课程： 4.1 书写、比较和顺序比率

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书写、比较和顺序比率
Mintgrove Middle School has a total of 300 students. If the ratio of boys to girls is 31 to 44, how many more girls are there in the school?
::Mintgrove中学共有300名学生,如果男孩与女孩的比例是31比44,学校里还有多少女孩?

In this concept, you will learn to write, compare and order ratios.
::在此概念中,您将学习写作、比较和顺序比率。

Ratios
::比率比率比率

A ratio is a comparison of two things or two quantities.
::A比率是两种物或两种物的比较。

Let’s look at an example where you are comparing ratios.
::让我们看看一个比较比率的例子。

Imagine that there are 25 students in a class; 12 boys and 13 girls.
::想象一个班级有25名学生:12名男孩和13名女孩。

You could say that the ratio of boys to girls is 12 to 13. You could also write this ratio as $\begin{aligned} 12 : 13 or \frac{12}{13} \end{aligned}$ .
::你可以说,男孩与女孩的比例是12对13,也可以写12对13,或者12对13。

You could also say that the ratio of boys to the total number of students can be written as 12 to 25 or $\begin{aligned} 12 : 25 or \frac{12}{25} \end{aligned}$ since the total number of students is $\begin{aligned} 12 + 13 = 25. \end{aligned}$
::您也可以说,男生占学生总数的比例可以写为12至25、12至25或12至25,因为学生总数是12+13=25。

The ratio of girls to the total number of students can then be written as 13 to 25 or $\begin{aligned} 13 : 25 or \frac{13}{25} \end{aligned}$ .
::然后,女孩占学生总数的比例可以写为13-25或13-25或13-25或13-25。

When you use ratios to compare, the order makes a difference . If you look at the previous three statements, comparing boys to girls, the number of boys is in numerator . When you compared the number of boys to the total number of students, the number of boys was in the numerator. When you compared the number of girls to the total number of students, the number of girls was in the numerator because it was first in the statement " The ratio of girls to the total number of students... "
::当您使用比对比率比较时,顺序就不同了。如果您查看前三次对男女孩的比较,男孩的数量在分子中。当您将男孩的数量与学生总数进行比较时,男孩的数量在分子中。当您将女孩的数量与学生总数进行比较时,女孩的数量在分子中,因为“女孩与学生总数的比例”的第一个词是“女孩与学生总数的比例......” 。

Th is also shows that there are three different ways to write ratios. You can write them with a colon between the two values that you are comparing, you can write them using the word “to”, and you can write them by putting the values in fraction form. You can choose which way you want to write a ratio and these ways are interchangeable too.
::这还显示有三种不同的写率方法。您可以在比较的两个值之间用冒号来写, 您可以用“ to” 字来写, 也可以用分数形式来写。您可以选择您想要写一个比例的方式, 这些方式也可以互换。

Now let’s look at an example.
::现在让我们来举一个例子。

There are 32 red and yellow candies in a bag. There are 10 yellow candies. What is the ratio of red candies to total candies in the bag?
::包里有32个红色和黄色糖果,10个黄色糖果。包里红色糖果与全部糖果的比例是多少?

First, find the number of red candies by subtracting the number of yellow candies from the total.
::首先,通过从总数中减去黄色糖果的数量,找到红色糖果的数量。

$\begin{aligned} 32 - 10 = 22 \end{aligned}$

Next, write the ratio of red candies to the total number of candies.
::接下来,写下红糖果与糖果总数的比例。

$\begin{aligned} \begin{array}{rcl} \frac{red candies}{total candies} & = & \frac{22}{32} \end{array} \end{aligned}$

::=2232

Then, simplify the fraction.
::然后,简化分数。

$\begin{aligned} \begin{array}{rcl} \frac{red candies}{total candies} & = & \frac{22}{32} \\ \frac{red candies}{total candies} & = & \frac{11}{16} \end{array} \end{aligned}$

::=2232 =1116 =1116

The answer is $\begin{aligned} \frac{11}{16} \end{aligned}$ .
::答案是1116号

The ratio of red candies in the bag is $\begin{aligned} \frac{11}{16} \end{aligned}$ .
::包里红糖的比例是1116

You could also write this ratio as 11 to 16 or $\begin{aligned} 11 : 16 \end{aligned}$ .
::您也可以写为 11 至 16 或 11 至 16 。

You can also order ratios. When you have more than two ratios, you can write them in order from least to greatest or from greatest to least.
::您也可以订购比率。当您拥有两个以上的比率时,您可以从最小到最大或者从最大到最小写成。

Let’s look at an example.
::让我们举个例子。

Order the following ratios from least to greatest:
::将以下比例从最低到最高顺序排列如下:

$\begin{aligned} 10 to 15 \end{aligned}$ , $\begin{aligned} \frac{16}{36} \end{aligned}$ , and $\begin{aligned} 12 : 48 \end{aligned}$ .
::10至15、1636和12:48。

