5.15使用单元分析进行转换

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section 使用单元分析进行转换

  Collapse Expand

  使用单元分析进行转换

  

  [Figure 1]

  播放段落

  Munni and Caras were excited about their upcoming whale-watching trip. They did their research and discovered that some whales can weigh up to 150 tons. Munni’s home scale said that he weighed 120. Munni told everyone that he weighed only 30  pounds  less than a whale. Was he correct?
  ::蒙尼和卡拉斯对即将到来的观鲸考察旅行感到兴奋。 他们进行了研究,发现一些鲸鱼的重量可达150吨。 曼尼的家庭规模说,他体重120磅。 曼尼告诉所有人,他比鲸鱼少了30磅。 他是否正确?

  播放段落

  In this concept, you will learn to use proportions and  unit analysis . 
  ::在这个概念中,你将学会使用比例和单位分析。

  播放段落

  Using Proportions and Unit Analysis
  ::使用比例和单位分析

  播放段落

  A  proportion  is created when two ratios are found to be  equivalent  or equal.
  ::当发现两个比率对等或相等时,就形成一个比例。

  播放段落

    $\begin{array}{r}\frac{1}{2}=\frac{3}{6}\end{array}$   is a proportion.
  ::1 2 = 3 6 是一个比例。

  播放段落

  A  unit rate  is a comparison of two measurements, one of which has a value of 1.
  ::A单位费率是两项测量的比较,其中一项测量值为1。

  播放段落

    $\begin{array}{r}\frac{55miles}{1hour}\end{array}$  is a unit rate. 
  ::55 m i e s 1 h o u r 是单位费率。

  播放段落

  A  unit scale  is a ratio that compares the dimensions of an actual object to the dimensions of a scale drawing or model that represents the actual object.
  ::单位比例尺是指将实际物体的尺寸与代表实际物体的尺度图象或模型的尺寸进行比较的比率。

  播放段落

  $\begin{array}{r}\frac{1inch}{11feet}\end{array}$    is a unit scale.
  ::1 n c h 11 f e e e t 是单位比例表。

  播放段落

  A  standard unit  is one that is most commonly used.
  ::标准单位是最常用的单位。

  播放段落

  Both the customary and metric systems of measurement include weight (or mass),  length , and liquid volume standard units.
  ::习惯和衡量系统包括重量(或质量)、长度和液体体积标准单位。

  播放段落

  Some examples are:
  ::举例如下:

  播放段落

  Customary standard weight:  16  ounces  = 1 pound
  ::习惯标准重量:16盎司=1磅

  播放段落

  Metric standard length:  1  meter  = 100 
  ::计量标准长度:1米=100

  播放段落

  Customary standard liquid: 4 quarts = 1 gallon 
  ::习惯标准液体:4夸特=1加仑

  播放段落

  These are called  conversion factors  and can be defined as the ratio of a measurement in one unit to the equivalent numerical value in another unit. 
  ::这些称为换算系数,可界定为一个单位的计量与另一个单位的等值数值之比。

  播放段落

  The ratio for ounces to pounds is 16:1 or  $\begin{array}{r}\frac{16ounces}{1pound}\end{array}$ .
  ::盎司对磅的比率为16:1或16 u n c e s 1 p o u n d。

  播放段落

  Standard units of measurement and proportions can be used to convert one unit of measurement to another. The process of converting one unit to another is called  unit conversion . Units can be converted by using proportions, setting ratios equal to one another, or by unit analysis, multiplying times a standard unit of measure.
  ::标准计量单位和比例单位可用于将一个计量单位转换为另一个计量单位。将一个单位转换为另一个单位的过程称为单位转换过程。单位可以使用比例转换,设定等同比率,或单位分析,乘以标准计量单位的乘数。

  播放段落

  Here is an example using proportions.
  ::以下是一个使用比例的例子。

  播放段落

  A pitcher holds 4 liters of water. Determine how many milliliters of water the pitcher holds. Use the unit conversion: 1 liter = 1000 milliliters 
  ::投手持有4升水。确定投手持有多少毫升水。使用单位转换:1升=1000毫升

