Course: 4.3 使用等同率写和解决比例

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使用等值率写入和解决比例比例
Jamie is participating in her local county’s reading challenge. She has to keep a log of all the books she reads, and how long it takes her to read each one. If Jamie can read 15 pages in 25 minutes, how many pages can she read in 65 minutes?
::杰米正在参与本县的阅读挑战。她必须保存一份她读过的所有书籍的日志,以及她阅读每本书需要多长时间。如果杰米能在25分钟内读15页,那么在65分钟内她能读到多少页?

In this concept, you will learn to write and solve proportions by using equivalent rates.
::在此概念中,您将学会使用等价利率来撰写和解决比例问题。

Equivalent Rates
::等等值利率

A ratio is a comparison between two quantities or numbers. Ratios can be written in fraction form, with a colon or by using the word “to”. Sometimes, you will compare ratios. Sometimes one ratio will be greater than another, and other times they can be equal or equivalent. When you have two equal ratios, you have a proportion . A proportion is created when two ratios are equal, or we can say that two equal ratios form a proportion.
::一个比率是两个数量或数字的比较。比率可以以分数形式写成, 有结肠, 或者使用“ to” 。有时, 您会比较比率。有时一个比率会大于另一个比率, 其它时间它们可以相等或等同。当两个比率相同时, 您就会有一个比例。当两个比率相同时, 一个比例会形成一个比例, 或者我们可以说两个相等比率构成一个比例。

You can write a proportion when we know that two ratios are equivalent.
::当我们知道两个比例相等时,你可以写一个比例。

$\begin{aligned} 1 : 2 = 2 : 4 \end{aligned}$

These two ratios are equivalent. You can say that the two ratios form a proportion.
::这两个比率相等。你可以说这两个比率构成一个比例。

Let’s look at an example.
::让我们举个例子。

Do these two ratios, $\begin{aligned} \frac{3}{4} \end{aligned}$ and $\begin{aligned} 4 : 24 \end{aligned}$ form a proportion?
::34和4:24这两个比率是否构成比例?

First, put the ratio $\begin{aligned} 4 : 24 \end{aligned}$ into fraction form.
::首先,将比率4: 24化为分数形式。

$\begin{aligned} 4 : 24 = \frac{4}{24} \end{aligned}$
Next, reduce the fraction.
::4:24=424 下一步,减少分数。

$\begin{aligned} \frac{4}{24} = \frac{1}{6} \end{aligned}$
Then, compare the two fractions.
::424=16 然后,比较两个分数

$\begin{aligned} \frac{1}{6} \neq \frac{3}{4} \end{aligned}$
The answer is $\begin{aligned} \frac{1}{6} \neq \frac{3}{4} \end{aligned}$ .
::1634 答案是1634

If the ratios are equivalent, they form a proportion. Since the ratios are not equivalent, the ratios do not form a proportion.
::如果比率相等,则构成一个比例,由于比率不相等,因此比率不构成一个比例。

To write a proportion, set two equivalent fractions equal to each other, using the information in the problem.
::写一个比例,使用问题中的信息,设定两个等同的分数,彼此等同。

Let’s do another example.
::让我们再举一个例子。

If you know the ratio of girls to boys in a class is $\begin{aligned} 2 : 3 \end{aligned}$ , and you know there are 24 boys in the class, you can write a proportion in order to find the number of girls in the class.
::如果你知道一班女生与男生的比例是2: 3, 你知道班里有24个男生, 你可以写一个比例, 以便找到班里女生的人数。

First, write the ratio of the girls to boys.
::首先,写下女孩与男孩的比例。

$\begin{aligned} \frac{girls}{boys} = \frac{2}{3} \end{aligned}$

::男女孩=23人

Next, write the proportion statement knowing there are 24 boys in the class.
::接下来,写比例说明,知道班上有24个男孩。

$\begin{aligned} \frac{2}{3} = \frac{x}{24} \end{aligned}$
Then, cross multiply to solve for $\begin{aligned} x \end{aligned}$ .
::23=x24 然后,交叉乘以解答 x。

$\begin{aligned} \begin{array}{rcl} \frac{2}{3} & = & \frac{x}{24} \\ 3 x & = & 2 \times 24 \\ 3 x & = & 48 \\ x & = & 19 \end{array} \end{aligned}$
The answer is 19.
::23=x243x=2x243x=48x=19 答案是19

The class has 19 girls and 24 boys in the class.
::该班有19名女孩和24名男孩。

Let’s use equivalent rates to solve a proportion.
::让我们使用等值利率解决一个比例问题。

The ratio of teachers to students in a certain school is $\begin{aligned} 2 : 25 \end{aligned}$ . If there are 400 students in the eighth-grade class, how many teachers are there?
::某一学校的教师与学生的比例是2:25,如果八年级班有400名学生,有多少教师?

