Course: 5.1等效比率 | The Way To Learn

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等效比率
On the first day of school, the students in the seventh grade Language arts class started a reading contest to see how many books they could read during the year. Kayla and Torrey have been participating in the challenge since day one. By October, eight weeks into school, Kayla had already finished 6 books. Torrey said she was reading 3 books every four weeks. The girls concluded that at the present point in time, they had both read the same number of books. Are they correct?
::上学第一天,七年级语言艺术班的学生开始了阅读竞赛,看他们一年能读到多少本书。凯拉和托雷自第一天起就参与了这项挑战。到10月,上学8周,凯拉已经完成6本书。托雷说,她每4周读3本书。女孩们得出结论说,目前她们都读了同样数量的书。它们是否正确?

In this concept, you will learn how to work with equivalent ratios.
::在这个概念中,你们将学会如何以同等比率工作。

Working with Equivalent Ratios
::与等同比率合作

A ratio is the relationship, or comparison, of a part to the whole.
::A比率是一部分与整个部分的关系或比较。

A ratio can be expressed:
::可以表示一个比率:

in words, 3 to 4, or 3 of 4
::3至4, 或4 中 3

with a colon, 3:4
::结肠3: 4

as a fraction, $\begin{aligned} \frac{3}{4} \end{aligned}$
::分数, 3 4

The easiest way to work with ratios is by thinking of them as fractions.
::使用比率最容易的方法就是将比率视为分数。

For example, think about the three lights on a traffic signal.
::例如,考虑一下交通信号上的三盏灯。

One of three is red, $\begin{aligned} \frac{1}{3} \end{aligned}$
::三个中一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色,一个是红色

2:3 are primary colors, $\begin{aligned} \frac{2}{3} \end{aligned}$
::2:3是主要颜色,2 3

$\begin{aligned} \frac{3}{3} \end{aligned}$ are colors of the rainbow
::3是彩虹的颜色

Equivalent means equal.
::等同意味着平等。

Equivalent ratios show the same, or equal, relationship between two quantities.
::等值比率显示两个数量之间的关系相同或相等。

Using the traffic lights again, suppose there are two signals on one street and one on another.
::假设一条街和另一条街上有两个信号,

The first street has 2 of 6 lights that are red. The second has 1 of 3.
::第一街有红色的6个灯光中的2个,第二街有3个灯光中的1个灯光。

The ratio of red lights to the total lights per signal is still the same.
::红灯与每个信号总光灯的比率保持不变。

Writing ratios in fractional form and simplifying can determine whether or not they are equivalent.
::以分数形式和简化形式的书写比率可以决定它们是否相等。

Here is an example:
::以下是一个例子:

Determine if the two ratios are equivalent.
::确定这两个比率是否相等。

$\begin{aligned} \frac{2}{3} \end{aligned}$ and $\begin{aligned} \frac{10}{15} \end{aligned}$
::2 3和10 15

First, recognize that $\begin{aligned} \frac{2}{3} \end{aligned}$ is in its simplest form .
::首先,承认2 3是最简单的形式。

Next, simplify $\begin{aligned} \frac{10}{15} \end{aligned}$ .
::接下来,简化10 15。

$\begin{aligned} \frac{10 \div 5}{15 \div 5} = \frac{2}{3} \end{aligned}$

Then, compare the two fractions, or ratios.
::然后,比较两个分数,或比率。

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

The answer is that the two ratios are equal, or equivalent.
::答案是这两个比率相同或相当。

Another way to check is to change $\begin{aligned} \frac{2}{3} \end{aligned}$ into a fraction with 15 as the denominator .
::另一种检查方式是将23变成一个分数,以15为分母。

First, write an equation.
::首先,写一个方程。

$\begin{aligned} \frac{2}{3} \end{aligned}$ = $\begin{aligned} \frac{x}{15} \end{aligned}$
::2 3 = x 15

Next, recognize that $\begin{aligned} 3 \times 5 = 15 \end{aligned}$ and multiply both the numerator and denominator times 3. Remember that whatever is done to one term must also be done to the other.
::接下来,请确认 3 × 5 = 15 = 15 并乘以分子和分母乘以 3 。请记住,无论对某一个术语做什么,也必须对另一个术语做。