First, let’s write them in the same form first of all. Let’s work with fraction form.
::首先,让我们首先以同样的形式写出来。让我们以分数的形式工作。

$\begin{aligned} 10 to 15 = \frac{10}{15} \frac{16}{36} 12 : 48 = \frac{12}{48} \end{aligned}$

Next, put the fractions in simplest form .
::下一个,将分数以最简单的形式排列。

$\begin{aligned} \frac{10}{15} = \frac{2}{3} \frac{16}{36} = \frac{4}{9} \frac{12}{48} = \frac{1}{4} \end{aligned}$

Then, get a common denominator so you can order these. The lowest common denominator of 3, 9, and 4 is 36.
::然后,获得一个共同的分母,这样你就可以订购这些。3、9和4的最低共同分母是36。

$\begin{aligned} \frac{2}{3} \times \frac{12}{12} = \frac{24}{36} \end{aligned}$

$\begin{aligned} \frac{4}{9} \times \frac{4}{4} = \frac{16}{36} \end{aligned}$

$\begin{aligned} \frac{1}{4} \times \frac{9}{9} = \frac{9}{36} \end{aligned}$

The order from least to greatest then is $\begin{aligned} \frac{1}{4}, \frac{4}{9}, \frac{2}{3} or 12 : 48, \frac{16}{36}, 10 to 15 \end{aligned}$ .
::从最起码到最大的顺序是14,49,23或12:48,1636,10至15。

You can also . This is when you have two or more ratios and you want to figure out which ones are larger and which ones are smaller.
::您也可以这样做。这是当您有两个或两个以上的比例, 并且想找出哪些更大,哪些较小的时候。

Let’s look at an example.
::让我们举个例子。

Mr. Collin’s class has 30 total students. Of these, 12 are boys. Mrs. Peterson’s class has 25 students. Of these, 11 are boys. Which class has a higher ratio of boys to total students?
::科林先生的班级共有30名学生,其中12名是男孩,彼得森夫人的班级有25名学生,其中11名是男孩,哪个班的男生占学生总数的比例较高?

First, find the ratio of boys to total students for both classes.
::首先,找出这两个班的男生占学生总数的比例。

$\begin{aligned} \begin{array}{rcl} Mr.Collin's class : \frac{12}{30} Mr.Peterson's class : \frac{11}{25} \end{array} \end{aligned}$
Next, simplify the ratio for Mr. Collin’s class.
::Collin先生的班级:1230 Peterson先生的班级:1125Next,简化Collin先生的班级比例。

$\begin{aligned} \begin{array}{rcl} \frac{12}{30} & = & \frac{2}{5} \end{array} \end{aligned}$

Then, compare the ratios the same way you compare fractions. Find a common denominator and compare the numerators. The least common denominator is 25.
::然后以比较分数的相同方式比较比率。找到一个共同的分母, 比较分子。最小的分母是25。

$\begin{aligned} Mr.Collin's class : \frac{2}{5} \end{aligned}$
::科林先生的班级: 25

$\begin{aligned} \frac{2}{5} \times \frac{5}{5} = \frac{10}{25} \end{aligned}$

$\begin{aligned} Mr.Peterson's class : \frac{11}{25} \end{aligned}$
::佩特森先生的班级:1125

The answer is that Mr. Peterson’s class has the higher ratio of boys to total students.
::答案是Peterson先生的班级男生占学生总数的比例较高。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the Mintgrove Middle School’s girls.
::更早之前,你曾遇到Mintgrove中学女孩的问题。

There are 300 students and the ratio of boys to girls is $\begin{aligned} 31 : 44 \end{aligned}$ .
::有300名学生,男孩与女孩的比例为31:44。

First, since you don’t know the exact number of either boys or girls, you can’t set up the ratio right away. Therefore, let $\begin{aligned} x \end{aligned}$ be the number of girls; then the number of boys will be $\begin{aligned} 300 - x \end{aligned}$ , since there are 300 students overall.
::首先,既然你不知道男孩或女孩的确切人数,你就不能马上设置这个比率。因此,让x是女孩人数;那么男孩人数将是300-x,因为学生总数是300人。

Next, set up the ratio by comparing the number of boys to the number of girls.
::其次,将男孩人数与女孩人数进行比较,以此确定这一比率。

$\begin{aligned} \frac{31}{44} = \frac{300 - x}{x} \end{aligned}$

Then, cross multiply.
::3144=300 - xx 然后,交叉乘以。

$\begin{aligned} \begin{array}{rcl} \frac{31}{44} & = & \frac{300 - x}{x} \\ 31 x & = & 44 (300 - x) \\ 31 x & = & 13200 - 44 x \end{array} \end{aligned}$

::3144=300-xx31x=44(300-xx)31x=13-200-44x

Then, simplify by adding $\begin{aligned} 44 x \end{aligned}$ to both sides.
::然后,简化,在双方增加44x。

