Course: 4.4 使用交叉产品进行书写和解决比例比例

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使用交叉产品撰写和解决比例比例
Sandra is doing a survey to discover students’ favorite type of movie. She asks 20 students and 7 say that science fiction is their favorite genre. At this rate , if Sandra asks all 460 students in the school, how many will choose science fiction as their favorite type of movie?
::桑德拉正在做一项调查,以发现学生最喜欢的电影类型。她问20个学生,7个说科幻小说是他们的最爱。以这个速度,如果桑德拉问学校所有460名学生,有多少人会选择科幻小说作为他们最喜欢的电影类型?

In this concept, you will learn to write and solve proportions by using cross-products and algebra.
::在此概念中,您将学会通过使用交叉产品和代数来撰写和解决比例问题。

Cross Products
::交叉产品

A proportion is created when two ratios are equal. Sometimes, you will know three parts of a proportion and there will be one missing part. When this happens, you will need to solve a proportion.
::当两个比例相等时, 比例就会被创建。有时, 您将会知道一个比例的三分之一, 将会有一个缺失部分。当这种情况发生时, 您需要解决一个比例。

A way of solving a proportion is called cross-multiplying , and this involves algebra. The rule for cross multiplying is:
::一种解决比例的方法被称为交叉乘法,这涉及代数。

If $\begin{aligned} \frac{a}{b} = \frac{c}{d} \end{aligned}$ , then $\begin{aligned} a d = b c \end{aligned}$ .
::如果ab=cd,则ad=bc。

This is also called “the product of the means is equal to the product of the extremes .” The values in the ‘ $\begin{aligned} b \end{aligned}$ ’ and ‘ $\begin{aligned} c \end{aligned}$ ’ positions are called the means , and the values in the ‘ $\begin{aligned} a \end{aligned}$ ’ and ‘ $\begin{aligned} d \end{aligned}$ ’ positions are called the extremes .
::这也被称为 " 手段的产物等于极端的产物 " 。 `b ' 和`c ' 立场中的价值观被称为手段,`a ' 和`d ' 立场中的价值观被称为极端。

Let’s look at an example.
::让我们举个例子。

Solve for $\begin{aligned} x \end{aligned}$ .
::解决x。

$\begin{aligned} \frac{x}{5} = \frac{9}{10} \end{aligned}$

::x5=910 x5=910

First, cross multiply.
::首先,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{x}{5} & = & \frac{9}{10} \\ 10 x & = & 9 \times 5 \\ 10 x & = & 45 \end{array} \end{aligned}$

::x5=91010x=9x510x=45

Next, solve for $\begin{aligned} x \end{aligned}$ by dividing both sides by 10.
::接下来,通过将两边除以10来解决 x 问题。

$\begin{aligned} \begin{array}{rcl} 10 x & = & 45 \\ \frac{10 x}{10} & = & \frac{45}{10} \\ x & = & 4.5 \end{array} \end{aligned}$

::10x=4510x10=4510x=4.5

The answer is 4.5.
::答案是4.5。

Here is another example.
::下面是另一个例子。

Solve for $\begin{aligned} x \end{aligned}$ .
::解决x。

$\begin{aligned} \frac{4}{5} = \frac{16}{x} \end{aligned}$

::45=16x 45=16x

First, cross multiply.
::首先,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{4}{5} & = & \frac{16}{x} \\ 4 x & = & 5 \times 16 \\ 4 x & = & 80 \end{array} \end{aligned}$

::45=16x4x=5x164x=80

Next, solve for $\begin{aligned} x \end{aligned}$ by dividing both sides by 4.
::下一步,通过将两边除以4来解决 x 问题。

$\begin{aligned} \begin{array}{rcl} 4 x & = & 80 \\ \frac{4 x}{4} & = & \frac{80}{4} \\ x & = & 20 \end{array} \end{aligned}$

::4x=804x4=804x4=804x=20

The answer is 20.
::答案是20岁

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Sandra’s science fiction study.
::Sandra的科幻小说研究有问题。