First, write the ratio of the teachers to students.
::首先,写出教师与学生的比例。

$\begin{aligned} \frac{teacher}{students} = \frac{2}{25} \end{aligned}$

::教师=225

Next, write the proportion statement knowing there are 400 students in the 8 ^th grade.
::接下来,写比例说明,知道八年级有400名学生。

$\begin{aligned} \frac{2}{25} = \frac{x}{400} \end{aligned}$
Then, cross multiply to solve for $\begin{aligned} x \end{aligned}$ .
::225=x400 然后,交叉乘以 x 解析。

$\begin{aligned} \begin{array}{rcl} \frac{2}{25} & = & \frac{x}{400} \\ 25 x & = & 2 \times 400 \\ 25 x & = & 800 \\ x & = & 32 \end{array} \end{aligned}$
The answer is 32.
::225=x40025x=2x40025x=800x=32 答案是32。

There are 32 8 ^th grade teachers.
::有32名八年级教师。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Jamie’s robust reading challenge.
::Jamie的阅读挑战颇具挑战性,

Jamie reads 15 pages in 25 minutes and wants to know how many pages she can read in 65 minutes.
::Jamie在25分钟内读了15页想知道65分钟后她能读到多少页

First, write a proportion to represent this problem.
::首先,写一个比例来代表这个问题。

$\begin{aligned} \frac{15}{25} = \frac{x}{65} \end{aligned}$

::1525=x65

Next, cross multiply.
::下一个,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{15}{25} & = & \frac{x}{65} \\ 25 x & = & 15 \times 65 \\ 25 x & = & 975 \end{array} \end{aligned}$

::1525=x6525x=15×6525x=975

Then, divide by 25 to solve for $\begin{aligned} x \end{aligned}$ .
::然后除以25 解决x

$\begin{aligned} \begin{array}{rcl} 25 x & = & 975 \\ \frac{25 x}{25} & = & \frac{975}{25} \\ x & = & 39 \end{array} \end{aligned}$

::25x=97525x25=97525x39

The answer is 39.
::答案是39岁

Therefore Jamie can read 39 pages in 65 minutes.
::因此Jamie可以在65分钟内读到39页

Example 2
::例2

Write a proportion to describe this situation. The proportion of red paper to white paper in a stack is 2 to 7. If there are 32 red pieces of paper, what proportion could be used to find the number of pieces of white paper?
::红纸与白纸在堆叠中的比例是2比7,如果有32张红纸,那么用什么比例来找到白纸的份数?

First, write the ratio of the teachers to students.
::首先,写出教师与学生的比例。

$\begin{aligned} \frac{red paper}{white paper} = \frac{2}{7} \end{aligned}$

::红纸白纸=27

Next, write the proportion statement knowing there are 32 pieces of red paper.
::接下来,写比例说明明知有32件红色纸。

$\begin{aligned} \frac{2}{7} = \frac{32}{x} \end{aligned}$
Then, cross multiply to solve for $\begin{aligned} x \end{aligned}$ .
::27=32x后, x 的交叉乘数解析。

$\begin{aligned} \begin{array}{rcl} \frac{2}{7} & = & \frac{32}{x} \\ 2 x & = & 7 \times 32 \\ 2 x & = & 224 \\ x & = & 112 \end{array} \end{aligned}$

::27=32x2x=7x322x=224x=112

The answer is 112.
::答案是112。

There are 112 white pieces of paper.
::有112张白纸

Example 3
::例3

Solve for $\begin{aligned} x \end{aligned}$ in the proportion $\begin{aligned} \frac{3}{4} = \frac{6}{x} \end{aligned}$ by using equal ratios.
::使用同等比率为34=6x比例的 x 解决。

First, cross multiply.
::首先,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{3}{4} & = & \frac{6}{x} \\ 3 x & = & 4 \times 6 \\ 3 x & = & 24 \end{array} \end{aligned}$
Next, divide by 3 $\begin{aligned} x \end{aligned}$ .
::34=6x3x=4x63x=24 下一步,除以 3 x 。

$\begin{aligned} \begin{array}{rcl} 3 x & = & 24 \\ \frac{3 x}{3} & = & \frac{24}{3} \\ x & = & 8 \end{array} \end{aligned}$
The answer is 8.
::3x=243x3=243x=8 答案是8

