3.6 用两层解决现实世界问题

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  解决两个两等解决现实世界问题

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  Solving Real-World Problems with Two-Step Equations 
  ::解决两个两等解决现实世界问题

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  The hardest part of solving word problems is translating from words to an equation . First, you need to identify the unknown quantity in the word problem . What is the value for which you have to solve? That will be your variable . What operations are described in the problem, and how do they affect the unknown value? Your equation should use mathematics to describe two equal quantities, with the unknown value (in the form of a variable letter) on at least one side of the equation.  Once you have restated the information in the word problem as a mathematical  statement , the actual computation is often the easy part.
  ::解决单词问题最困难的部分是将单词转换为方程。 首先, 您需要确定单词问题中未知的数量。 您需要解决的值是什么 ? 这将是你的变量 。 在问题中描述哪些操作, 它们如何影响未知值 ? 您的方程式应该使用数学来描述两个相等的数量, 至少在方程的一边有未知值( 以可变字母的形式) 。 一旦您将单词问题中的信息重述为数学语句, 实际计算通常比较容易 。

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  Real-World Application: Repair Job 
  ::Real-World 应用程序:修理工作

  

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  An emergency plumber charges $65 as a call-out fee plus an additional $75 per hour. He arrives at a house at 9:30 and works to repair a water tank. If the total repair bill is $196.25, at what time was the repair completed?
  ::紧急水管工在9: 30抵达一栋房屋,修水箱。 如果总修理费为196.25美元,修理何时完成?

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  In order to solve this problem, we collect the information from the text and convert it to an equation.
  ::为了解决这一问题,我们从文本中收集信息,并将其转换成等式。

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  Unknown: time taken in hours – this will be  $\begin{array}{r}x\end{array}$
  ::未知时间: 时间以小时计 - 这将是 x

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  The bill is made up of two parts: a call out fee and a per-hour fee. The call out is a flat fee, and independent of $\begin{array}{r}x\end{array}$ , it’s the same no matter how many hours the plumber works. The per-hour part depends on the number of hours,  $\begin{array}{r}x.\end{array}$  So the total fee is $65 ( regardless of the number of hours ) plus $\begin{array}{r}\mathrm{\$}75x\end{array}$ (where $\begin{array}{r}x\end{array}$ is the number of hours), or $\begin{array}{r}65+75x\end{array}$ .
  ::账单由两部分组成:电话费和每小时费。电话费是平价费,与x无关,无论水管工工作多少小时都是相同的。 每小时费用取决于小时数,x。 所以总费用是65美元(无论小时数多少)加上75美元(在x是小时数的地方),或65+75美元。

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  Looking at the problem again, we also can see that the total bill is $196.25. So our statement of equal quantities (our  equation) is:  $\begin{array}{r}196.25=65+75x\end{array}$ .
  ::再看这个问题,我们也可以看到总帐单是196.25美元。所以我们关于同等数量(我们的方程式)的表述是:196.25=65+75x。

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  Solving for $\begin{array}{r}x\end{array}$ :
  ::x 的解决方案 :

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  $$\begin{array}{rl}196.25& =65+75x\text{Subtract 65 from both sides.}\\ 131.25& =75x\text{Divide both sides by 75.}\\ 1.75& =x\text{The job took 1.75 hours.}\end{array}$$

  
  ::196.25=65+75x65 双方65。 131.25=75x75=75.75=x 双方75.75=x

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  The repair job was completed 1.75 hours after 9:30, so it was completed at 11:15AM.
  ::9时30分后1时75分,修复工作完成,因此于上午11时15分完成。

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  Real-World Application: Predicting Height 
  ::现实世界应用:预测高度

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  When Asia was young, her Daddy marked her height on the door frame every month. Asia’s Daddy noticed that between the ages of one and three, he could predict her height (in inches) by taking her age in months, adding 75 inches, and multiplying the result by one-third. Use this information to answer the following:
  ::当亚洲年轻时,她的爸爸每月在门框上标记她的身高。 亚洲的爸爸注意到,在一到三岁之间,他可以预测她的身高(英寸 ) , 以月计她的年龄,增加75英寸,并将结果乘以三分之一。 使用这些信息回答如下:

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  a) Write an equation linking her predicted height, $\begin{array}{r}h\end{array}$ , with her age in months,  $\begin{array}{r}m\end{array}$ . 
  :a) 写一个方程,将她的预测身高、h(以月计)和她的年龄(以月计)联系起来,m(以月计)。