Sandra wants to figure out how many of the 460 students will choose science fiction, given that 7 out of 20 students already selected science fiction as their favorite movie genre.
::Sandra想知道460名学生中有多少人会选择科幻小说, 因为20名学生中有7人已经选择科幻小说作为他们最喜欢的电影流派。

First, write your proportion.
::首先,写下你的比例

$\begin{aligned} \frac{7}{20} = \frac{x}{460} \end{aligned}$

::720=x460

Next, cross multiply.
::下一个,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{7}{20} & = & \frac{x}{460} \\ 20 x & = & 7 \times 460 \\ 20 x & = & 3220 \end{array} \end{aligned}$

::720=x46020x=7x46020x=3220

Then, solve for $\begin{aligned} x \end{aligned}$ by dividing both sides by 20.
::然后,通过将双方除以20来解决x问题。

$\begin{aligned} \begin{array}{rcl} 20 x & = & 3220 \\ \frac{20 x}{20} & = & \frac{3220}{20} \\ x & = & 161 \end{array} \end{aligned}$

::20x=3222020x20=32220xx=161

The answer is 161.
::答案是161

Sandra, using the ratio of $\begin{aligned} 7 : 20 \end{aligned}$ would find 161 of the 46- students choose science fiction as their favorite type of movie.
::Sandra,使用7:20的比例,46名学生中有161人选择科幻小说作为他们最喜欢的电影类型。

Example 2
::例2

The ratio of apples to bananas at a store is 3 to 8. If there are 90 apples, how many bananas are there?
::一家商店的苹果与香蕉的比例是3比8,如果有90个苹果,有多少香蕉?

First, write your proportion.
::首先,写下你的比例

$\begin{aligned} \frac{3}{8} = \frac{90}{x} \end{aligned}$

::38=90x

Next, cross multiply.
::下一个,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{3}{8} & = & \frac{90}{x} \\ 3 x & = & 8 \times 90 \\ 3 x & = & 720 \end{array} \end{aligned}$

::38=90x3x=8x903x=720

Then, solve for $\begin{aligned} x \end{aligned}$ by dividing both sides by 3.
::然后,通过将两边除以3来解决 x 问题。

$\begin{aligned} \begin{array}{rcl} 3 x & = & 720 \\ \frac{3 x}{3} & = & \frac{720}{3} \\ x & = & 240 \end{array} \end{aligned}$
The answer is 240.
::3x=7203x3=7203x3=2203x=240 答案是240。

There were 240 bananas.
::有240个香蕉。

Example 3
::例3

Solve for $\begin{aligned} x \end{aligned}$ .
::解决x。

$\begin{aligned} \frac{x}{9} = \frac{18}{27} \end{aligned}$

::x9=1827

First, cross multiply.
::首先,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{x}{9} & = & \frac{18}{27} \\ 27 x & = & 9 \times 18 \\ 27 x & = & 162 \end{array} \end{aligned}$

::x9=182727x=9x1827x=162

Next, solve for $\begin{aligned} x \end{aligned}$ by dividing both sides by 27.
::接下来,通过将双方除以27来解决x问题。

$\begin{aligned} \begin{array}{rcl} 27 x & = & 162 \\ \frac{27 x}{27} & = & \frac{162}{27} \\ x & = & 6 \end{array} \end{aligned}$
The answer is 6.
::27x=16227x27=16227x=6 答案是6

Example 4
::例4

Solve for $\begin{aligned} y \end{aligned}$ .
::解决你。

$\begin{aligned} \frac{3}{7} = \frac{33}{y} \end{aligned}$

::37=33y

First, cross multiply.
::首先,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{3}{7} & = & \frac{33}{y} \\ 3 y & = & 7 \times 33 \\ 3 y & = & 231 \end{array} \end{aligned}$

::37=33y3y=7×333y=231

Next, solve for $\begin{aligned} y \end{aligned}$ by dividing both sides by 3.
::接下来,通过将双方除以3来解决y。

$\begin{aligned} \begin{array}{rcl} 3 y & = & 231 \\ \frac{3 y}{3} & = & \frac{231}{3} \\ y & = & 77 \end{array} \end{aligned}$

