5.4等效费率的确定和书写

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• 选择节等效费率的确定和书写

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  等效费率的确定和书写

  

  [Figure 1]

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  Keling is a good golfer. He is trying out for the golf team at school next week, so he headed out for some practice. The pro at the driving range told Keling that he could get a bucket of 45 balls for $9 or pay $1 for 4 balls at a time. Are these two prices at the same  rate ? If not, which rate is the better deal?
  ::Keling是个好高尔夫球手。他下周要参加学校的高尔夫球队的比赛,所以他要参加一些练习。驾驶场的专业人士告诉Keling,他可以得到一桶45个球9美元,或者一次支付一美元4个球。这两种价格是相同的价格吗?如果不是,哪一种费率更好呢?

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  In this concept, you will learn to identify and write equivalent  . 
  ::在此概念中,您将学会识别和写作等值 。

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  Identifying and Writing Equivalent Rates
  ::确定和写写等效率

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  A  rate  is a ratio that compares quantities in different units. The word "per" is used when talking about rates and is sometimes abbreviated with a slash, /.
  ::a 比率是比较不同单位数量的比率。在谈论费率时使用“per”一词,有时用斜线缩写, / 。

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  Some common rates are used all the time, like miles  per  hour, dollars per gallon, and days per week. Others are more specific to an occasion, like cats per household or chocolate chips per cookie.
  ::有些常见的利率总是被使用,比如每小时英里、每加仑美元和每星期的天数。 其他的则比较具体,比如每家每户猫或每饼干巧克力薯片。

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  Two rates are  equivalent  if they show the same relationship between two identical units of measure.
  ::两个比率如果表明两个相同的计量单位之间的关系相同,则等值。

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  The same strategies used to find equivalent ratios can be used to find equivalent rates.
  ::用于确定等同比率的相同战略可用于确定等同比率。

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  Here is an example.
  ::举一个例子。

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  Two cars are traveling. One car goes 40 miles in 2 hours, and the other goes 80 miles in 4 hours. Determine whether or not the rates are equivalent.
  ::两辆车正在行驶,一辆车在2小时内行驶40英里,另一辆车在4小时内行驶80英里。确定费率是否相等。

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  First, write the rates as fractions. Remember to make sure the terms are the same, in this case, miles per hour.
  ::首先, 将利率写成分数 。 记住要确定条件相同, 在此情况下, 以每小时英里计 。

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  40 miles per 2 hours =  $\begin{array}{r}\frac{40}{2}\frac{miles}{hours}=\frac{40}{2}\end{array}$
  ::每2小时40英里=40 2米 = 40 2米 i l e s h o u r s = 40 2

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  80 miles per 4 hours =  $\begin{array}{r}\frac{80}{4}\frac{miles}{hours}=\frac{80}{4}\end{array}$  
  ::每4小时80英里=80 4米 4米 i l e s h o u r s = 80 4

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   Next,  reduce  both fractions to their lowest terms.
  ::其次,将两个分数降低到最低值。

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  $\begin{array}{r}\frac{40}{2}=\frac{40÷2}{2÷2}=\frac{20}{1}\end{array}$ $\begin{array}{r}\frac{miles}{hour}\end{array}$
  
  ::40 2 = 40 = 40 2 2 = 2 2 = 2 = 20 1米 i i e s h o u r

   

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  $\begin{array}{r}\frac{80}{4}=\frac{80÷4}{4÷4}=\frac{20}{1}\end{array}$ $\begin{array}{r}\frac{miles}{hour}\end{array}$
  
  ::80 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

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   Then, compare the fractions in their lowest terms. Remember to include the units.
  ::然后以最小值比较分数。 记住要包含单位 。

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  20 miles/hour = 20 miles/hour
  ::20英里/小时=20英里/小时

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  The answer is that the rates are equivalent. Both cars traveled at a rate of 20 miles per hour.
  ::答案是价格相当,两辆汽车每小时走20英里。

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  Other methods that can be used to determine whether or not rates are equal include changing one or both fractions so that the denominators are equal, and cross multiplying.
  ::可用于确定税率是否相等的其他方法包括改变一个或两个分数,使分母相等,并交叉乘法。

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  Cross multiplication does not yield a rate. Only the numerical values are used as a check.
  ::交叉乘不产生一个率。只有数值作为检查使用。

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  When using rates, it does not matter which unit is in the  numerator  and which is in the  denominator , as long as they match one another.
  ::当使用率时,如果分子中是哪个单位,分母中是哪个单位,只要它们相互匹配,就无关紧要。