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  To convert the text to an equation, first determine the type of equation we have. We are going to have an equation that links two variables . Our unknown will change, depending on the information we are given. For example, we could solve for height given age, or solve for age given height. However, the text gives us a way to determine height , so o ur equation will start with “ $\begin{array}{r}h=\end{array}$ ”.
  ::要将文本转换成方程, 首先决定我们拥有的方程类型。 我们将会有一个连接两个变量的方程。 我们未知的方程将会改变, 取决于我们所得到的信息 。 例如, 我们可以解决给定的身高年龄, 或者给定的身高年龄 。 但是, 文本为我们提供了一个确定身高的方法, 因此我们的方程将以“ h=” 开始 。

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  The text tells us that we can predict her height by taking her age in months, adding 75, and multiplying by $\begin{array}{r}\frac{1}{3}\end{array}$ . So our equation is $\begin{array}{r}h=(m+75)\cdot \frac{1}{3}\end{array}$ , or $\begin{array}{r}h=\frac{1}{3}(m+75)\end{array}$ .
  ::文本告诉我们,我们可以预测她的身高, 以月数计她的年龄, 加75, 乘以13。 所以我们的方程是h=(m+75)/13, 或h=13(m+75)。

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  b) Determine her predicted height on her second birthday 
  :b) 确定她第二个生日时的预计身高

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  To predict Asia’s height on her second birthday, we substitute $\begin{array}{r}m=24\end{array}$ into our equation (because 2 years is 24 months) and solve for $\begin{array}{r}h\end{array}$ .
  ::为了预测亚洲在二岁生日时的身高, 我们用m=24来代替我们的等式(因为2年是24个月),

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  $$\begin{array}{rl}h& =\frac{1}{3}(24+75)\\ h& =\frac{1}{3}(99)\\ h& =33\end{array}$$

  
  ::h=13( 24+75) h=13( 99) h=33

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  Asia’s height on her second birthday was predicted to be 33 inches.
  ::亚洲在二岁生日时的身高预计为33英寸。

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  c) Determine at what age she is predicted to reach three feet tall. 
  :c) 确定她预计达到三英尺高的年龄。

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  To determine the predicted age when she reached three feet, substitute $\begin{array}{r}h=36\end{array}$ into the equation and solve for $\begin{array}{r}m\end{array}$ .
  ::为了确定她达到三英尺的预测年龄,在方程中替换 h=36, 并解决 m。

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  $$\begin{array}{rl}36& =\frac{1}{3}(m+75)\\ 108& =m+75\\ 33& =m\end{array}$$

  
  ::36=13(m+75)108=m+7533=m

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  Asia was predicted to be 33 months old when her height was three feet.
  ::亚洲预计33个月大,当时她的身高是3英尺。

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  Real-World Application: Converting Temperatures 
  ::Real- World 应用程序: 转换温度

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  To convert temperatures in Fahrenheit to temperatures in Celsius, take the temperature in degrees Fahrenheit, subtract 32, then divide the result by 1.8. This gives the temperature in degrees Celsius.
  ::将华氏温度转换为摄氏温度,将华氏温度乘以华氏温度,减去32,然后将结果除以1.8。这表示温度以摄氏度计算。

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  a) Write an equation that shows the conversion process.
  :a) 写一个表示转换过程的方程式。

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  The text gives the process to convert Fahrenheit to Celsius. We can write an equation using two variables. We will use $\begin{array}{r}f\end{array}$ for temperature in Fahrenheit, and $\begin{array}{r}c\end{array}$ for temperature in Celsius.
  ::文本给出了将华氏转换为摄氏的流程。 我们可以使用两个变量写入一个方程式 。 我们用 Fahrenheit 的温度使用 f , C 的温度使用 c 。

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  $$\begin{array}{rlrl}& \text{First we take the temperature in Fahrenheit and subtract 32.}& & f-32\\ & \text{Then divide by 1.8.}& & \frac{f-32}{1.8}\\ & \text{This equals the temperature in Celsius.}& & c=\frac{f-32}{1.8}\end{array}$$

  
  ::首先,我们以法赫里内海温取温度,再减去32-32,再除以1.8.f-321.8,这等于摄氏.c=f-321.8的温度。

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  In order to convert from one temperature scale to another, simply substitute in for whichever temperature you know, and solve for the one you don’t know.
  ::为了从一个温度尺度转换到另一个温度尺度, 简单地取代你所知道的温度, 解决你所不知道的温度。

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  b) Convert 50 degrees Fahrenheit to degrees Celsius.
  :b) 将华氏50度转换为摄氏度。

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  To convert 50 degrees Fahrenheit to degrees Celsius, substitute $\begin{array}{r}f=50\end{array}$ into the equation.
  ::将50华氏度转换为摄氏度,在方程中替换 F=50。

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  $$\begin{array}{rl}c& =\frac{50-32}{1.8}\\ c& =\frac{18}{1.8}\\ c& =10\end{array}$$

