Course: 5.9比例 | The Way To Learn

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比例
Blake volunteered to be part of a "living flag" at the state fairgrounds on Independence Day. He had a small flag and wondered how the organizers would convert the dimensions to accommodate hundreds of people. Blake's flag measured 4 inches by 6 inches. If the length of the living flag was to be 120 feet, what should the width be?
::Blake自愿成为独立日国家竞技场“活旗”的一部分。他有一个小旗,并想知道组织者将如何转换维度以容纳数百人。 Blake的国旗乘以4英寸乘以6英寸。如果活旗的长度是120英尺,那么宽度应该有多大?

In this concept, you will learn how to write and work with proportions.
::在这个概念中,你将学习如何以比例写作和工作。

Working with Proportions
::与比例工作

A ratio represents a comparison between two quantities. Equivalent ratios are ratios that are equal. A proportion is made up of two equivalent ratios.
::A比率是两个数量之间的比较,等值比率是相等比率,一个比例由两个等值比率组成。

An example of a proportion is $\begin{aligned} \frac{1}{2} = \frac{2}{4} \end{aligned}$
::比例的一个例子是 1 2 = 2 4

$\begin{aligned} \frac{1}{2} \end{aligned}$ and $\begin{aligned} \frac{2}{4} \end{aligned}$ are in proportion to one another. They are proportional.
::2和24是比例对等的,是比例对等的。

When given three parts of a proportion, the fourth can be determined.
::当给定比例的三部分时,可以确定第四部分。

Here is an example.
::举一个例子。

$\begin{aligned} \frac{1}{2} = \frac{x}{12} \end{aligned}$
::1 2 = x 12

In this proportion, x is unknown.
::在这一比例中,x为未知值。

Proportional reasoning , or examining the relationship between two numbers, can be used to determine the value of x.
::比例推理或审查两个数字之间的关系,可用来确定x的价值。

First, since both denominators are given, begin with those two numbers.
::首先,由于给出了两个分母,从这两个数字开始。

2 and 12
::2和12

Next, recognize the ratio that is given. In this problem it is $\begin{aligned} \frac{1}{2} \end{aligned}$ . Think about what can be done to the denominator of the given ratio to make it equal to the denominator of the ratio with the unknown in it. In this case, what can be done to 2 to make it equal to 12.
::下一步, 承认给定的比例。在此问题上, 给定的比例是 12 。想想给定比例的分母可以做些什么, 使它与未知比例的分母相等。在这种情况下, 要让给定比例的分母等于 12 , 可以对 2 做什么。

2 x 6 = 12
::2 x 6 = 12

2 can be multiplied times 6 to equal 12.
::2 可以乘以 6 乘以 6 等于 12。

Then, multiply both the numerator and denominator of the given ratio times 6.
::然后,乘以给定比率乘以6的分子和分母。

$\begin{aligned} \frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} = \frac{x}{12} \end{aligned}$
::1 2 = 1 = 1 × 6 2 × 6 = 6 12 = x 12

The answer is 6.
::答案是6个

$\begin{aligned} \frac{1}{2} = \frac{6}{12} \end{aligned}$

The ratio $\begin{aligned} \frac{1}{2} \end{aligned}$ is equal to the ratio $\begin{aligned} \frac{6}{12} \end{aligned}$
::12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁 12岁

The two ratios are proportional to one another. They have the same proportions.
::两者的比例是成比例的,比例相同。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Blake and the living flag.
::早些时候,你得到一个问题关于布莱克和活旗。

He wondered how his little 4" by 6" parade flag could be used to calculate the width of a giant human flag that has a length of 120 feet.
::他想知道如何用他的小四四乘六的游行旗来计算长达120英尺的巨型人类旗帜的宽度。

First, write an equation for the proportion.
::首先,为比例写一个方程。

$\begin{aligned} \frac{4}{6} = \frac{x}{120} \end{aligned}$
::4 6 = x 120

Next, work with the denominators.
::接下来,与分母合作。

6 and 120
::6和120

Then, using the given ratio of $\begin{aligned} \frac{4}{6} \end{aligned}$ , determine what can be done to 6 to make it equal to 120.
::然后,使用4 6的比例,确定可以做到6的比例,使其等于120。

6 x 20 = 120
::6 x 20 = 120

Next, multiply both the numerator and denominator of the given ratio times 20.
::下一步,乘以给定比率的分子数和分母乘以20。

$\begin{aligned} \frac{4 \times 20}{6 \times 20} = \frac{80}{120} \end{aligned}$

The answer is y = 80 feet.
::答案是y=80英尺

$\begin{aligned} \frac{4}{6} = \frac{80}{120} \end{aligned}$

Example 2
::例2

Use proportional reasoning to solve for x.
::使用比例推理来解决 x 。

$\begin{aligned} \frac{15}{35} = \frac{x}{7} \end{aligned}$
::15 35 = x 7

First, since both denominators are given, start with those two numbers.
::首先,由于给出了两个分母,从这两个数字开始。

35 and 7
::35和7

Next, recognize the given ratio, $\begin{aligned} \frac{15}{35} \end{aligned}$ , and determine what can be done to 35 to make it equal to 7.
::其次,承认给定比率,15 35,并确定对35可以做些什么,使其等于7。

$\begin{aligned} 35 \div 5 = 7 \end{aligned}$

Dividing 35 by 5 equals 7.
::35比5等于7,35比5等于7

Then, divide both the numerator and denominator of the given ratio by 5.
::然后,将给定比率的分子数和分母除以5。

