Course: 5.10使用交叉积的比例

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使用交叉积的比例
Jamaya dropped bottles and wrappers whenever and wherever she wanted to. In order to break this bad habit, Jamaya’s mother made her help with a local clean-up effort. For every piece of litter that Jamaya had dropped in and around her home, she had to pick up 10 pieces with the group. If Jamaya dropped a total of 15 bottles, cans, and wrappers, how many pieces of litter does she have to clean up?
::Jamaya随时随地随地投下瓶子和包装纸。为了打破这种不良习惯,Jamaya的母亲帮助她在当地清理垃圾。对于Jamaya在家中和周围投下的每块垃圾,她不得不与这群人一起捡起10块。如果Jamaya总共扔下15瓶罐子、罐子和包装纸,她需要清理多少垃圾呢?

In this concept, you will learn how to solve proportions using cross products.
::在这个概念中,你将学会如何使用交叉产品解决比例问题。

Solving Proportions Using Cross Products
::使用交叉产品解决比例

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。

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。

A proportion can be expressed as two equivalent fractions.
::一个比例可以表示为两个等效分数。

$\begin{aligned} \frac{a}{b} = \frac{c}{d} \end{aligned}$

A proportion can be expressed with colons.
::比例可以用冒号表示。

$\begin{aligned} a : b = c : d \end{aligned}$

In a proportion, the means are the two terms that are closest together when the proportion is written with colons. So, in $\begin{aligned} a : b = c : d \end{aligned}$ , the means are $\begin{aligned} b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ .
::比例而言,手段是两个最接近的术语,当比例与结肠写成时。因此,在 a:b = c: d 中,手段是 b 和 c 。

The extremes are the terms in the proportion that are furthest apart when the proportion is written with colons. So, in $\begin{aligned} a : b = c : d \end{aligned}$ , the extremes are $\begin{aligned} a \end{aligned}$ and $\begin{aligned} d \end{aligned}$ .
::最极端是比例在用冒号写成时最相距最远的语句。因此,在 a:b = c: d:中,最极端是 a 和 d。

The diagram below shows how to identify the means and the extremes in a proportion.
::下图显示如何按比例确定手段和极端。

The Cross Products Property of Proportions states that the product of the means is equal to the product of the extremes.
::《比例制的交叉产品属性》指出,这些手段的产品与极端产品相同。

$\begin{aligned} \frac{a}{b} & = \frac{c}{d} \\ b \cdot c & = a \cdot d \end{aligned}$
::b = c d b b c = a d

Here is an example.
::举一个例子。

Solve for a.
::解决一个。

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

First, multiply the means and the extremes and set them equal to one another.
::首先,使手段和极端倍增,使它们彼此等同。

$\begin{aligned} a \cdot 8 & = 4 \cdot 6 \\ 8 a & = 24 \end{aligned}$

Next, solve the equation for the missing variable .
::下一步,解析缺失变量的方程式。

$\begin{aligned} \frac{8 a}{8} & = \frac{24}{8} \\ a & = 3 \end{aligned}$

The answer is a = 3.
::答案是a=3。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Jamaya and her littering habit.
::早些时候,你被问及贾马亚和她撒垃圾习惯的问题。

She had dropped a total of 15 pieces of litter in and around her home, and her mother said she had to pick up 10 pieces for every piece that she dropped. How many pieces of litter does Jamaya have to clean up?
::她在家里和周围总共扔了15块垃圾,她母亲说,她每扔一块,她必须捡到10块。 Jamaya要清理多少垃圾?

First, write a proportion.
::首先,写一个比例。

$\begin{aligned} \frac{10 p i e c e s t o p i c k u p}{1 p i e c e d r o p p e d} = \frac{x p i e c e s t o p i c k u p}{15 p i e c e s d r o p p e d} \end{aligned}$

Next, cross multiply.
::下一个,交叉乘数。

$\begin{aligned} x = 150 \end{aligned}$

The answer is x=150.
::答案是 x=150 。

Jamaya must pick up 150 pieces of litter.
::Jamaya必须捡起150块垃圾。

Example 2
::例2

Solve for x.
::解决x。

$\begin{aligned} \frac{x}{5} = \frac{6}{10} \end{aligned}$

First, cross multiply and set the products equal to one another.
::首先,交叉乘数,然后将产品设置为等量。