::3y=2313y3=2313y=77

The answer is 77.
::答案是77岁

Example 5
::例5

Solve for $\begin{aligned} x \end{aligned}$ .
::解决x。

$\begin{aligned} \frac{x}{2} = \frac{49.5}{99} \end{aligned}$

::x2=49.599

First, cross multiply.
::首先,交叉乘数。

$\begin{aligned} \begin{array}{rcl} \frac{x}{2} & = & \frac{49.5}{99} \\ 99 x & = & 2 \times 49.5 \\ 99 x & = & 99 \end{array} \end{aligned}$

::x2=49.59999x=2×49.599x=99

Next, solve for $\begin{aligned} x \end{aligned}$ by dividing both sides by 99.
::接下来,通过将双方除以99来解决x问题。

$\begin{aligned} \begin{array}{rcl} 99 x & = & 99 \\ \frac{99 x}{99} & = & \frac{99}{99} \\ x & = & 1 \end{array} \end{aligned}$
The answer is 1.
::99x=999999x99=9999x=1 回答是1

Review
::回顾

Solve each proportion by using cross–multiplying with algebra. You may round to the nearest tenth when necessary.
::使用乘以代数的交叉倍数来解决每一比例。必要时您可以圆到最接近的十分之一。

1. $\begin{aligned} \frac{3}{5} = \frac{y}{2.5} \end{aligned}$
::1. 35=y2.5

2. $\begin{aligned} \frac{6}{7} = \frac{2.5}{y} \end{aligned}$
::2. 67=2.5y

3. $\begin{aligned} \frac{4}{5} = \frac{2}{x} \end{aligned}$
::3. 45=2x

4. $\begin{aligned} \frac{9}{11} = \frac{14}{x} \end{aligned}$
::4. 911=14x

5. $\begin{aligned} \frac{2}{3} = \frac{5}{y} \end{aligned}$
::5. 23=5y

6. $\begin{aligned} \frac{12}{3} = \frac{4}{y} \end{aligned}$
::6. 123=4y

7. $\begin{aligned} \frac{22}{40} = \frac{11}{x} \end{aligned}$
::7. 2240=11x

8. $\begin{aligned} \frac{60}{x} = \frac{5}{10} \end{aligned}$
::8. 60x=510

9. $\begin{aligned} \frac{12}{50} = \frac{3}{y} \end{aligned}$
::9. 1250=3y

10. $\begin{aligned} \frac{42}{36} = \frac{7}{y} \end{aligned}$
::10. 4236=7y

11. $\begin{aligned} \frac{56}{63} = \frac{x}{9} \end{aligned}$
::11. 5663=x9

12. $\begin{aligned} \frac{120}{130} = \frac{1.2}{y} \end{aligned}$
::12. 120130=1.2y

Solve each problem.
::解决每一个问题。

13. The ratio of fiction to nonfiction books at a library is 5 to 3. If there are 480 nonfiction books, find $\begin{aligned} f \end{aligned}$ , the number of fiction books.
::13. 图书馆小说书与非小说书之比为5比3,如果有480本非小说书,见 f,小说书的数量。

14. The ratio of cherry trees to apple trees at an orchard is 4 to 9. If there are 184 cherry trees, find $\begin{aligned} a \end{aligned}$ , the number of apple trees.
::14. 果园樱树与苹果树的比例为4比9,如果果园有184棵樱树,则找到a,苹果树的数量。

15. The ratio of cars to SUVs in a parking lot is 10 to 7. If there are 84 SUVs, find $\begin{aligned} c \end{aligned}$ , the number of cars in the lot.
::15. 在停车场,汽车与SUV的比率是10比7,如果有84辆SUV,则在停车场找到C,汽车的数量。

Review (Answers)
::回顾(答复)

To see the answer key for this book, go to the and click on the Answer Key under the ' ' option.
::要查看本书的答案键, 请在“ ” 选项下点击答案键。

Resources
::资源

Section outline

General

使用交叉产品撰写和解决比例比例

Cross Products ::交叉产品

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

Cross Products
::交叉产品

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