  播放段落

  First, write the unit conversion as a fraction. 
  ::首先,将单位转换写成一个分数。

  $$\begin{array}{r}\frac{1liter}{1000millimeters}\end{array}$$

  播放段落

   Next, write a ratio that compares the unit as it is given to the unknown converted unit.
  ::下一步,写出一个比例,将单位与给给给给未知转换单位的单位进行比较。

  播放段落

  Remember that a ratio standing alone can be written with either value in the numerator or denominator, but once it is used in a calculation, it must be formatted accordingly.
  ::请注意,单立比率可以用分子或分母中的值来写出,但一旦在计算中使用,则必须相应地格式化。

  播放段落

  Unit to be converted = from 4 liters to x milliliters
  ::要转换的单位 = 从4升转换为x毫升

  播放段落

  This could be written as  $\begin{array}{r}\frac{4liters}{xmillimeters}or\frac{xmillimeters}{4liters}\end{array}$ . 
  ::这可以写成4 l i t e r s x m i i i i e t e r r r o r x i i i i i e e r r s 4 l i t e r r 。

  播放段落

  Then, write a proportion using the correct format. When writing a proportion, the units must be consistent.
  ::然后,使用正确的格式写一个比例。在写一个比例时,单位必须一致。

  播放段落

    $\begin{array}{r}\frac{1liter}{1000milliliters}=\frac{4liters}{xmilliliters}\end{array}$ is correct.
  ::1 i i t r 1000 m i i i i i t r r s = 4 l i i t e r s i i i i i i t e r s 正确。

  播放段落

  Next, solve by cross multiplication.
  ::下一步,通过交叉乘法解决。

  播放段落

  $\begin{array}{r}x=4000\end{array}$
  ::x = 40000

  播放段落

  The answer is x = 4,000 ml.
  ::答案是x=4,000毫升。

  播放段落

  Another way to solve the same problem is to use unit analysis.
  ::解决同一问题的另一种办法是使用单位分析。

  播放段落

  First, write the unit conversion as a fraction.
  ::首先,将单位转换写成一个分数。

  $$\begin{array}{r}\frac{1liter}{1000milliliters}or\frac{1000milliliters}{1liter}\end{array}$$

  播放段落

  Next, write a ratio for the unit to be converted.
  ::下一步,写出要转换的单位的比率 。

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

  播放段落

  When using unit analysis, the correct format will be such that after multiplication, the “from” units will cancel, and the answer will equal the “to” units.
  ::当使用单位分析时,正确的格式是,乘以后,“自”单位将取消,回答等于“至”单位。

   

  $$\begin{array}{r}\frac{4liters}{1}\times \frac{1000milliliters}{1liters}\end{array}$$

  播放段落

  Then, cancel liters, and multiply.
  ::然后,取消公升数,再乘数。

  $$\begin{array}{r}\frac{4\times 1000ml}{1\times 1}=4000ml\end{array}$$

  播放段落

  The answer is 4,000 ml.
  ::答案是4,000毫升。

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Earlier, you were given a problem about Munni and Caras, who are going on a whale-watching trip.
  ::早些时候,你被问及 蒙尼和卡拉斯的问题, 他们要去观鲸旅行。

  播放段落

  Munni said a whale that weighed 150 tons was only 30 more than his weight of 120.  Munni forgot the whale’s weight was in tons, and his was in pounds. Find the whale’s weight in pounds.
  ::Munni说,一头体重150吨的鲸鱼只比120吨的重量多出30倍。 Munni忘了鲸鱼的重量是吨,而他的体重是磅。 找到鲸鱼的重量是磅。

  播放段落

  First, write the unit conversion as a fraction. 
  ::首先,将单位转换写成一个分数。

  $$\begin{array}{r}\frac{2000pounds}{1ton}\end{array}$$

  播放段落

  Next, write a ratio that compares the unit as it is given to the unknown converted unit.
  ::下一步,写出一个比例,将单位与给给给给未知转换单位的单位进行比较。

  $$\begin{array}{r}\frac{xpounds}{150tons}\end{array}$$

   