  
  ::c=50-321.8c=181.8c=10

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  50 degrees Fahrenheit is equal to 10 degrees Celsius.
  ::50华氏度等于10摄氏度。

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  Examples 
  ::实例

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  To convert temperatures in Fahrenheit to temperatures in Celsius, t ake the temperature in degrees Fahrenheit ,  subtract 32 , t hen divide the result by 1.8.
  ::要将华氏温度转换为摄氏温度,以华氏度为温度,减去32,然后将结果除以1.8。

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  Example 1
  ::例1

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  Convert 25 degrees Celsius to degrees Fahrenheit.
  ::将25摄氏度转换为华氏度

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  To convert 25 degrees Celsius to degrees Fahrenheit, substitute $\begin{array}{r}c=25\end{array}$ into the equation:
  ::将25摄氏度转换为华氏度,以 c=25 替代方程:

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  $$\begin{array}{rl}25& =\frac{f-32}{1.8}\\ 45& =f-32\\ 77& =f\end{array}$$

  
  ::25=f-321.845=f-3277=f

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  25 degrees Celsius is equal to 77 degrees Fahrenheit.
  ::25摄氏度等于77华氏度。

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  Example 2
  ::例2

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  Convert -40 degrees Celsius to degrees Fahrenheit.
  ::转换 -40摄氏度到华氏度

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  To convert -40 degrees Celsius to degrees Fahrenheit, substitute $\begin{array}{r}c=-40\end{array}$ into the equation.
  ::将 - 40 摄氏度转换为华氏度, 将c 40 转换为方程 。

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  $$\begin{array}{rl}-40& =\frac{f-32}{1.8}\\ -72& =f-32\\ -40& =f\end{array}$$

  
  ::-40=f-321.8-72=f-32-40=f

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  -40 degrees Celsius is equal to -40 degrees Fahrenheit. (No, that’s not a mistake! This is the one temperature where they are equal.)
  ::-40摄氏度等于-40华氏度。 (不,这不是一个错误。 这是它们相等的温度之一。 )

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  Review 
  ::回顾

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  1. Jade is stranded downtown with only $10 to get home. Taxis cost $0.75 per mile, but there is an additional $2.35 hire charge. Write a formula and use it to calculate how many miles she can travel with her money.
     ::杰德被困在市区,只有10美元回家。出租车费用为每英里0.75美元,但还有2.35美元的租金。写一个公式,用它来计算她能用钱旅行多少英里。
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  2. Jasmin’s dad, Andrew, is planning a surprise birthday party for her. He will hire a bouncy castle, and will provide party food for all the guests. The bouncy castle costs $150 for the afternoon, and the food will cost $3 per person. Andrew has a budget of $300. Write an equation and use it to determine the maximum number of guests he can invite.
     ::Jasmin的爸爸安德鲁(Andrew)正在计划为她举办一个惊喜生日派对。 他将雇用一个充气城堡,并为所有客人提供聚会食物。 充气城堡下午花费150美元,每人3美元。 安德鲁的预算为300美元。 写一个方程式,用它来决定他可以邀请的游客的最多人数。

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  The local amusement park sells summer memberships for $50 each. Normal admission to the park costs $25; admission for members costs $15.
  ::当地游乐园销售夏季会员每人50美元,正常入园费25美元;会员入场费15美元。

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  3. If Darren wants to spend no more than $100 on trips to the amusement park this summer, how many visits can he make if he buys a membership with part of that money?
     ::如果达伦今年夏天想花不到100美元去游乐园, 如果他用一部分钱购买会员资格, 他能进行多少次访问?
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  4. How many visits can he make if he does not?
     ::如果他没有,他能进行多少次访问?
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  5. If he increases his budget to $160, how many visits can he make as a member?
     ::如果他将预算增加到160美元,
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  6. And how many as a non-member?
     ::有多少个非会员?

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  For an upcoming school field trip, there must be one adult supervisor for every five children.
  ::即将到来的学校实地考察中,每五名儿童必须有一名成人监督员。

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  7. Write an expression for the number of children and adults who will go on the field trip.
     ::写下参加实地考察的儿童和成人人数。
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  8. If the bus seats 40 people, how many children can go on the trip?
     ::如果巴士载着40人, 有多少孩子能去旅行?
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  9. How many children can go if a second 40-person bus is added?
     ::如果加上第二辆40人的公共汽车,有多少儿童可以去?
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  10. Four of the adult chaperones decide to arrive separately by car. Now how many children can go in the two buses?
      ::4名成年监护人决定分别乘车到达,现在有多少儿童可以乘坐两辆公共汽车?

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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 。