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  For example:
  ::例如:

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   gumdrops per person is not the same as persons per gumdrop
  ::每人口香糖与每人口香糖不同

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  $\begin{array}{r}\frac{gumdrops}{person}\ne \frac{persons}{gumdrop}\end{array}$
  ::g m d d d r o p p e e r r o o o p e r r r r y y s g u m d o p

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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 Keling, who was at the driving range.
  ::早些时候,有人给了你Keling一个问题, Keling当时在车场。

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  He was trying to figure out if he should get a bucket of 45 balls for $9 or pay $1 for 4 balls at a time.
  ::他想弄清楚他是不是应该拿一桶45球9美元 或者一次付一美元4球1美元

  First, write the rates as fractions being sure to keep the units consistent.

    $\begin{array}{r}\frac{9}{45}\frac{dollars}{balls}\end{array}$

      and

    $\begin{array}{r}\frac{1}{4}\frac{dollar}{balls}\end{array}$

  Next, determine a method. Since  $\begin{array}{r}\frac{1}{4}\end{array}$ is already in lowest terms, reduce  $\begin{array}{r}\frac{9}{45}\end{array}$  

    $\begin{array}{r}\frac{9}{45}=\frac{9÷9}{45÷9}=\frac{1}{5}\end{array}$
  

   

  Then, compare the two fractions.

  $\begin{array}{r}\frac{1}{4}\ne \frac{1}{5}\end{array}$  

   The answer is that the rates are not equivalent.

   Keling would be better off paying   $\begin{array}{r}\frac{1}{5}\frac{dollar}{balls}\end{array}$  which is the rate of 45 balls for $9, rather than  $\begin{array}{r}\frac{1}{4}\frac{dollar}{balls}\end{array}$

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   Example 2
  ::例2

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  Determine if two machines are winding thread at the same speed. One machine winds at a rate of 5 meters every 3 seconds. The other takes 18 seconds to wind 20 meters.
  ::确定两台机器是否以同样的速度在刮线。每三秒一台机器风速为5米,另一台需要18秒风速为20米。

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  First, write the rates as fractions, making sure the units are identical.
  ::首先,将费率写成分数,确保单位相同。

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    $\begin{array}{r}\frac{5}{3}\frac{meters}{second}\end{array}$  and   $\begin{array}{r}\frac{20}{18}\frac{meters}{seconds}\end{array}$
  ::5米 3米 e t e r s s e c o n d 和 20米 18 e t e e e e e e r s s e e c c n d s

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  Next, assume the rates are equal to one another and write an equation.
  ::接下来,假设利率是等同的,然后写一个方程。

    $\begin{array}{r}\frac{5}{3}=\frac{20}{18}\end{array}$

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  Then cross multiply.
  ::然后乘以乘数

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  5 x 18 = 3 x 20
  ::5 x 18 = 3 x 20

  $\begin{array}{r}90\ne 60\end{array}$

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  The answer is that the rates are not equivalent.
  ::答案是,汇率不等同。

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  Example 3
  ::例3

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  Are the rates equivalent? 3 feet in 9 seconds and 6 feet in 18 seconds
  ::9秒3英尺,18秒6英尺

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  First, write the rates as fractions.
  ::首先,将费率写成分数。

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    $\begin{array}{r}\frac{3}{9}\end{array}$  and  $\begin{array}{r}\frac{6}{18}\end{array}$
  ::9、9和6 18

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  Next, determine a method. Multiply the first fraction times 2 to convert to, and work with, 18ths.
  ::下一步确定一个方法。乘以第一个分数乘以 2,然后转换为 18 。

  $\begin{array}{r}\frac{3}{9}=\frac{3\times 2}{9\times 2}=\frac{6}{18}\end{array}$
  

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  Then, compare the fractions.
  ::然后,比较分数。

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  $\begin{array}{r}\frac{6}{18}\frac{feet}{seconds}=\frac{6}{18}\frac{feet}{seconds}\end{array}$  
  ::6 e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e c n d s

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  The answer is that the rates are equivalent.
  ::答案是,利率是等值的。

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  Example 4
  ::例4

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  Compare the rates: 5 miles in 30 minutes and 42 minutes to go 6 miles
  ::比较价格: 30分钟5英里 6英里42分钟

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  First, write the rates as fractions being sure to keep the units consistent.
  ::首先,将费率写为分数,以确保单位保持一致。

