3.1515%变化百分比

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  变化百分比

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  Percent of Change 
  ::变化百分比

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  A useful way to express changes in quantities is through percents. You’ve probably seen signs such as “20% extra free,” or “save 35% today.” When we use percents to represent a change, we generally use the formula
  ::表示数量变化的有用方法就是通过百分比。 您可能已经看到了“ 20%的额外免费 ” 或“今天保存35 % ” 等迹象。 当我们使用百分比来表示变化时,我们通常使用公式。

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  $$\begin{array}{r}\text{Percent change}=\frac{\text{final amount - original amount}}{\text{original amount}}\times 100\mathrm{\%}\end{array}$$

  
  ::百分比变化=最终数额 -- -- 原始数额x100%

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  or
  ::或

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  $$\begin{array}{r}\frac{\text{percent change}}{100}=\frac{\text{actual change}}{\text{original amount}}\end{array}$$

  
  ::百分比变化100=实际变化100=实际变化

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  This means that a positive percent change is an increase , while a negative change is a decrease .
  ::这意味着正百分比变化是增长,而负变化是下降。

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  Real-World Application: School Enrollment   
  ::现实世界应用:学校入学

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  A school of 500 students is expecting a 20% increase in students next year. How many students will the school have?
  ::有500名学生的学校预计明年学生人数将增加20%。 学校将有多少学生?

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  First let’s solve this using the first formula. Since the 20% change is an increase, we represent it in the formula as 20 (if it were a decrease, it would be -20.) Plugging in all the numbers, we get
  ::首先,让我们用第一个公式来解决这个问题。 由于20%的改变是增加的, 我们在公式中以 20 表示它(如果是减少,就是 - 20。 ) 在所有数字中插入, 我们得到的是

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  $$\begin{array}{r}20\mathrm{\%}=\frac{\text{final amount}-500}{500}\times 100\mathrm{\%}\end{array}$$

  
  ::20 最终数额-500500×100%

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  Dividing both sides by 100%, we get $\begin{array}{r}0.2=\frac{\text{final amount}-500}{500}\end{array}$ .
  ::双方百分之百分裂 我们得到0.2=500500

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  Multiplying both sides by 500 gives us $\begin{array}{r}100=\text{final amount}-500\end{array}$ .
  ::以500乘以两边 等于100=500

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  Then adding 500 to both sides gives us 600 as the final number of students.
  ::然后在双方增加500名学生,

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  How about if we use the second formula? Then we get $\begin{array}{r}\frac{20}{100}=\frac{\text{actual change}}{500}\end{array}$ . ( Reducing the first fraction to $\begin{array}{r}\frac{1}{5}\end{array}$ will make the problem easier, so let’s rewrite the equation as $\begin{array}{r}\frac{1}{5}=\frac{\text{actual change}}{500}.\end{array}$
  ::如果我们使用第二个公式呢?然后我们得到20100=实际变化500。 (将第一个分数减为15将使问题更容易解决,让我们将方程式改写为15=实际变化500。)

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  Cross multiplying is our next step; that gives us $\begin{array}{r}500=5\times (\text{actual change})\end{array}$ . tells us the change is equal to 100. We were told this was an increase, so if we start out with 500 students, after an increase of 100 we know there will be a total of 600.
  ::交叉乘法是我们的下一个步骤; 这给了我们500=5x( 实际变化) 。 告诉我们, 变化等于100。 告诉我们,我们被告知这是增加, 所以如果我们从500个学生开始, 在增加100个学生之后, 我们知道总共会有600个学生。

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  Markup
  ::标记标记标记

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  A markup is an increase from the price a store pays for an item from its supplier to the retail price it charges to the public. For example, a 100% mark-up (commonly known in business as keystone ) means that the price is doubled. Half of the retail price covers the cost of the item from the supplier, half is profit.
  ::标记是指从商店从供应商购买物品的价格到它向公众收取的零售价格的上涨。 例如,100%的加价(通常在商业界中称为“基石 ” ) 意味着价格翻一番。 零售价格的一半支付供应商的物品成本,一半是利润。

