3.6 以比率书写线性方

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• 选择节以比率写入的线性方

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  以比率写入的线性方

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  Determining Discount
  ::确定折扣

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  Jack DaSilva owns and manages  a shoe store called ShoeSpot. He buys shoes in large quantities from whole distributors and then sells them at market value. Most businesses that don’t make their products operate this way. When dealing with markups and discounts, linear equations with ratios will be likely to appear. Before you take a look at a situation that Jack encountered, you will first learn how to solve linear equations with ratios.
  ::Jack DaSilva拥有并管理着一家叫ShoeSpot的鞋店。 他从所有分销商那里大量购买鞋子,然后以市场价值出售。 大多数不生产其产品的企业都以这种方式经营。 当处理印记和折扣时,可能会出现带有比率的线性方程。 在你审视Jack遇到的情况之前,你将首先学会如何用比率解决线性方程。

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  Solving Linear Equations with Ratios
  ::具有比率的溶解线性线性公式

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  To solve equations with ratios, you will take the same approach as you would use to solve a proportion .  You will cross-multiply the two fractions and solve the resulting equation .
  ::要用比率解析方程式, 您将采取与您用来解析比例相同的方法。 您将交叉乘以两个分数, 并解析结果方程式 。

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  Example
  ::示例示例示例示例

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  $\begin{array}{r}\frac{6}{2x-1}=\frac{2}{5}\end{array}$
  ::62x-1=25

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  First, you will cross multiply.
  ::首先,你们将乘以乘以乘以。

  

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  When you cross multiply, you obtain the following multi-step equation:
  ::乘以乘数后,获得以下多步方程:

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  $$\begin{array}{r}5\cdot 6=2\cdot (2x-1)\end{array}$$

  
  ::56=2(2x-1)

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  You can solve this as follows:
  ::您可通过以下方式解决这个问题:

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  $$\begin{array}{rl}30& =2(2x-1)\\ 30& =2\cdot 2x-2\cdot 1\\ 30& =4x-2\\ 32& =4x\\ 8& =x\end{array}$$

  
  ::30=2(2x-1-1)30=2(2x-2x)-2=(130)=4x-232=4x8=x

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  Discussion Questions
  ::讨论问题 讨论问题

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  1. Why do we have to write the $\begin{array}{r}2x-1\end{array}$   in parenthesis? The 2 is being multiplied by the whole expression .
     ::我们为什么要在括号中写 2x-1 ? 2 正乘以整个表达式 。
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  2. What is the value of the denominator of the fraction $\begin{array}{r}\frac{6}{2x-1}?\end{array}$  
     ::分数 62x-1 的分母值是多少?

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  Determining Discount Continued
  ::确定贴现

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  Now that you know how to solve with ratios, you can solve Jack’s discount question.
  ::现在你知道如何解决比率问题,你可以解决杰克的折扣问题。

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  Example
  ::示例示例示例示例

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  Jack buys a new line of basketball sneakers for $57 and plans to sell them for $95. To bring customers into the store, Jack wants to offer a sale for the first week that the sneakers are on sale. Jack wants to know what the largest percentage discount he can offer without losing money is. To determine this Jack can use the following discount formula :
  ::杰克为57美元买一排新的篮球运动鞋,并计划以95美元卖掉这些运动鞋。为了把顾客带进商店,杰克想在运动鞋销售的第一周出售这些运动鞋。杰克想知道在不亏钱的情况下他能提供的最大百分比折扣是什么。为了确定这个杰克可以使用以下折扣公式:

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  $$\begin{array}{r}\frac{WholesalePrice}{SalesPrice}=\frac{100-PercentageDiscount}{100}\end{array}$$

  
  ::批发原价=100 - 百分比

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  $\begin{array}{r}\bullet \end{array}$  Wholesale Price = $57
  ::* 批发价格=57美元

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  $\begin{array}{r}\bullet \end{array}$  Sales Price = $95
  ::* 销售价格=95美元

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  $\begin{array}{r}\bullet \end{array}$  Percentage Discount =  $\begin{array}{r}x\end{array}$
  ::* 贴现百分比=x

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  When you substitute these values into the equation above, you get the following equation:
  ::当您将这些值替换为以上方程式时,您可以得到以下方程式:

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  $$\begin{array}{r}\frac{57}{95}=\frac{100-x}{100}\end{array}$$

  
  ::5795=100-x100

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  You  can then cross multiply to get the following:
  ::然后,您可以交叉乘以获得以下结果:

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  $$\begin{array}{rl}95\cdot (100-x)& =57\cdot 100\\ 95\cdot (100-x)& =5700\\ 9500-95x& =5700\\ -95x& =-3800\\ x& =40\end{array}$$