  播放段落

  Then, write a proportion making sure that the units are consistent.
  ::然后,写一个比例以确保单位的一致性。

  $$\begin{array}{r}\frac{2000pounds}{1ton}=\frac{xpounds}{150tons}\end{array}$$

    

  播放段落

  Next, solve by cross multiplication.
  ::下一步,通过交叉乘法解决。

  $$\begin{array}{r}x=300,000pounds\end{array}$$

  播放段落

  The answer is the whale weighs 300,000 pounds.
  ::答案是鲸鱼重30万磅

  播放段落

  Example 2
  ::例2

  播放段落

  How many milliliters are there in 2.5 liters? Write a proportion and solve.
  ::2.5升中有多少毫升?

  播放段落

  First, write the unit conversion as a fraction.
  ::首先,将单位转换写成一个分数。

  $$\begin{array}{r}\frac{1liter}{1000milliliters}\end{array}$$

  播放段落

  Next, write a ratio that compares the unit as it is given to the unknown converted unit.
  ::下一步,写出一个比例,将单位与给给给给未知转换单位的单位进行比较。

  $$\begin{array}{r}\frac{2.5liters}{xmilliliters}\end{array}$$

  播放段落

  Then, write a proportion making sure that the units are consistent.
  ::然后,写一个比例以确保单位的一致性。

  $$\begin{array}{r}\frac{1liter}{1000milliliters}=\frac{2.5liters}{xmilliliters}\end{array}$$

   

  播放段落

  Next, solve by cross multiplication.
  ::下一步,通过交叉乘法解决。

  $$\begin{array}{r}X=2500ml\end{array}$$

  播放段落

  The answer is x = 2,500 ml
  ::答案是x=2 500毫升=2 500毫升

  播放段落

  Example 3
  ::例3

  播放段落

  How many   in 11 kilometers? Write a proportion and solve. 
  ::11公里内有多少人?

  播放段落

  First, write the unit conversion as a fraction.
  ::首先,将单位转换写成一个分数。

  $$\begin{array}{r}\frac{1000m}{km}\end{array}$$

   Next, write a ratio that compares the unit as it is given to the unknown converted unit.

  $$\begin{array}{r}\frac{xm}{11km}\end{array}$$

  播放段落

  Then, write a proportion making sure that the units are consistent.
  ::然后,写一个比例以确保单位的一致性。

  $$\begin{array}{r}\frac{1000m}{km}=\frac{xm}{11km}\end{array}$$

  播放段落

  Next, solve by cross multiplication.
  ::下一步,通过交叉乘法解决。

  $$\begin{array}{r}X=11,000m\end{array}$$

  播放段落

  The answer is x = 11,000 m
  ::答案是x=11 000米

  播放段落

  Example 4
  ::例4

  播放段落

  How many  inches  are there in 18 feet? Use unit analysis.
  ::18英尺内有多少英寸 使用单位分析

  播放段落

  First, write the unit conversion as a fraction.
  ::首先,将单位转换写成一个分数。

  $$\begin{array}{r}\frac{12inches}{1foot}\end{array}$$

   

  播放段落

  Next, write a ratio for the unit to be converted.
  ::下一步,写出要转换的单位的比率 。

  $$\begin{array}{r}\frac{18feet}{1}\end{array}$$

  播放段落

  Then, cancel units, and multiply.
  ::然后,取消单位, 乘数。

  $$\begin{array}{r}\frac{12inches}{1foot}\times \frac{18feet}{1}=216inches\end{array}$$

  播放段落

   The answer is 18 feet = 216 inches.
  ::答案是18英尺 = 216英寸

  播放段落

  Example 5
  ::例5

  播放段落

  Convert 3 gallons to cups.
  ::将3加仑转换为杯子。

  播放段落

  First, choose a method.
  ::首先,选择一种方法。

  播放段落

  Since there is more than one standard unit between gallons and cups, setting up one proportion is not possible without intermediate steps. Use unit analysis.
  ::由于加仑和杯子之间有一个以上的标准单位,因此,没有中间步骤,就不可能设定一个比例。 使用单位分析 。