$\begin{aligned} \begin{array}{rcl} 31 x & = & 13200 - 44 x \\ 31 x + 44 x & = & 13200 - 44 x + 44 x \\ 75 x & = & 13200 \end{array} \end{aligned}$

::31x=13200-44x31x+44x13200-44x44x4x4475x13200

Then, solve for $\begin{aligned} x \end{aligned}$ by dividing both sides by 75.
::然后,通过将双方除以75来解决x问题。

$\begin{aligned} \begin{array}{rcl} 75 x & = & 13200 \\ \frac{75 x}{75} & = & \frac{13200}{75} \\ x & = & 176 \end{array} \end{aligned}$

::75x=1320075x75=1320075x15x176

The answer is 176.
::答案是176个

There are 176 girls in the school.
::学校里有176名女生。

Example 2
::例2

Write a ratio in simplest form to describe this situation. Marcy ate 14 cookies in 28 minutes. Write a ratio to compare cookies to minutes.
::以最简单的形式写一个比例来描述这种情况。 Marcy 在28分钟内吃了14个饼干。写一个比例来比较饼干和分钟。

First, find the ratio of the number of cookies to time.
::首先,找到饼干数量与时间的比例。

$\begin{aligned} \begin{array}{rcl} \frac{number of cookies}{time} & = & \frac{14}{28} \end{array} \end{aligned}$

::饼干时间数=1428

Next, simplify the ratio.
::接下来,简化比例。

$\begin{aligned} \begin{array}{rcl} \frac{14}{28} & = & \frac{1}{2} \end{array} \end{aligned}$

The answer is $\begin{aligned} \frac{1}{2} \end{aligned}$ .
::答案是12岁

You could also write this ratio as $\begin{aligned} 14 : 28 or 1 : 2 \end{aligned}$ .
::您也可以写为 14: 28 或 1: 2 。

Example 3
::例3

Simplify the ratio.
::简化比例。

$\begin{aligned} \frac{7}{21} \end{aligned}$

First, find a number that divides into both 7 and 21.
::首先,找到一个分为7和21的数值。

Both 7 and 21 are divisible by 7.
::7 和21 均除以 7 。

Next, divide each of the numbers in the fraction by 7.
::接下来,将分数中的每个数字除以 7。

$\begin{aligned} \frac{7}{21} = \frac{1}{3} \end{aligned}$

The answer is $\begin{aligned} \frac{1}{3} \end{aligned}$ .
::答案是13岁

Example 4
::例4

Simplify the ratio.
::简化比例。

$\begin{aligned} 5 : 30 \end{aligned}$

First, write the ratio as a fraction.
::首先,将比率写成一个分数。

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

Next, simplify the fraction. Both 5 and 30 are divisible by 5.
::接下来,简化分数。 5 和 30 都可除以 5 。

$\begin{aligned} \frac{5}{30} = \frac{1}{6} \end{aligned}$

Then, rewrite the fraction as a ratio.
::然后,将分数重写为比例。

$\begin{aligned} \frac{1}{6} = 1 : 6 \end{aligned}$

The answer is $\begin{aligned} 1 : 6 \end{aligned}$ .
::答案是1:6。

Example 5
::例5

Simplify the ratio.
::简化比例。

$\begin{aligned} 24 to 36 \end{aligned}$
::24至36

First, write the ratio as a fraction.
::首先,将比率写成一个分数。

$\begin{aligned} 24 to 36 = \frac{24}{36} \end{aligned}$

::24至36=2436

Next, simplify the fraction. Both 24 and 36 are divisible by 12.
::接下来,简化分数。 24和36的分数都可除以 12 。

$\begin{aligned} \frac{24}{36} = \frac{2}{3} \end{aligned}$

Then, rewrite the fraction as a ratio.
::然后,将分数重写为比例。

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

::23=2至3

The answer is 2 to 3.
::答案是2比3

Review
::回顾

Look at each ratio. Then write it in the other two ways that it can be written.
::看看每个比例,然后用另外两种写法写出来。

1. 16 to 3.
::1. 16至3岁。

2. 4 to 5.
::2. 4至5岁。

3. 1: 4 .

4. $\begin{aligned} \frac{12}{1} \end{aligned}$ .

5. 6:11.

6. 33 to 100.
::6. 33至100个。

7. $\begin{aligned} \frac{4}{9} \end{aligned}$ .

8. 3 to 4.
::8. 3至4个。

9. 45 to 12.
::9. 45至12岁。

10. 12:12.

Simplify each ratio and write your answer in fraction form.
::简化每个比例, 并以分数形式写入您的回答。

11. 4 to 12.
::11. 4至12岁。

12. 5:20.

13. 36 to 6.
::13. 36至6岁。

14. 18:36.

15. 20 to 100.
::15. 20至100岁。

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。

章节大纲

常规

书写、比较和顺序比率

Ratios ::比率比率比率

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)

Ratios
::比率比率比率

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)