7.2整数和分数的乘积估计

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  整数和分数的乘积估计

  
  

  

  [Figure 1]

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  Johnny is thinking about how much he sleeps. He estimates that a person spends about a third of the day asleep. About how many days does a a person spend asleep over the course of one year? 
  ::Johnny在想他睡了多少觉。他估计一个人睡了大约三分之一的一天。一个人睡了多少天?一个人在一年的时间里睡了多少天?

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  In this concept, you will learn how to estimate the products of  whole numbers  and fractions.
  ::在这个概念中,你将学会如何估计整个数字和分数的产品。

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  Estimating Products with Whole Numbers and Fractions
  ::带有整数和分数的估算产品

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  You can estimate products of whole numbers and fractions. When you  estimate , you are looking for an answer that is reasonable but need not be exact.
  ::您可以估计整个数字和分数的产品。 当您估算时, 您正在寻找一个合理但不需要准确的答案 。

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  Round  the whole number to the closest  compatible number  to the  denominator .  Compatible numbers  are numbers that are close in value to the real number that would make it easier to find an estimate calculation. 
  ::将整数舍入到与分母最接近的相容数字。相容数字是接近于实际数字的数字,这样就更容易找到估算值计算方法。

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  Here is a multiplication problem. 
  ::这是一个乘法问题。

  $$\begin{array}{r}395\times 103\end{array}$$

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  You can find an estimate of the product by  rounding  the numbers to the nearest hundreds place. Compatible numbers for 395 and 103 are 400 and 100. 
  ::通过四舍五入到最接近的数百个地方,可以找到产品估计值。395和103的对应数字是400和100。

  $$\begin{array}{r}400\times 100=40,000\end{array}$$

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  To estimate the product of a whole number and a fraction, round the whole number to a number compatible to the denominator of the fraction or round the fraction a fraction compatible to the whole number.
  ::要估计整数和分数的产值,将整数折成一个与分数分母相容的数字,或将分数折成一个与整数相容的分数。

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  Here is a multiplication problem involving a fraction and whole number. 
  ::这是一个涉及一个分数和整数的乘法问题。

  $$\begin{array}{r}\frac{3}{9}\times 11=\underset{\_}{}\end{array}$$

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  First, think about the fraction three-ninths. Three-ninths simplifies to one-third.
  ::首先,考虑一下三十九分之一,三十九分之一简化到三分之一

  $$\begin{array}{r}\frac{1}{3}\times 11=\underset{\_}{}\end{array}$$

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  Note that multiplying by one-third is also the same as  . You can round the whole number to a number that is compatible to the denominator of the fraction, 3. A compatible whole number would be a  multiple  of 3 that is close to 11. 11 is close to 9 and 12. 12 is the closer compatible number. 
  ::注意乘以 三分之一 也和 相同。 您可以将整数舍入到一个与分数分母相容的数, 3 。 一个兼容的整数将是3的倍数, 接近11 11 11 接近9, 12 接近12 。

  $$\begin{array}{r}\frac{1}{3}\times 12=\underset{\_}{}\end{array}$$

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  Now, multiply to find an estimate product. 
  ::现在,乘以找到估计产品。

  $$\begin{array}{r}\frac{1}{3}\times 12=4\end{array}$$

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  An estimate product is 4.
  ::估计产值为4。

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  Here is another problem. 
  ::还有一个问题。

  $$\begin{array}{r}\frac{5}{16}\times 20=\underset{\_}{}\end{array}$$

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  First, round the fraction to a fraction that is easy to multiply with 20. Four-sixteenths is close to five-sixteenths and simplifies to one-fourth. 20 is divisible by four. Round five-sixteenths to one-fourth. 
  ::首先,将分数回合到一个容易乘以20的分数,四十六分之一接近五十六分之一,简化到十四分之一。 二十分之一除以四。 回合五十六分之一到十四分之一。

  $$\begin{array}{rl}\frac{5}{16}\approx \frac{4}{16}& =\frac{1}{4}\\ \frac{1}{4}\times 20\end{array}$$

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  Then, multiply to find an estimate product.
  ::然后,乘以找到估计产品。

  $$\begin{array}{r}\frac{1}{4}\times 20=5\end{array}$$

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  An estimate product is 5.
  ::估计产品为5。

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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 Johnny's sleep.
  ::早些时候,你得到一个问题 关于强尼的睡眠。

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  Johnny wants to know about how many days a person spends asleep in a year. There are 365 days in a year. To find an estimate, multiply 365 by one-third. 
  ::Johnny想知道一个人在一年中睡了多少天。一年中有365天。为了找到估计数字,365乘以三分之一。

  $$\begin{array}{r}365\times \frac{1}{3}\end{array}$$

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  First, round the whole number to a number compatible with 3. Remember that the  divisibility rules  for 3 states that a number is divisible by 3 if the  sum  of all the digits is also divisible by 3. Round 365 to 366. 
  ::首先,将整个数字转换为与3. 相符的数字。 记住, 3 的可分性规则规定,如果所有数字的总和除以3 365至366, 一个数字可以除以3 。

  $$\begin{array}{r}365\approx 366\end{array}$$

  $$\begin{array}{r}366\times \frac{1}{3}\end{array}$$

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  Then, multiply to find the product. Multiply by one-third is the same as dividing by 3.
  ::然后,乘以找到产品。乘以三分之一等于除以3。