$\begin{aligned} 10 x = 30 \end{aligned}$

Next, solve for x.
::下一个,解答 x。

$\begin{aligned} x = 3 \end{aligned}$

The answer is x=3.
::答案是 x=3 。

Example 3
::例3

Solve for a.
::解决一个。

$\begin{aligned} \frac{a}{9} = \frac{15}{27} \end{aligned}$

First, cross multiply and set the products equal to one another.
::首先,交叉乘数,然后将产品设置为等量。

$\begin{aligned} 27 a = 135 \end{aligned}$

Next, solve for a.
::下一个,解决一个。

$\begin{aligned} a = 5 \end{aligned}$

The answer is a = 5.
::答案是a=5。

Example 4
::例4

Solve for b.
::解决b.

$\begin{aligned} \frac{b}{4} = \frac{12}{16} \end{aligned}$

First, cross multiply and set the products equal to one another.
::首先,交叉乘数,然后将产品设置为等量。

$\begin{aligned} 16 b = 48 \end{aligned}$

Next, solve for b.
::下一个,解决b。

$\begin{aligned} b = 3 \end{aligned}$

The answer is b = 3.
::答案是b=3。

Example 5
::例5

Rudy was a silly kitten who loved to play with ping pong balls. It took him just 15 minutes to swat 12 of them underneath the sofa. How many ping pong balls could Rudy hit under the sofa in 1 hour?
::鲁迪是个傻逼小猫,喜欢玩乒乓球,他只花了15分钟才在沙发下面打12个球。

First, write a proportion.
::首先,写一个比例。

$\begin{aligned} \frac{15 m i n u t e s}{12 b a l l s} = \frac{60 m i n u t e s}{x b a l l s} \end{aligned}$

Next, cross multiply.
::下一个,交叉乘数。

$\begin{aligned} 15 x = 720 \end{aligned}$

Then, solve for x.
::那么,解决x。

$\begin{aligned} x = 48 \end{aligned}$

The answer is 48. In one hour, 60 minutes, Rudy could swat 48 ping pong balls under the sofa.
::答案是48 一小时60分钟后鲁迪就可以在沙发下面击打48个乒乓球

Review
::回顾

Use cross products to find the value of the variable in each proportion.
::使用交叉产品查找每个比例变量的值。

$\begin{aligned} \frac{6}{10} = \frac{x}{5} \end{aligned}$
::6 10 = x 5

$\begin{aligned} \frac{2}{3} = \frac{x}{9} \end{aligned}$
::2 3 = x 9

$\begin{aligned} \frac{4}{9} = \frac{a}{45} \end{aligned}$
::4 9=a 45

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

$\begin{aligned} \frac{b}{8} = \frac{5}{16} \end{aligned}$
::b 8 = 5 16

$\begin{aligned} \frac{6}{3} = \frac{x}{9} \end{aligned}$
::6 3 = x 9

$\begin{aligned} \frac{4}{x} = \frac{8}{10} \end{aligned}$
::4 x = 8 10

$\begin{aligned} \frac{1.5}{y} = \frac{3}{9} \end{aligned}$
::1.5y = 3 9

$\begin{aligned} \frac{4}{11} = \frac{c}{33} \end{aligned}$
::4 11 = c 33

$\begin{aligned} \frac{2}{6} = \frac{5}{y} \end{aligned}$
::2 6 = 5 y

$\begin{aligned} \frac{2}{10} = \frac{5}{x} \end{aligned}$
::2 10 = 5 x

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

$\begin{aligned} \frac{5}{r} = \frac{70}{126} \end{aligned}$
::5 r = 70 126

$\begin{aligned} \frac{4}{14} = \frac{14}{k} \end{aligned}$
::4 14 = 14 k

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

$\begin{aligned} \frac{2}{5} = \frac{17}{a} \end{aligned}$
::2 5 = 17 a

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

使用交叉积的比例

Solving Proportions Using Cross Products ::使用交叉产品解决比例

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Solving Proportions Using Cross Products
::使用交叉产品解决比例

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源