比例属性属性

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Proportions
::比例比例

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A proportion is two ratios that are set equal to each other. Usually the ratios in proportions are written in fraction form. An example of a proportion is $\begin{align*}\frac{2}{x}={5}{10}\end{align*}$ . To solve a proportion, you need to cross-multiply. The Cross-Multiplication Theorem , which allows us to solve proportions using this method, states that if $\begin{align*}a, b, c,\end{align*}$ and $\begin{align*}d\end{align*}$ are real numbers, with $\begin{align*}b \neq 0\end{align*}$ and $\begin{align*}d \neq 0\end{align*}$ and if $\begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}$ , then $\begin{align*}ad=bc\end{align*}$ . Cross-multiplying allows us to get rid of the fractions in our equation. The Cross-Multiplication Theorem has several sub-theorems, called corollaries.
::一个比例是两个对等的比例。 通常比例的比例以分数形式写成。 一个比例的示例是 2x=510。 要解决比例, 您需要交叉乘以。 交叉重叠参数允许我们使用这个方法来解析比例, 它告诉我们如果a、 b、 c和 d是真实数字, 加上 b0 和 d0 , 如果 ab=cd, 则ad=bc。 交叉叠加允许我们清除方程式中的分数。 交叉重叠参数有数个子参数, 叫做线条 。

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Corollary #1: If $\begin{align*}a, b, c,\end{align*}$ and $\begin{align*}d\end{align*}$ are nonzero and $\begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}$ , then $\begin{align*}\frac{a}{c} = \frac{b}{d}\end{align*}$ .
::#1:如果a、b、c和d为非零和ab=cd,则ac=bd。

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Switch $\begin{align*}b\end{align*}$ and $\begin{align*}c\end{align*}$ .
::b和c交换机

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Corollary #2: If $\begin{align*}a, b, c,\end{align*}$ and $\begin{align*}d\end{align*}$ are nonzero and $\begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}$ , then $\begin{align*}\frac{d}{b} = \frac{c}{a}\end{align*}$ .
::2: 如果 a、b、c和d为非零和ab=cd,则db=ca。

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Switch $\begin{align*}a\end{align*}$ and $\begin{align*}d\end{align*}$ .
::切换。

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Corollary #3: If $\begin{align*}a, b, c,\end{align*}$ and $\begin{align*}d\end{align*}$ are nonzero and $\begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}$ , then $\begin{align*}\frac{b}{a} = \frac{c}{d}\end{align*}$ .
::3:如果a、b、c和d为非零和ab=cd,则b=cd。

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Flip each ratio upside down.
::翻转每个比率颠倒下来。

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Corollary #4: If $\begin{align*}a, b, c,\end{align*}$ and $\begin{align*}d\end{align*}$ are nonzero and $\begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}$ , then $\begin{align*}\frac{a+b}{b} = \frac{c+d}{d}\end{align*}$ .
::卷轴 # 4: 如果 a、b、c和d为非零和ab=cd, 那么 a+bb=c+dd。

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Corollary #5: If $\begin{align*}a, b, c,\end{align*}$ and $\begin{align*}d\end{align*}$ are nonzero and $\begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}$ , then $\begin{align*}\frac{a-b}{b} = \frac{c-d}{d}\end{align*}$ .
::5:如果a、b、c和d为非零和ab=cd,则a-bb=c-dd。

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What if you were told that a scale model of a python is in the ratio of 1:24? If the model measures 0.75 feet long, how long is the real python?
::如果有人告诉你们,比方方的比重是1:24,那么,如果比方的比重是0.75英尺长,真正的比方是多久?

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Examples
::实例

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Example 1
::例1

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In the picture, $\begin{align*}\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}\end{align*}$ .
::在照片中,ABXY=BCY ACXZ。

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Find the measures of $\begin{align*}AC\end{align*}$ and $\begin{align*}XY\end{align*}$ .
::查找AC和XY的测量方法。

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Plug in the lengths of the sides we know.
::插在我们所知道的两面的长度中。

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Example 2
::例2

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In the picture, $\begin{align*}\frac{AB}{BE} = \frac{AC}{CD}\end{align*}$ . Find $\begin{align*}BE\end{align*}$ .
::照片里是ABBE=ACCD 找到BE

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Substitute in the lengths of the sides we know.
::代替我们所知道的两边的长度

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$$\begin{align*}\frac{12}{BE} = \frac{20}{25} \ \longrightarrow \ {20(BE)} & = 12(25)\\ BE & = 15\end{align*}$$
::12BE=2025 20(BE)=12(25)BE=15

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Example 3
::例3

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Solve the proportions.  Remember, to solve a proportion, you need to cross-multiply.
::解决比例问题 记住 要解决比例问题

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1. $\begin{align*}\frac{4}{5} = \frac{x}{30}\end{align*}$
   ::45=x30

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2. $\begin{align*}\frac{y+1}{8} = \frac{5}{20}\end{align*}$
   ::y+18=520 y+18=520

 

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3. $\begin{align*}\frac{6}{5} = \frac{2x+4}{x-2}\end{align*}$
   ::65=2x+4x-2

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Example 4
::例4

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Your parents have an architect’s drawing of their home. On the paper, the house’s dimensions are 36 in by 30 in. If the shorter length of the house is actually 50 feet, what is the longer length?
::你的父母有建筑师对房子的绘画。 在报纸上,房子的尺寸是36乘30英寸。 如果房子的短长实际上只有50英尺,那么长是多少?