$\begin{aligned} \frac{15 \div 5}{35 \div 5} = \frac{3}{7} \end{aligned}$

The answer is x = 3
::答案是 x = 3

$\begin{aligned} \frac{15}{35} = \frac{3}{7} \end{aligned}$

Example 3
::例3

$\begin{aligned} \frac{32}{36} = \frac{x}{9} \end{aligned}$
::32 36 = x 9

Solve for x:
::解决 x:

First, begin with the given denominators.
::首先,首先从给定的分母开始。

36 and 9
::36和9

Next, using the given ratio of $\begin{aligned} \frac{32}{36} \end{aligned}$ , determine what can be done to 36 to make it equal to 9.
::其次,采用32 36的既定比率,确定对36可以做些什么,使其等于9。

$\begin{aligned} 36 \div 4 = 9 \end{aligned}$

Then, divide both the numerator and denominator of the given ratio by 4.
::然后,将给定比率的分子数和分母除以4。

$\begin{aligned} \frac{32 \div 4}{36 \div 4} = \frac{8}{9} \end{aligned}$

The answer is x = 8
::答案是 x = 8

$\begin{aligned} \frac{32}{36} = \frac{8}{9} \end{aligned}$

Example 4
::例4

$\begin{aligned} \frac{12}{24} = \frac{4}{z} \end{aligned}$
::12 24 = 4 兹

Solve for z.
::解决z。

First, begin with the numerators as both are given.
::首先,首先从给定两个数字开始。

12 and 4
::12和4

Next, determine what can be done to the numerator 12 in the given ratio of $\begin{aligned} \frac{12}{24} \end{aligned}$ to make it equal to 4 in the unknown ratio.
::其次,确定对12-24这一给定比率中的分子12可做些什么,使该比率在未知比率中等于4。

$\begin{aligned} 12 \div 3 = 4 \end{aligned}$

Then, divide both sides of the given ratio by 3.
::然后,将给定比率的两边除以3。

$\begin{aligned} \frac{12 \div 3}{24 \div 3} = \frac{4}{8} \end{aligned}$

The answer is z = 8.
::答案是 Z = 8 。

$\begin{aligned} \frac{12}{24} = \frac{4}{8} \end{aligned}$

Example 5
::例5

Solve for y.
::解决你。

$\begin{aligned} \frac{y}{5} = \frac{12}{20} \end{aligned}$
::y 5 = 12 20 y 5 = 12

First, work with the denominators.
::首先,与分母合作。

5 and 20
::5和20

Next, use the denominator, 20, in the given ratio of $\begin{aligned} \frac{12}{20} \end{aligned}$ and determine what can be done to 20 to make it equal to the denominator, 5, in the unknown ratio.
::其次,使用分母20,在12 20的给定比率中,确定可以做到20什么才能使其与分母5等同,以未知比率计算。

$\begin{aligned} 20 \div 4 = 5 \end{aligned}$

Then, divide both sides of the given ratio by 4.
::然后,将给定比率的两边除以4。

$\begin{aligned} \frac{12 \div 4}{20 \div 4} = \frac{3}{5} \end{aligned}$

The answer is y = 3.
::答案是 Y = 3 。

$\begin{aligned} \frac{3}{5} = \frac{12}{20} \end{aligned}$

Review
::回顾

Tell whether or not each pair of ratios form a proportion.
::说明每一对比率是否构成一个比例。

$\begin{aligned} \frac{1}{2} and \frac{4}{8} \end{aligned}$
::1、2和4 8

$\begin{aligned} \frac{3}{7} and \frac{6}{14} \end{aligned}$
::3、7和6 14

$\begin{aligned} \frac{5}{2} and \frac{10}{6} \end{aligned}$
::5、2和10 6

$\begin{aligned} \frac{3}{1} and \frac{9}{3} \end{aligned}$
::1 3 和 9 3

$\begin{aligned} \frac{2}{9} and \frac{1.5}{4.5} \end{aligned}$
::9和1.54.5

$\begin{aligned} \frac{4}{9} and \frac{8}{10} \end{aligned}$
::9、9和8 10

$\begin{aligned} \frac{1}{4} and \frac{5}{20} \end{aligned}$
::1、4和5 20

$\begin{aligned} \frac{3}{4} and \frac{9}{10} \end{aligned}$
::3、4和9 10

Use proportional reasoning to find the value of the variable in each proportion.
::使用比例推理来找到每个比例的变量值。

$\begin{aligned} \frac{1}{4} = \frac{a}{20} \end{aligned}$
::1 4 = 20

$\begin{aligned} \frac{15}{30} = \frac{x}{2} \end{aligned}$
::15 30 = x 2

$\begin{aligned} \frac{2}{9} = \frac{n}{63} \end{aligned}$
::2 9 = N 63

$\begin{aligned} \frac{z}{7} = \frac{12}{21} \end{aligned}$
::z 7 z = 12 21

$\begin{aligned} \frac{3}{5} = \frac{t}{60} \end{aligned}$
::3 5 = T 60

$\begin{aligned} \frac{k}{72} = \frac{5}{12} \end{aligned}$
::k 72 = 5 12

$\begin{aligned} \frac{x}{32} = \frac{4}{8} \end{aligned}$
::x 32 = 4 8x 32 = 4 8

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

比例

Working with Proportions ::与比例工作

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 Proportions
::与比例工作

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