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    $\begin{array}{r}\frac{5}{30}\frac{miles}{minutes}\end{array}$ and   $\begin{array}{r}\frac{6}{42}\frac{miles}{minutes}\end{array}$  
  ::5 30米 i l e s m i i i i i y t e s 和 6 42米 i i i i i i i i i i i i i i i i i i t e s

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   Next, determine a method. Reduce both rates to their lowest terms.
  ::接下来,确定一种方法,将这两个比率降低到最低值。

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    $\begin{array}{r}\frac{5}{30}=\frac{5÷5}{30÷5}=\frac{1}{6}\end{array}$ $\begin{array}{r}\frac{mile}{minutes}\end{array}$  
  ::5 30 = 5 = 5 5 30 = 5 = 1 6米 i i i i i i n u t e s

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  $\begin{array}{r}\frac{6}{42}=\frac{6÷6}{42÷6}=\frac{1}{7}\end{array}$ $\begin{array}{r}\frac{mile}{minutes}\end{array}$  
  ::6 42 = 6 = 6 6 42 = 6 = 1 7米 i i i i i i n u t e s

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   Then, compare the fractions.
  ::然后,比较分数。

  $\begin{array}{r}\frac{1}{6}\ne \frac{1}{7}\end{array}$  

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   The answer is that the rates are not equal. 
  ::答案是利率不相等。

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  Example 5
  ::例5

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  Compare 5 pounds for $20.00 and 8 pounds for $32.00
  ::相较5磅 $20.00,$32.00,$8磅

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  First, write the rates as fractions.
  ::首先,将费率写成分数。

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    $\begin{array}{r}\frac{5}{20}\frac{pounds}{dollars}\end{array}$   and   $\begin{array}{r}\frac{8}{32}\frac{pounds}{dollars}\end{array}$
  ::5 20 p un un d d d d d l l r s r s 和 8 32 p o un d d d d d d d l l r s

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  Next, determine a method. Reduce each fraction to its lowest terms. 
  ::接下来,确定一种方法,将每一分数减到最低值。

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  $\begin{array}{r}\frac{5}{20}=\frac{5÷5}{20÷5}=\frac{1}{4}\end{array}$ $\begin{array}{r}\frac{pounds}{dollars}\end{array}$
  ::5 20 = 5 = 5 5 20 = 5 = 1 4 p o u n d d d d d d d d l d l r r s

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    $\begin{array}{r}\frac{8}{32}=\frac{8÷8}{32÷8}=\frac{1}{4}\end{array}$ $\begin{array}{r}\frac{pounds}{dollars}\end{array}$
  ::8 32 = 8 = 8 32 = 8 32 = 8 = 1 4 p o 4 o d d d d d d d d d l o l r r s

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  Then, compare the rates.
  ::然后,比较费率。

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

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  The answer is that the rates are equivalent.
  ::答案是,利率是等值的。

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    $\begin{array}{r}\frac{1}{4}\frac{pound}{dollars}\end{array}$ can also be written  $\begin{array}{r}\frac{4}{1}\frac{dollars}{pound}\end{array}$ or $4 per pound.
  ::1,4 p u d d d d l l r s 也可以写成4,1 d o l r p u d d 或 4 美元/磅。

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  Review
  ::回顾

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  Write an equivalent rate for each rate.
  ::每一费率写一个等值的费率。

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  1. 2 for $10.00
     ::2美元$10.00
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  2. 3 for $15.00
     ::3美元15.00美元
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  3. 5 gallons for $12.50
     ::5加仑,12.50美元
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  4. 16 pounds for $40.00
     ::16磅$40.00$40.00$ 16磅$40.00$
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  5. 18 inches for $2.00
     ::18英寸 $2.00
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  6. 5 pounds of blueberries for $20.00
     ::5磅蓝莓$20.00
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  7. 40 miles in 80 minutes
     ::80分钟40英里
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  8. 20 miles in 4 hours
     ::4小时20英里
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  9. 10 feet in 2 minutes
     ::2分钟内10英尺
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  10. 12 pounds in 6 weeks
      ::6周12磅

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  Write each rate as a fraction and simplify it.
  ::将每种费率作为一个分数写下来并简化。

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  11. 2 for $10.00
      ::2美元$10.00
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  12. 3 for $15.00
      ::3美元15.00美元
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  13. 5 gallons for $25.00
      ::5加仑,25美元,00美元
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  14. 40 pounds for $80.00
      ::40镑$80.00
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  15. 18 inches for $2.00
      ::18英寸 $2.00