2.11检查小数乘法的估计

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  检查小数乘法的估计

  

  [Figure 1]

    

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  When Dylan‘s baby sister was born last month she weighed 6 pounds 8.7 ounces, or 6.54375 pounds. Dylan‘s mom said that his sister will probably triple her weight by the time she is a year old. How could Dylan estimate what his sister will weigh when she is one year old?
  ::当Dylan的妹妹上个月出生时,她体重6磅8.7盎司,或6.54375磅。 Dylan的妈妈说,他妹妹在一岁时可能会将体重提高两倍。 Dylan如何估计她一岁时的体重?

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  In this concept, you will learn to estimate decimal products by multiplying leading digits.
  ::在此概念中,您将学会通过乘以前两位数来估计小数点的产品。

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  Estimating Decimal Products
  ::估计十进制产品

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  Recall that when you  estimate  you are finding an approximate solution to a problem. One good time to use  estimation  is when you want to check your answer to a problem. If you estimate a solution to a problem either before or after you solve it, you can see whether or not your answer is realistic.
  ::记得当您估计您正在找到一个近似问题的解决方案时。 使用估算的好时机是您想要检查一个问题的答案。 如果您在解决问题之前或之后估计一个问题的解决方案, 您可以看到答案是否现实。

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  When you are working with decimal numbers with many digits, one way to estimate their product is to multiply only their two leading digits.
  ::当您使用多位数的十进制数字计算小数时,一种估计其产品的方法就是只乘以其两位前两位数。

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  Here are the steps for multiplying decimal numbers using leading digits.
  ::这是使用前两位数乘以小数的步数。

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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. Multiply the leading digits using your steps for decimal multiplication.
     ::使用小数乘法的步数乘以前两位数。

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  Here is an example.
  ::举一个例子。

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  Estimate the product of  $\begin{array}{r}6.42\times 0.383\end{array}$ .
  ::估计产值为6.42×0.383。

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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 6.42 are 6.4
    ::6.42的前位数字为6.4。
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  • The leading digits of 0.383 are 0.38
    ::0.383位前数为0.38

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  Note that because 0 is the only digit to the left of the decimal point in the second number, it does not count as one of the leading digits.
  ::请注意,由于 0 是第二个数字中小数点左侧唯一的位数, 它不计为前位数之一 。

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  Now, multiply just as if you were multiplying  whole numbers . Because the original numbers have 3 digits total after their decimal points, insert a decimal point into your answer so that it has 3 digits to its right.
  ::现在, 乘以整个数字。 因为原数字在小数点后总共有三个位数, 而在您的答复中插入一个小数点, 以便右侧有三个位数 。

  $$\begin{array}{r}\begin{array}{rc}& 6.4\\ \times & \underset{\_}{.38}\\ & 512\\ +& \underset{\_}{1920}\\ & 2.432\end{array}\end{array}$$

  The answer is the product of 6.42 and 0.383 is approximately 2.432.

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

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  Find the product. Then estimate to confirm your solution. $\begin{array}{r}22.17\times 4.45\end{array}$  .
  ::寻找产品。然后估算确认您的解决方案。 22. 17 × 4. 45 。

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  First, notice that this problem asks you to do two things. You will need to multiply and then estimate the solution. Start by multiplying just as if you were multiplying whole numbers. Because the original numbers have 4 digits total after their decimal points, insert a decimal point into your answer so that it has 4 digits to its right.
  ::首先,请注意这个问题要求您做两件事。 您需要乘数, 然后估计解决方案。 开始的乘数就像您正在乘整数一样。 因为最初的数字在小数点之后有四位数的总和, 请在您的答案中插入一个小数点, 以便右侧有四位数 。

  $$\begin{array}{r}\begin{array}{rc}& 22.17\\ \times & \underset{\_}{4.45}\\ & 11085\\ & 88680\\ +& \underset{\_}{886800}\\ & 98.6565\end{array}\end{array}$$

  Next, estimate the answer by multiplying only the leading digits. Identify the two leading digits of each number.

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  • The leading digits of 22.17 are 22
    ::22.17位前数字为22位
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  • The leading digits of 4.45 are 4.4
    ::4.45位前数为4.4位。

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  Now, multiply the leading digits.
  ::现在,乘以前两位数。

  $$\begin{array}{r}\begin{array}{rc}& 22\\ \times & \underset{\_}{4.4}\\ & 88\\ +& \underset{\_}{880}\\ & 96.8\end{array}\end{array}$$

  Next, compare your answer with your estimate. Your answer of 98.6565 is close to your estimate of 96.8. This means you can be confident that your answer is correct.