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  Real-World Application: Furniture Store 
  ::真实世界应用程序:家具储存

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  A furniture store places a 30% markup on everything it sells. It offers its employees a 20% discount from the sales price. The employees are demanding a 25% discount, saying that the store would still make a profit. The manager says that at a 25% discount from the sales price would cause the store to lose money. Who is right?
  ::一家家具商店在其销售的所有物品上都加了30%的印记。 它向雇员提供20%的销售价格折扣。 雇员们要求25%的折扣, 说商店仍能赢利。 经理说, 以25%的销售价格折扣会使商店损失钱财。 谁是正确的?

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  We’ll consider this problem two ways. First, let’s consider an item that the store buys from its supplier for a certain price, say $1000. The markup would be 30% of 1000, or $300, so the item would sell for $1300 and the store would make a $300 profit.
  ::我们从两个角度来考虑这个问题。 首先,让我们来考虑一家商店以一定的价格从供应商那里购买的物品,比如1000美元。 涨价为1000美元(即300美元 ) 的30 % , 也就是说,300美元(即300美元 ) , 这样这个商店就能卖1300美元(即1300美元),并且可以赚300美元(即300美元 ) 。

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  And what if an employee buys the product? With a discount of 20%, the employee would pay 80% of the $1300 retail price, or $\begin{array}{r}0.8\times \mathrm{\$}1300=\mathrm{\$}1040\end{array}$ .
  ::如果一个雇员购买了产品呢?如果折价20%,该雇员将支付1300美元零售价的80%,即0.8x100美元=1040美元。

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  But with a 25% discount, the employee would pay 75% of the retail price, or $\begin{array}{r}0.75\times \mathrm{\$}1300=\mathrm{\$}975\end{array}$ .
  ::但是,如果打折扣25%,雇员将支付零售价的75%,即0.75×100美元=975美元。

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  So with a 20% employee discount, the store still makes a $40 profit on the item they bought for $1000—but with a 25% employee discount, the store loses $25 on the item.
  ::以20%的雇员折扣, 商店仍能赚到40美元的利润, 他们以1 000美元的价格购买该物品, 但以25%的雇员折扣, 商店损失了25美元。

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  Now let’s use algebra to see how this works for an item of any price. If $\begin{array}{r}x\end{array}$ is the price of an item, then the store’s markup is 30% of $\begin{array}{r}x\end{array}$ , or $\begin{array}{r}0.3x\end{array}$ , and the retail price of the item is $\begin{array}{r}x+0.3x\end{array}$ , or $\begin{array}{r}1.3x\end{array}$ . An employee buying the item at a 20% discount would pay $\begin{array}{r}0.8\times 1.3x=1.04x\end{array}$ , while an employee buying it at a 25% discount would pay $\begin{array}{r}0.75\times 1.3x=0.975x\end{array}$ .
  ::现在,让我们使用代数来查看任何价格的物品是如何运作的。 如果x是物品的价格,那么商店的加价为x的30%或0.3x,而该物品的零售价为x+0.3x或1.3x。 以20%的折扣购买该物品的雇员将支付0.8x1.3x=1.04x,而以25%的折扣购买该物品的雇员将支付0.75xxx=0.975x。

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  So the manager is right: a 20% employee discount still allows the store to make a profit, while a 25% employee discount would cause the store to lose money.
  ::经理说得没错:20%的雇员折扣仍然允许商店盈利,而25%的雇员折扣会使商店损失钱财。

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  It may not seem to make sense that the store would lose money after applying a 30% markup and only a 25% discount. The reason it does work out that way is that the discount is bigger in absolute dollars after the markup is factored in. That is, an employee getting 25% off an item is getting 25% off the original price plus 25% off the 30% markup, and those two numbers together add up to more than 30% of the original price.
  ::商店在申请30%的加价和25%的折扣后会损失钱,这可能似乎不合理。 这样做的理由是,在计入加价后,以绝对美元计价的折扣会更大。 也就是说,从一个项目中获得25%的雇员从原价中获得了25%的折扣,加上30%的加价的25%,而这两个数字加起来加起来等于原价的30%以上。