$\begin{aligned} \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \end{aligned}$

Then, compare the ratios.
::然后,比较比率。

$\begin{aligned} \frac{10}{15} \end{aligned}$ = $\begin{aligned} \frac{10}{15} \end{aligned}$

The answer is that the two ratios are equivalent.
::答案是这两个比率相等。

The last way to determine whether or not two ratios are equivalent is to cross multiply.
::最后确定两个比率是否相等的方法是乘以乘数。

First, write an equation that assumes the fractions are equal.
::首先,写一个假设分数相等的方程。

$\begin{aligned} \frac{2}{3} = \frac{10}{15} \end{aligned}$

Next, multiply the numerator of one fraction times the denominator of the other and vice versa.
::接下来,乘以一个分数乘以另一个分母的分母,反之亦然。

$\begin{aligned} 2 \times 15 = 3 \times 10 \end{aligned}$

Then, compare the two products.
::然后,比较这两个产品。

30=30

The products are equal.
::产品是平等的。

The answer is that the ratios are equivalent.
::答案是比率相等。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Kayla and Torrey and the reading challenge.
::之前,你被问及Kayla和Torrey的问题以及阅读的挑战。

Kayla had read 6 books in 8 weeks and Torrey had read 3 books every four weeks. Had they read the same number of books? Are the ratios equivalent?
::Kayla在8周内读了6本书,Torrey每4周读了3本书,他们读的书数是否相同?比率是否相等?

First, write the ratios as fractions.
::首先,将比率写成分数。

Kayla $\begin{aligned} \frac{6 b o o k s}{8 w e e k s} \end{aligned}$ or $\begin{aligned} \frac{6}{8} \end{aligned}$
::Kayla 6 b o k s 8 w e e e e k s 或 6 8

Torrey $\begin{aligned} \frac{3 b o o k s}{4 w e e k s} \end{aligned}$ or $\begin{aligned} \frac{3}{4} \end{aligned}$
::托雷 3 b o k s 4 w e e e e e k s 或 3 4

Next, determine a method, cross multiplication, and write an equation.
::下一步,确定一个方法,交叉乘法,然后写一个方程。

$\begin{aligned} \frac{6}{8} = \frac{3}{4} \end{aligned}$

Then cross multiply and compare.
::然后交叉乘数和比较。

24 = 24

The answer is that the girls have read the same number of books. The ratios are equivalent.
::答案是,女孩读的书数量相同,比例相等。

Example 2
::例2

Determine if 7:6 and 13:12 are equivalent ratios.
::确定7:6和13:12是等值比率。

First, rewrite the ratios as fractions.
::首先,将比率重写为分数。

$\begin{aligned} \frac{7}{6} \end{aligned}$ and $\begin{aligned} \frac{13}{12} \end{aligned}$ .
::7、6和13 12 。

Next, recognize that 6 must be multiplied by 2 in order to equalize the denominators and perform the same operation to both terms.
::其次,承认6必须乘以2,才能使分母等同,并按两个条件执行同样的操作。

$\begin{aligned} \frac{7}{6} = \frac{7 \times 2}{6 \times 2} = \frac{14}{12} \end{aligned}$

Then, compare the ratios.
::然后,比较比率。

$\begin{aligned} \frac{13}{12} \neq \frac{14}{12} \end{aligned}$

The answer is that the two ratios are not equivalent.
::答案是这两个比率不相等。

Example 3
::例3

Determine if the ratios are equivalent.
::确定比率是否相等。

$\begin{aligned} \frac{3}{4} \end{aligned}$ and $\begin{aligned} \frac{9}{12} \end{aligned}$
::3、4和9 12

First, since a method is not specified, choose a method one. Determine if it is easiest to work with fourths, twelfths, or cross multiply. In this problem, it is easy to cross multiply.
::首先,由于没有指定一种方法, 请选择一种方法。确定是否最容易使用四、十二或交叉乘法。在此问题上, 很容易交叉乘法。