   

  $$\begin{array}{r}366\times \frac{1}{3}=122\end{array}$$

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  A person spends about 122 days asleep in a year. 
  ::一个人在一年中睡了大约122天。

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  Example 2
  ::例2

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  Estimate the following product.
  ::估计以下产品。

  $\begin{array}{r}\frac{1}{2}\times 281\end{array}$

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  First, round the whole number or the fraction. You can round 281 to a number compatible to the denominator 2.
  ::首先,将整数或分数四舍五入。您可以将281圆为与分母 2 兼容的数。

  $$\begin{array}{r}281\approx 280\end{array}$$

  $$\begin{array}{r}\frac{1}{2}\times 280\end{array}$$

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  Then, multiply 280 by one-half. 
  ::然后将280乘以一半

  $$\begin{array}{r}\frac{1}{2}\times 280=140\end{array}$$

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  An estimate product is 140.
  ::估计产品为140。

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  Example 3
  ::例3

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  Estimate the product:  $\begin{array}{r}10\times \frac{3}{8}=\underset{\_}{}\end{array}$ .
  ::估计产品:10×3-8=_。

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  First, look at the fraction. Round three-eighths to a compatible number.
  ::首先,看看分数,第三回合八分之三到一个匹配的数字

  $$\begin{array}{r}\frac{3}{8}\approx \frac{4}{8}=\frac{1}{2}\end{array}$$

      

  $$\begin{array}{r}10\times \frac{1}{2}\end{array}$$

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  Then, multiply to find the product. 
  ::然后,乘以找到产品。

  $$\begin{array}{r}10\times \frac{1}{2}=5\end{array}$$

  
  

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  An estimate product is 5. 
  ::估计产品为5。

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  Example 4
  ::例4

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  Estimate the product:  $\begin{array}{r}\frac{1}{2}\times 19=\underset{\_}{}\end{array}$ .
  ::估计产品:1 2×19=_。

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  First, round the whole number to a number compatible to the denominator, 2.
  ::首先,将整数转换为一个与分母相容的数, 2。

   

  $$\begin{array}{r}\frac{1}{2}\times 18\end{array}$$

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  Then, multiply to find the product.  
  ::然后,乘以找到产品。

  $$\begin{array}{r}\frac{1}{2}\times 18=\frac{1}{\cancel{2}}\times \frac{\cancel{18}}{1}=\frac{1}{1}\times \frac{9}{1}=9\end{array}$$

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  An estimate product is 9.
  ::估计产值为9。

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  Example 5
  ::例5

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  Estimate the product:  $\begin{array}{r}\frac{3}{4}\times 78=\underset{\_}{}\end{array}$ .
  ::估计产品:3 4×78=_。

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  First, round the whole number to a number compatible to the denominator, 4. 
  ::首先,将整个数字转成一个与分母4兼容的数字。

  $$\begin{array}{r}\frac{3}{4}\times 80\end{array}$$

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  Then, multiply to find the product. 
  ::然后,乘以找到产品。

  $$\begin{array}{r}\frac{3}{4}\times 80=\frac{3}{\cancel{4}}\times \frac{\cancel{80}}{1}=\frac{3}{1}\times \frac{20}{1}=60\end{array}$$

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  An estimate product is 60.
  
  ::估计产品为60。

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  Review
  ::回顾

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  Estimate the products. 
  ::估计产品。

  1. $\begin{array}{r}6\times \frac{1}{2}=\underset{\_}{}\end{array}$
  2. $\begin{array}{r}16\times \frac{1}{2}=\underset{\_}{}\end{array}$
  3. $\begin{array}{r}26\times \frac{1}{2}=\underset{\_}{}\end{array}$
  4. $\begin{array}{r}36\times \frac{1}{3}=\underset{\_}{}\end{array}$
  5. $\begin{array}{r}40\times \frac{1}{5}=\underset{\_}{}\end{array}$
  6. $\begin{array}{r}20\times \frac{1}{4}=\underset{\_}{}\end{array}$
  7. $\begin{array}{r}30\times \frac{1}{2}=\underset{\_}{}\end{array}$
  8. $\begin{array}{r}100\times \frac{1}{10}=\underset{\_}{}\end{array}$
  9. $\begin{array}{r}60\times \frac{1}{3}=\underset{\_}{}\end{array}$
  10. $\begin{array}{r}90\times \frac{1}{3}=\underset{\_}{}\end{array}$
  11. $\begin{array}{r}33\times \frac{1}{11}=\underset{\_}{}\end{array}$
  12. $\begin{array}{r}44\times \frac{1}{4}=\underset{\_}{}\end{array}$
  13. $\begin{array}{r}36\times \frac{1}{12}=\underset{\_}{}\end{array}$
  14. $\begin{array}{r}50\times \frac{1}{25}=\underset{\_}{}\end{array}$
  15. $\begin{array}{r}75\times \frac{3}{4}=\underset{\_}{}\end{array}$

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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