9 和 5 之间。

  $$\begin{array}{r}39.500\end{array}$$

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  The answer is the peaches will cost $39.50.
  ::答案是桃子将花费39.50美元

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  Example 2
  ::例2

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  Round the numbers to the nearest tenth and then multiply.
  ::将数字四舍五入到最接近的十分之一,然后乘以乘数。

  $\begin{array}{r}5.68\times 1.38\end{array}$

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  First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point.
  ::首先,将每个数字四舍五入到最接近的十分之一。记住,十分位数是小数点右侧的第一个位数。

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  • 5.68 rounds to 5.7
    ::5.68至5.7子弹
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  • 1.38 rounds to 1.4
    ::1.38至1.4发子弹

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  Now, multiply the rounded numbers. Start by arranging the numbers vertically so that their right-most digits line up.
  ::现在,乘以四舍五入的数字。从垂直排列数字开始,这样他们最右侧的数字就会排成一行。

  $$\begin{array}{r}\begin{array}{rcl}& & 5.7\\ & & \underset{\_}{\times 1.4}\end{array}\end{array}$$

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  Next, temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.
  ::下一步,暂时忽略小数点,然后像乘整数一样乘以乘以整数。

  $$\begin{array}{r}\begin{array}{rcl}& & 5.7\\ & & \underset{\_}{\times 1.4}\\ & & 228\\ & & \underset{\_}{+570}\\ & & 798\end{array}\end{array}$$

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  Now, count the total number of places after the decimal points in the original two numbers. There are 2 digits total after the decimal points in the original numbers.
  ::现在,在原两个数字中计算小数点后的总位数。在原数字中,小数点后的总位数为2位数。在原数字中,小数点后的总位数为2位数。

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  Finally, count 2 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 7 and the 9.
  ::最后,在答案中从右到左倒数两个位数,然后插入小数点。您将把小数点放在7和9之间。

  $$\begin{array}{r}7.98\end{array}$$

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  The answer is  $\begin{array}{r}5.7\times 1.4=7.98\end{array}$ .
  ::答案是5.7×1.4=7.98。

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  Example 3
  ::例3

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  Find the product of 1.23 and 6.7.
  ::查找1.23和6.7的产物。

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  First, arrange the numbers vertically so that their right-most digits line up.
  ::首先,垂直排列数字,以便他们的最右位数排成一排。

  $$\begin{array}{r}\begin{array}{rcl}& & 1.23\\ & & \underset{\_}{\times 6.7}\end{array}\end{array}$$

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  Next, temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.
  ::下一步,暂时忽略小数点,然后像乘整数一样乘以乘以整数。

  $$\begin{array}{r}\begin{array}{rcl}& & 1.23\\ & & \underset{\_}{\times 6.7}\\ & & 861\\ & & \underset{\_}{+7380}\\ & & 8241\end{array}\end{array}$$

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  Now, count the total number of places after the decimal points in the original two numbers.
  ::现在,在最初的两个数字中,计算小数点之后的位数总数。

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  • 1.23 has 2 digits after the decimal point.
    ::1.23小数点后有两个位数。
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  • 6.7 has 1 digit after the decimal point.
    ::6.7小数点后为1位数。

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  $\begin{array}{r}2+1=3\end{array}$ , so there are 3 digits total after the decimal points in the original numbers.
  ::2 + 1 = 3, 所以在原数小数点之后, 总共有三个位数。

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  Finally, count 3 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 8 and the 2.
  ::最后,在答案中从右到左倒数三位数,然后插入小数点。您将把小数点放在8和2之间。

  $$\begin{array}{r}8.241\end{array}$$

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  The answer is  $\begin{array}{r}1.23\times 6.7=8.241\end{array}$ .
  ::答案是1.23×6.7=8.241。

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  Example 4
  ::例4

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  Find the product of 4.56 and 1.34.
  ::查找4.56和1.34的产物。

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  First, arrange the numbers vertically so that their right-most digits line up.
  ::首先,垂直排列数字,以便他们的最右位数排成一排。

  $$\begin{array}{r}\begin{array}{rcl}& & 4.56\\ & & \underset{\_}{\times 1.34}\end{array}\end{array}$$