Next, write an equation assuming the ratios are equal.
::接下来,写一个方程假设比率是相等的。

$\begin{aligned} \frac{3}{4} = \frac{9}{12} \end{aligned}$

Then, cross multiply.
::然后,交叉倍增。

$\begin{aligned} 3 \times 12 = 9 \times 4 \end{aligned}$

Next, compare.
::下一个,比较。

36 = 36

The answer is that the ratios are equivalent.
::答案是比率相等。

Example 4
::例4

Change both fractions to 30ths to determine whether or not the ratios $\begin{aligned} \frac{5}{6} \end{aligned}$ and $\begin{aligned} \frac{20}{30} \end{aligned}$ are equivalent.
::将两个分数都改为30,以确定5 6和20 30的比率是否相等。

First, write an equation.
::首先,写一个方程。

$\begin{aligned} \frac{5}{6} = \frac{x}{30} \end{aligned}$
::5 6 =x 30

Next, recognize that 6 must be multiplied times 5 to get 30, and multiply both terms times 5.
::其次,承认6乘以5乘以5才能得到30, 并同时乘以两个条件乘以5。

$\begin{aligned} \frac{5 \times 5}{6 \times 5} = \frac{25}{30} \end{aligned}$

Then, compare the ratios expressed as 30ths.
::然后,比较以30秒表示的比率。

$\begin{aligned} \frac{25}{30} \neq \frac{20}{30} \end{aligned}$

The answer is that the ratios are not equivalent.
::答案是比率不相等。

Example 5
::例5

Are the ratios 4 to 5 and 8 to 10 equivalent?
::4至5和8至10的比率是否相等?

First, write the ratios as fractions.
::首先,将比率写成分数。

$\begin{aligned} \frac{4}{5} \end{aligned}$ and $\begin{aligned} \frac{8}{10} \end{aligned}$
::4、5和8 10

Next, determine a method. In this case, cross multiplication.
::下一步,确定一个方法。在这种情况下,交叉乘法。

$\begin{aligned} 4 \times 10 = 5 \times 8 \end{aligned}$

Then, compare.
::那么,比较。

40 = 40

The answer is that the ratios are equivalent.
::答案是比率相等。

Review
::回顾

Determine whether each of the following ratio pairs is equal. Write yes if they are equal and no if they are not equal.
::确定以下各对比例是否相等。如果是相等的, 请写是, 如果不是等的, 请写是。

$\begin{aligned} \frac{1}{2} and \frac{6}{12} \end{aligned}$
::1、2和6 12

$\begin{aligned} \frac{3}{8} and \frac{1}{4} \end{aligned}$
::3 3 和 1 4

$\begin{aligned} \frac{6}{7} and \frac{2}{3} \end{aligned}$
::6-7和2 3

$\begin{aligned} \frac{6}{7} and \frac{12}{14} \end{aligned}$
::6、7和12 13

$\begin{aligned} \frac{2}{3} and \frac{10}{15} \end{aligned}$
::2 3和10 15

$\begin{aligned} \frac{17}{21} and \frac{6}{7} \end{aligned}$
::17、21和6 7

$\begin{aligned} \frac{24}{48} and \frac{12}{24} \end{aligned}$
::24、48和12 24

$\begin{aligned} \frac{16}{18} and \frac{32}{38} \end{aligned}$
::16、18和32 38

$\begin{aligned} \frac{9}{45} and \frac{1}{9} \end{aligned}$
::9、45和1 9

$\begin{aligned} \frac{4}{6} and \frac{44}{66} \end{aligned}$
::4、6和44 64

$\begin{aligned} \frac{6}{9} and \frac{4}{6} \end{aligned}$
::6、9和4 6、9和4 6

$\begin{aligned} \frac{14}{16} and \frac{20}{24} \end{aligned}$
::14、16和20 24

$\begin{aligned} \frac{12}{16} and \frac{24}{32} \end{aligned}$
::12、12和24 31

$\begin{aligned} \frac{24}{48} and \frac{1}{2} \end{aligned}$
::24 48和1 2

$\begin{aligned} \frac{84}{96} and \frac{3}{4} \end{aligned}$
::84 96和3 4

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

等效比率

Working with Equivalent Ratios ::与等同比率合作

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

Working with Equivalent Ratios
::与等同比率合作

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