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  The answer is that the exact product of  $\begin{array}{r}22.17\times 4.45\end{array}$  is 98.6565, while the estimate of the product is 96.8.
  ::答案是22.17×4.45的准确产值是98.665,而产品估计值是96.8。

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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 Dylan and his baby sister.
  ::早些时候,你得到一个问题 关于迪伦和他的小妹妹。

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  When she was born she weighed 6.54375 pounds and she will probably triple her weight by the time she is a year old. Dylan wants to estimate how much she will weigh when she is a year old.
  ::当她出生时,她体重为6.54375磅,到她一岁时,她可能将体重提高三倍。 迪伦想估计她一岁时的体重。

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  First, Dylan needs to realize that if his sister triples her weight that means her original weight will be multiplied by 3. Dylan is trying to find the product of  $\begin{array}{r}6.54375\times 3\end{array}$ .
  ::首先,迪伦需要意识到 如果他的妹妹的体重是她的三倍 这意味着她的原体重会乘以3 迪伦正在寻找6.54375×3的产物

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  Now, Dylan can estimate the product by multiplying the leading digits. He should identify the two leading digits of each number.
  ::现在,迪伦可以通过乘以前两位数来估计产品。 他应该确定每个数字的两位前两位数。

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  • The leading digits of 6.54375 are 6.5
    ::6.54375的前位数字为6.5。
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  • The leading digits of 3 are 3
    ::3位的前三位数是3位

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  Note that because 3 only had one digit to start with, it only has one leading digit!
  ::请注意,因为3只有一个数字要开始,它只有一个头号数字!

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  Next, Dylan should multiply just as if he was multiplying whole numbers. Because the original numbers have 1 digit total after their decimal points, he should insert a decimal point into his answer so that it has 1 digit to its right.
  ::接下来,迪伦应该像他正乘以整个数字一样进行乘法。因为原数字在小数点后的总和是1位数,因此他应该在回答中插入一个小数点,以便右侧有一个小数点。

  $$\begin{array}{r}\begin{array}{rc}& 6.5\\ \times & \underset{\_}{3}\\ & 19.5\end{array}\end{array}$$

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  The answer is that Dylan's baby sister will likely weigh approximately 19.5 pounds when she is a year old.
  ::答案是,迪伦的妹妹的妹妹 可能会体重约19.5磅 当她一岁。

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  Example 2
  ::例2

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  Estimate the product of  $\begin{array}{r}0.4561\times 0.32109\end{array}$ .
  ::估计产值为0.4561×0.32109。

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  First, identify the two leading digits of each number. Remember that when the only digit to the left of the decimal point is a 0, it does not count as a  leading digit .
  ::首先, 标明每个数字的两个前两位数。 记住, 当小数点左侧的唯一一位数是 0 时, 它不会算作一个前两位数 。

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  • The leading digits of 0.4561 are 0.45
    ::0.4561的前位数字为0.455
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  • The leading digits of 0.32109 are 0.32
    ::0.32109的前位数字为0.32109,前位数字为0.32

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  Now, multiply just as if you were multiplying whole numbers. Because the original numbers have 4 digits total after their decimal points, insert a decimal point into your answer so that it has 4 digits to its right.
  ::现在, 乘以您正在乘以整个数字。 因为原始数字在小数点之后总和为4位数, 请在您的回答中插入一个小数点, 这样它右侧就有4位数 。

  $$\begin{array}{r}\begin{array}{rc}& .45\\ \times & \underset{\_}{.32}\\ & 90\\ +& \underset{\_}{1350}\\ & .1440\end{array}\end{array}$$

  Your estimate is .1440 which is the same as 0.144.

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  The answer is that the product of 0.4561 and 0.32109 is approximately 0.1440.
  ::答案是0.4561和0.32109的产物约为0.1440。

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  Example 3
  ::例3

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  Find the product. Then estimate to confirm your solution. $\begin{array}{r}6.79\times 1.2\end{array}$  .
  ::寻找产品。然后估计确认你的溶液。 6.79 × 1. 2 。

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  First, multiply to find the exact answer. You will multiply just as if you were multiplying whole numbers. Because the original numbers have 2 digits total after their decimal points, insert a decimal point into your answer so that it has 2 digits to its right.
  ::首先,乘以找到准确的答案。你会像乘以整个数字一样倍增。因为原数字在小数点后的总和为2位数,所以在回答中插入一个小数点,以便右侧有2位数。

  $$\begin{array}{r}\begin{array}{rc}& 67.9\\ \times & \underset{\_}{1.2}\\ & 1358\\ +& \underset{\_}{6790}\\ & 81.48\end{array}\end{array}$$

   Next, estimate the answer by multiplying the leading digits. Identify the two leading digits of each number.