  
  ::95-(100-x)=57-100-95-(100-x)=57-009-500-95x=5700-95x=5700-95x3800x=40

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  The largest percentage discount that Jack can offer without losing money is 40%. Use the interactive below to explore the idea of markups, discounts and break even.
  ::杰克可以提供最大百分率的折扣而不会亏损的是40 % 。 使用下面的交互式折扣来探索加价、折扣和平均折价的想法 。

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  Discussion Questions
  ::讨论问题 讨论问题

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  1. Does this answer make sense within the context? Explain what it means. Why is Jack calculating this value if he selling the shoes for more than he bought them for?
     ::这个答案在上下文中是否合理?解释它的含义。为什么杰克要计算这个价值?如果他卖鞋的价比买鞋的价还高呢?
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  2. Now, check to make sure the answer makes sense mathematically. Does it check out?
     ::现在,检查一下,确保答案在数学上是有道理的。它检查过吗?

  

  INTERACTIVE

  Sales Pricing

  

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  Drag the slider to change the percent markup or discount and see how it affects sales price and profit margin.
  ::拖动滑块以改变百分数加分或折价,看看它如何影响销售价格和利润率。

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  Discussion Questions
  ::讨论问题 讨论问题

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  1. What percentage discount should Jack choose?
     ::杰克应该选择何种百分比的折扣?
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  2. Could a business ever gain anything by selling something at the break-even price?
     ::企业能以平衡价格卖东西 得到什么好处吗?
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  3. Do stores ever offer discounts that bring the price below the break-even price?
     ::商店是否提供折扣,使价格低于平衡价格?

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  Ratio Equations With More Than One Variable
  ::有不止一个变量的方位比率

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  Linear e quations with r atios can sometimes lead to . To solve a linear equation like this, you will take the same approach by cross multiplying the ratios and solving the resulting equation.
  ::具有比率的线性方程式有时可能导致 。 要解决像这样的线性方程式, 您将采取相同的方法, 交叉乘以比率, 并解析由此产生的方程式 。

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  Example
  ::示例示例示例示例

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  Marisa is an employee at ShoeSpot looking at a sales report for the new line of basketball sneakers. The sneakers come in two colors red and blue. The report states that 36% of the shoes that were sold were red. The report also states that there were 48 pairs of blue sneakers sold. How can she determine how many red sneakers were sold?
  ::Marisa是ShoeSpot的雇员,在看篮球运动鞋新系列的销售报告。运动鞋以红色和蓝色两种颜色出现。报告指出,36%的鞋子是红色的。报告还指出,有48双蓝运动鞋出售。她如何确定有多少红色运动鞋出售?

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  One way to solve this problem is by using the following equation:
  ::解决这一问题的一个办法是使用以下方程式:

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  $$\begin{array}{r}\frac{RedShoes}{Total}=\frac{PercentageofRedShoesSold}{100}\end{array}$$

  
  ::红鞋 Total = 红鞋百分比100

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  $\begin{array}{r}\bullet \end{array}$  Red Shoes =  $\begin{array}{r}x\end{array}$
  ::* 红鞋=x

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  $\begin{array}{r}\bullet \end{array}$  Total =  $\begin{array}{r}x+48\end{array}$
  ::* 共计=x+48

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  $\begin{array}{r}\bullet \end{array}$  Percentage of Red Shoes Sold =  $\begin{array}{r}36\end{array}$
  ::* 红鞋出售的百分比=36

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  When you substitute these values into the equation above, you get the following:
  ::当您将这些值替换为以上方程式时,可获得以下数值:

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  $$\begin{array}{r}\frac{x}{x+48}=\frac{36}{100}\end{array}$$

  
  ::xx+48=36100

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  Cross multiply the ratios
  ::交叉乘以比率

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  $$\begin{array}{r}x\cdot 100=36\cdot (x+48)\end{array}$$

  
  ::x100=36(x+48)

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  Solve
  ::解决

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  $$\begin{array}{rl}100x& =36(x+48)\\ 100x& =36x+1728\\ 64x& =1728\\ x& =27\end{array}$$

  
  ::100x=36(x+48)100x=36x+172864x=1728x=27

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  There were 27 red sneakers sold.
  ::卖了27个红色运动鞋。

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  Discussion Question
  ::讨论问题

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  What can you say about the structure of the equation? What does it tell you about what you are solving for?
  ::关于方程的结构,你能说些什么?它告诉了你什么是 你正在解决什么?

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     Summary
  ::摘要

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  • Use  cross multiplication to solve linear equations with ratios on both sides.
    ::使用交叉乘法解析线性方程式,使两侧比率都达到比例。
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  • Solve percentage problems by writing an equation with ratios, cross multiplying and solving the resulting two-step equation.
    ::解决百分率问题,方法是以比率来写方程式,交叉乘法和解决由此产生的两步方程式。