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To solve, first set up a proportion. If the shorter length is 50 feet, then it lines up with 30 in, the shorter length of the paper dimensions.
::为了解决问题,首先设定一个比例。如果短的长度为50英尺,那么它的长度为30英寸,纸张尺寸的长度较短。

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$$\begin{align*}\frac{30}{36} = \frac{50}{x} \longrightarrow \ 30x &=1800\\ x &=60 \quad \quad \text{The longer length is 60 feet.}\end{align*}$$
::3036=50x30x=1800x=60 长为60英尺。

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Example 5
::例5

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Suppose we have the proportion $\begin{align*}\frac{2}{5} = \frac{14}{35}\end{align*}$ . Write three true proportions that follow.
::假设我们拥有25=1435的比例 写三个真正的比例

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First of all, we know this is a true proportion because you would multiply $\begin{align*}\frac{2}{5}\end{align*}$ by $\begin{align*}\frac{7}{7}\end{align*}$ to get $\begin{align*}\frac{14}{35}\end{align*}$ . Using the first three corollaries:
::首先,我们知道这个比例是真实的 因为你会乘以25乘以77 乘以1435

1. $\begin{align*}\frac{2}{14} = \frac{5}{35}\end{align*}$
2. $\begin{align*}\frac{35}{5} = \frac{14}{2}\end{align*}$
3. $\begin{align*}\frac{5}{2} = \frac{35}{14}\end{align*}$

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Review
::回顾

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Solve each proportion.
::解决每一比例。

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1. $\begin{align*}\frac{x}{10} = \frac{42}{35}\end{align*}$
   ::x10=4235x10=4235
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2. $\begin{align*}\frac{x}{x-2} = \frac{5}{7}\end{align*}$
   ::xx-2=57
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3. $\begin{align*}\frac{6}{9} = \frac{y}{24}\end{align*}$
   ::69=y24
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4. $\begin{align*}\frac{x}{9} = \frac{16}{x}\end{align*}$
   ::x9=16x
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5. $\begin{align*}\frac{y-3}{8} = \frac{y+6}{5}\end{align*}$
   ::y- 38=y+65
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6. $\begin{align*}\frac{20}{z+5} = \frac{16}{7}\end{align*}$
   ::20z+5=167
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7. Shawna drove 245 miles and used 8.2 gallons of gas. At the same rate, if she drove 416 miles, how many gallons of gas will she need? Round to the nearest tenth.
   ::Shawna开往245英里,使用8.2加仑的气体。同样,如果开往416英里,她需要多少加仑的气体?圆到最近的10加仑。
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8. The president, vice-president, and financial officer of a company divide the profits is a 4:3:2 ratio. If the company made $1,800,000 last year, how much did each person receive?
   ::公司总裁、副总裁和财长将利润分成4:3:2的比例。 如果公司去年赚了1800,000美元,每人得到多少?

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Given the true proportion, $\begin{align*}\frac{10}{6}= \frac{15}{d} = \frac{x}{y}\end{align*}$ and $\begin{align*}d, x,\end{align*}$ and $\begin{align*}y\end{align*}$ are nonzero, determine if the following proportions are also true.
::根据真实比例,106=15d=xy和d,x,y为非零,确定以下比例是否也属实。

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9. $\begin{align*}\frac{10}{y} = \frac{x}{6}\end{align*}$
   ::10y=x6
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10. $\begin{align*}\frac{15}{10} = \frac{d}{6}\end{align*}$
    ::1510=d6 1510=d6
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11. $\begin{align*}\frac{6+10}{10} = \frac{y+x}{x}\end{align*}$
    ::6+101010=y+xxx
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12. $\begin{align*}\frac{15}{x} = \frac{y}{d}\end{align*}$
    ::15x=yd 15x=yd

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For questions 13-16, $\begin{align*}\frac{AE}{ED} = \frac{BC}{CD}\end{align*}$ and $\begin{align*}\frac{ED}{AD} = \frac{CD}{DB} = \frac{EC}{AB}\end{align*}$ .
::关于问题13-16,AEED=BCCD,EDAD=CDDB=ECAB。

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13. Find $\begin{align*}DB\end{align*}$ .
    ::寻找 DB 。
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14. Find $\begin{align*}EC\end{align*}$ .
    ::寻找欧盟委员会。
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15. Find $\begin{align*}CB\end{align*}$ .
    ::找到CB
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16. Find $\begin{align*}AD\end{align*}$ .
    ::寻找AD。

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Review (Answers)
::回顾(答复)

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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 。

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Resources
::资源