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  • The leading digits of 67.9 are 67
    ::67.9个前位数为67.9个前位数为67个
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  • The leading digits of 1.2 are 1.2
    ::1.2位前数为1.2位前数

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  Now, multiply the leading digits.
  ::现在,乘以前两位数。

  $$\begin{array}{r}\begin{array}{rc}& 67\\ \times & \underset{\_}{1.2}\\ & 134\\ +& \underset{\_}{670}\\ & 80.4\end{array}\end{array}$$

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  Next, compare your answer with your estimate. Your answer of 81.48 is close to your estimate of 80.4. This means you can be confident that your answer is correct.
  ::接下来,请比较您的答案与您的估计。您对81.48的回答接近于您对80.4的估算。这意味着您可以确信您的回答是正确的。

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  The answer is that the exact product of  $\begin{array}{r}67.9\times 1.2\end{array}$  is 81.48, while the estimate of the product is 80.4.
  ::答案是67.9×1.2的准确产值为81.48,而产品的估计值为80.4。

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  Example 4
  ::例4

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  Estimate the product of  $\begin{array}{r}5.321\times 2.301\end{array}$ .
  ::估计产值为5.321 × 2.301。

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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 5.321 are 5.3
    ::5.321位前数为5.321位前数。
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  • The leading digits of 2.301 are 2.3
    ::2.31的前两位数是2.3

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  Now, multiply just as if you were multiplying whole numbers. Because the original numbers have 2 digits total after their decimal points, insert a decimal point into your answer so that it has 2 digits to its right.
  ::现在, 乘以整个数字。 因为原始数字在小数点后的总和为2位数, 所以在您的回答中插入一个小数点, 这样它右侧就有2位数 。

  $$\begin{array}{r}\begin{array}{rc}& 5.3\\ \times & \underset{\_}{2.3}\\ & 159\\ +& \underset{\_}{1060}\\ & 12.19\end{array}\end{array}$$

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  The answer is that the product of 5.321 and 2.301 is approximately 12.19.
  ::答案是5.321和2.301的产物大约为12.19。

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  Example 5
  ::例5

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  Find the product. Then estimate to confirm your solution.  $\begin{array}{r}9.1204\times 8.713\end{array}$ .
  ::寻找产品。然后估算确认你的溶液。 9.1204 × 8. 713 。

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  First, multiply to find the exact answer. You will multiply just as if you were multiplying whole numbers. Because the original numbers have 7 digits total after their decimal points, insert a decimal point into your answer so that it has 7 digits to its right.
  ::首先,乘以找到准确的答案。你会像乘以整个数字一样倍增。因为最初的数字在小数点之后总共是7位数,所以在回答中插入一个小数点,以便右边有7位数。

  $$\begin{array}{r}\begin{array}{rc}& 9.1204\\ \times & \underset{\_}{8.713}\\ & 273612\\ & 912040\\ & 63842800\\ +& \underset{\_}{729632000}\\ & 79.4660452\end{array}\end{array}$$

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  Next, estimate the answer by multiplying the leading digits. Identify the two leading digits of each number.
  ::接下来,通过乘以前两位数来估计答案。 确定每个数字的两位前两位数 。

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  • The leading digits of 9.1204 are 9.1
    ::9.1204位前数字为9.120位,前数字为9.1位。
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  • The leading digits of 8.713 are 8.7
    ::8.713位前数为8.77位数。

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  Now, multiply the leading digits.
  ::现在,乘以前两位数。

  $$\begin{array}{r}\begin{array}{rc}& 9.1\\ \times & \underset{\_}{8.7}\\ & 637\\ +& \underset{\_}{7280}\\ & 79.17\end{array}\end{array}$$

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  Next, compare your answer with your estimate. Your answer of 79.4660452 is close to your estimate of 79.17. This means you can be confident that your answer is correct.
  ::接下来,请比较您的答案与您的估计。 您的79. 4660452的回答接近您79.17的估算。 这意味着您可以确信您的回答是正确的 。

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  The answer is that the exact product of  $\begin{array}{r}9.1204\times 8.713\end{array}$  is 79.4660452, while the estimate of the product is 79.17.
  ::答案是,9.1204×8.713的准确产品为79.4660452,而产品的估计值为79.17。

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  Review
  ::回顾

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  Estimate the products by multiplying the leading digits.
  ::通过乘以前两位数来估计产品。

  1. $\begin{array}{r}7.502\times 0.9281\end{array}$
  2. $\begin{array}{r}46.14\times 2.726\end{array}$
  3. $\begin{array}{r}0.39828\times 0.16701\end{array}$
  4. $\begin{array}{r}83.243\times 6.517\end{array}$
  5. $\begin{array}{r}5.67\times .987\end{array}$
  6. $\begin{array}{r}7.342\times 1.325\end{array}$
  7. $\begin{array}{r}17.342\times .325\end{array}$
  8. $\begin{array}{r}.34291\times 1.525\end{array}$
  9. $\begin{array}{r}.5342\times .87325\end{array}$
  10. $\begin{array}{r}.38942\times .9825\end{array}$
  11. $\begin{array}{r}7.567\times 3.325\end{array}$
  12. $\begin{array}{r}12.342\times 11.325\end{array}$
  13. $\begin{array}{r}21.342\times 14.555\end{array}$
  14. $\begin{array}{r}.110342\times .098325\end{array}$
  15. $\begin{array}{r}37.1342\times 1.97325\end{array}$

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  Review (Answers)
  ::回顾(答复)