Therefore $\begin{aligned} \frac{3}{4} = \frac{6}{8} \end{aligned}$ .
::因此,34=68。

Example 4
::例4

Solve for $\begin{aligned} x \end{aligned}$ in the proportion $\begin{aligned} \frac{9}{50} = \frac{x}{100} \end{aligned}$ by using equal ratios.
::使用同等比例为 950=x100 比例的 x 解决。

First, cross multiply.
::首先,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{9}{50} & = & \frac{x}{100} \\ 50 x & = & 9 \times 100 \\ 50 x & = & 900 \end{array} \end{aligned}$

::950=x10050x=9x10050x=9x10050x=900

Next, divide by 50 to solve for $\begin{aligned} x \end{aligned}$ .
::下一个,除以50 解决x。

$\begin{aligned} \begin{array}{rcl} 50 x & = & 900 \\ \frac{50 x}{50} & = & \frac{900}{50} \\ x & = & 18 \end{array} \end{aligned}$

::50x=900050x50=900050x18

The answer is 18.
::答案是18岁

Therefore $\begin{aligned} \frac{9}{50} = \frac{18}{100} \end{aligned}$ .
::因此,950=18100。

Example 5
::例5

Solve for $\begin{aligned} x \end{aligned}$ in the proportion $\begin{aligned} \frac{3.5}{7} = \frac{x}{35} \end{aligned}$ by using equal ratios.
::3.57=x35乘以3.57=x35比例的x,使用相等比率解决。

First, cross multiply.
::首先,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{3.5}{7} & = & \frac{x}{35} \\ 7 x & = & 3.5 \times 35 \\ 7 x & = & 122.5 \end{array} \end{aligned}$

::3.57=x357x=3.5×357x=122.5

Next, divide by 7 to solve for $\begin{aligned} x \end{aligned}$ .
::下一步,除以 7 来解答 x 。

$\begin{aligned} \begin{array}{rcl} 7 x & = & 122.5 \\ \frac{7 x}{7} & = & \frac{122.5}{7} \\ x & = & 17.5 \end{array} \end{aligned}$

::7x=122.57x7=122.57x=17.5

The answer is 17.5.
::答案是17.5

Therefore $\begin{aligned} \frac{3.5}{7} = \frac{175}{35} \end{aligned}$ .
::因此,3.57=17535。

Review
::回顾

Solve each proportion using equal ratios.
::使用同等比率解决每一比例。

$\begin{aligned} \frac{3}{4} = \frac{x}{12} \end{aligned}$
::34=12 =34=12

$\begin{aligned} \frac{5}{6} = \frac{x}{12} \end{aligned}$
::56=x12

$\begin{aligned} \frac{4}{7} = \frac{8}{y} \end{aligned}$
::47=8y

$\begin{aligned} \frac{2}{3} = \frac{12}{y} \end{aligned}$
::23=12y

$\begin{aligned} \frac{4}{5} = \frac{44}{y} \end{aligned}$
::45=44y

$\begin{aligned} \frac{12}{13} = \frac{x}{26} \end{aligned}$
::1213=x26

$\begin{aligned} \frac{9}{10} = \frac{81}{y} \end{aligned}$
::910=81y

$\begin{aligned} \frac{6}{7} = \frac{18}{y} \end{aligned}$
::67=18y

$\begin{aligned} \frac{7}{8} = \frac{x}{56} \end{aligned}$
::78=x56

$\begin{aligned} \frac{12}{14} = \frac{36}{x} \end{aligned}$
::1214=36x

$\begin{aligned} \frac{6}{4} = \frac{x}{12} \end{aligned}$
::64=12 64=12

$\begin{aligned} \frac{12}{14} = \frac{24}{x} \end{aligned}$
::1214=24x

$\begin{aligned} \frac{13}{14} = \frac{x}{42} \end{aligned}$
::1314=x42

$\begin{aligned} \frac{1.5}{4} = \frac{x}{8} \end{aligned}$
::1.54=x8

$\begin{aligned} \frac{3.5}{4.5} = \frac{x}{9} \end{aligned}$
::3.54.5=x9

$\begin{aligned} \frac{9}{14} = \frac{108}{x} \end{aligned}$
::914=108x

Review (Answers)
::回顾(答复)

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。

Resources
::资源

Section outline

General

使用等值率写入和解决比例比例

Equivalent Rates ::等等值利率

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Equivalent Rates
::等等值利率

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源