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  Next, temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.
  ::下一步,暂时忽略小数点,然后像乘整数一样乘以乘以整数。

  $$\begin{array}{r}\begin{array}{rcl}& & 4.56\\ & & \underset{\_}{\times 1.34}\\ & & 1824\\ & & 13680\\ & & \underset{\_}{+45600}\\ & & 61104\end{array}\end{array}$$

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  Now, count the total number of places after the decimal points in the original two numbers. There are 4 digits total after the decimal points in the original numbers.
  ::现在,在原两个数字中计算小数点之后的位数总数。在原数字中,小数点之后的总位数为4位数。

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  Finally, count 4 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 6 and the 1.
  ::最后,在答案中从右到左倒数四个位数,然后插入小数点。您将把小数点放在6和1之间。

  $$\begin{array}{r}6.1104\end{array}$$

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  The answer is  $\begin{array}{r}4.56\times 1.34=6.1104\end{array}$ .
  ::答案是4.56×1.34=6.1104。

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  Example 5
  ::例5

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  Round the numbers to the nearest tenth and then multiply.
  ::将数字四舍五入到最接近的十分之一,然后乘以乘数。

  $\begin{array}{r}5.67\times 4.35\end{array}$

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  First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point.
  ::首先,将每个数字四舍五入到最接近的十分之一。记住,十分位数是小数点右侧的第一个位数。

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  • 5.67 rounds to 5.7
    ::5.67至5.7发子弹
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  • 4.35 rounds to 4.4
    ::4.35发至4.4

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  Now, multiply the rounded numbers. Start by arranging the numbers vertically so that their right-most digits line up.
  ::现在,乘以四舍五入的数字。从垂直排列数字开始,这样他们最右侧的数字就会排成一行。

  $$\begin{array}{r}\begin{array}{rcl}& & 5.7\\ & & \underset{\_}{\times 4.4}\end{array}\end{array}$$

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  Next, temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.
  ::下一步,暂时忽略小数点,然后像乘整数一样乘以乘以整数。

  $$\begin{array}{r}\begin{array}{rcl}& & 5.7\\ & & \underset{\_}{\times 4.4}\\ & & 228\\ & & \underset{\_}{+2280}\\ & & 2508\end{array}\end{array}$$

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  Now, count the total number of places after the decimal points in the original two numbers. There are 2 digits total after the decimal points in the original numbers.
  ::现在,在原两个数字中计算小数点后的总位数。在原数字中,小数点后的总位数为2位数。在原数字中,小数点后的总位数为2位数。

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  Finally, count 2 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 5 and the 0.
  ::最后,在答案中从右到左倒数两个位数,然后插入小数点。您将把小数点位于5和0之间。

  $$\begin{array}{r}25.08\end{array}$$

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  The answer is  $\begin{array}{r}5.7\times 4.4=25.08\end{array}$ .
  ::答案是5.7×4.4=25.08。

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  Review
  ::回顾

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  Find the products.
  ::找到产品。

  1. $\begin{array}{r}12.7\times 0.8\end{array}$  
  2. $\begin{array}{r}0.552\times 0.3\end{array}$  
  3. $\begin{array}{r}6.09\times 3.34\end{array}$  
  4. $\begin{array}{r}25.6\times 0.72\end{array}$  
  5. $\begin{array}{r}56.71\times .34\end{array}$  
  6. $\begin{array}{r}.45\times 4.3\end{array}$  
  7. $\begin{array}{r}1.234\times 7.8\end{array}$  

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  Find the product after rounding each decimal to the nearest tenth.
  ::将小数点后每一小数点四舍五入到最接近的十分之一后查找产品。

  8. $\begin{array}{r}33.076\times 5.228\end{array}$  
  9. $\begin{array}{r}9.29\times 0.6521\end{array}$  
  10. $\begin{array}{r}4.5513\times 4.874\end{array}$  
  11. $\begin{array}{r}12.48\times 7.95\end{array}$  
  12. $\begin{array}{r}14.56\times 4.52\end{array}$  
  13. $\begin{array}{r}8.76\times 1.24\end{array}$  
  14. $\begin{array}{r}9.123\times 6.789\end{array}$  
  15. $\begin{array}{r}9.3323\times 8.719\end{array}$  

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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 。

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  Resources
  ::资源