  播放段落

  First, write the unit conversions from gallons to cups as fractions. 
  ::首先,将单位从加仑转换成杯子作为分数。

  $$\begin{array}{r}\frac{1\text{ gallon}}{4\text{ quarts}}\frac{1\text{ quart}}{2\text{ pints}}\frac{1\text{ pint}}{4\text{ cups}}\end{array}$$

  播放段落

  Next, write a ratio for the unit to be converted.
  ::下一步,写出要转换的单位的比率 。

  $$\begin{array}{r}\frac{3\text{ gallons}}{1}\end{array}$$

  播放段落

  Then, set up a multiplication problem being sure to cancel unwanted units. 
  ::然后设置一个乘法问题, 以确保取消不想要的单位 。

  $$\begin{array}{r}\frac{3\text{ gallons}}{1}\times \frac{4\text{ quarts}}{1\text{ gallon}}\times \frac{2\text{ pints}}{1\text{ quart}}\times \frac{4\text{ cups}}{1\text{ pint}}\end{array}$$

  播放段落

  Next, cancel units and multiply.
  ::下一步,取消单位和乘法。

  $$\begin{array}{r}\frac{3\times 4\times 2\times 4\text{ cups}}{1\times 1\times 1\times 1}=96\text{ cups}\end{array}$$

  播放段落

  The answer is 3 gallons = 96 cups.
  ::答案是3加仑=96杯

  播放段落

  Review
  ::回顾

  播放段落

   Use unit analysis to solve each problem.
  ::使用单位分析来解决每个问题。

  播放段落
  1. If there are 5280 feet in a mile, how many feet in 18.5 miles?
     ::如果一英里内有5280英尺 18.5英里内有几英尺?
  播放段落
  2. How many milliliters in 3.75 liters?
     ::多少毫升在3.75升?
  播放段落
  3. How many milliliters in 18.25 liters?
     ::多少毫升在18.25升?
  播放段落
  4. How many pounds in 3 tons?
     ::3吨多少磅?
  播放段落
  5. How many pounds in 2.5 tons?
     ::2.5吨多少磅?
  播放段落
  6. How many pounds in 4.75 tons?
     ::4.75吨中有多少磅?
  播放段落
  7. How many feet in 18 yards?
     ::18码内有几英尺?
  播放段落
  8. How many inches in 4 feet?
     ::4英尺有几英寸长?
  播放段落
  9. How many inches in 8.75 feet?
     ::8.75英尺有几英寸?
  播放段落
  10. How many milliliters in 29.5 liters?
      ::多少毫升在29.5升?

  播放段落

  Solve each problem.
  ::解决每一个问题。

  播放段落
  11. Fred needs to buy vanilla extract to bake a cake. He could buy a 4-ounce bottle of vanilla extract for $8, or a 6-ounce bottle of vanilla extra for $15. Which bottle is the better buy?
      ::佛瑞德需要买香草提取物来烤蛋糕。他可以买一瓶4盎司香草提取物8美元,或一瓶6盎司香草额外15美元。哪个瓶子更好买?
  播放段落
  12. A rope is 3 yards long. How many inches long is the rope? Use these unit conversions: 1 yard = 3 feet and 1 foot = 12 inches.
      ::绳子长3码,绳子长多少英寸?使用这些装置转换:1码=3英尺,1英尺=12英寸。
  播放段落
  13. At the farmer's market, Maureen can buy 6 ears of corn for $3. At that price, how much would it cost to buy 9 ears of corn?
      ::在农民市场,莫琳可以买6耳朵玉米 3美元。以这个价格,买9耳朵玉米要花多少钱?
  播放段落
  14. James bought a 12unce bottle of apple juice. How many pints of apple juice did James buy? Use these unit conversions: 1 cup = 8 fluid ounces and 1 pint = 2 cups.
      ::James买了一瓶128盎司的苹果汁。James买了多少品脱苹果汁?使用这些单位转换:1杯=8液盎司,1品脱=2杯。