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  Real-World Application: Employee Demographics 
  ::现实世界应用:雇员人口统计

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  In 2004 the US Department of Agriculture had 112071 employees, of which 87846 were Caucasian. Of the remaining minorities, African-American and Hispanic employees had the two largest demographic groups, with 11754 and 6899 employees respectively. $\begin{array}{r}{}^{\ast }\end{array}$
  ::2004年,美国农业部有1 12071名雇员,其中87846人为高加索人,其余少数族裔中,非裔美国人和西班牙裔雇员是两个最大的人口群体,分别为11754人和6899人*。

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  a) Calculate the total percentage of minority (non-Caucasian) employees at the USDA
  ::a) 计算美国农业部少数(非高加索)雇员的总百分比

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  Use the percent equation $\begin{array}{r}\text{Rate}\times \text{Total}=\text{Part}\end{array}$ .
  ::使用百分比方程式 RatexTal=Part 。

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  The total number of employees is 112071. We know that the number of Caucasian employees is 87846, which means that there must be $\begin{array}{r}112071-87646=24225\end{array}$ non-Caucasian employees. This is the part . Plugging in the total and the part, we get $\begin{array}{r}\text{Rate}\times 112071=24225\end{array}$ .
  ::雇员总数为112071人。我们知道,高加索雇员人数为87846人,这意味着必须有112071-87646=24225名非高加索雇员。这就是部分。在总数和部分中,我们得到的是Rationsx112071=24225。

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  Divide both sides by 112071 to get $\begin{array}{r}\text{Rate}=\frac{24225}{112071}\approx 0.216\end{array}$ . Multiply by 100 to get this as a percent: 21.6%.
  ::将两边除以112071, 以获得 =242251120710. 216。 乘以100, 以获得这个百分比: 21.6%。

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  21.6% of USDA employees in 2004 were from minority groups.
  ::2004年,美国农业部21.6%的雇员来自少数群体。

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  b) Calculate the percentage of African-American employees at the USDA. 
  :b) 计算美籍非裔雇员在美援署的百分比。

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  Here, the total is still 112071 and the part is 11754, so we have $\begin{array}{r}\text{Rate}\times 112071=11754\end{array}$ . Dividing, we get $\begin{array}{r}\text{Rate}=\frac{11754}{112071}\approx 0.105\end{array}$ , or 10.5%.
  ::这里的总数仍为112071, 部分为11754, 所以我们有Rationsx1112071=11754。分解, 我们得到的Rations=117541120710.105, 或10.5%。

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  10.5% of USDA employees in 2004 were African-American.
  ::2004年,美国农业部雇员中10.5%是非裔美国人。

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  c) Calculate the percentage of minority employees who were neither African-American nor Hispanic. 
  :c) 计算既非非裔美国人也非西班牙裔少数族裔雇员的百分比。

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  Here, our total is just the number of non-Caucasian employees, which we found out is 24225. Subtracting the African-American and Hispanic employees leaves $\begin{array}{r}24225-11754-6899=5572\end{array}$ employees in the group we’re looking at.
  ::这里,我们的总数只是非高加索雇员的人数,我们发现有24225人。 减去非裔美国人和西班牙裔雇员后,我们所关注的群体中只剩下24225-11754-6899=5572名雇员。

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  So with 24225 for the whole and 5572 for the part, our equation is $\begin{array}{r}\text{Rate}\times 24225=5572\end{array}$ , or $\begin{array}{r}\text{Rate}=\frac{5572}{24225}\approx 0.230\end{array}$ , or 23%.
  ::因此,以整个24225和5572为单位, 我们的方程式是Rationsx24225=5572, 或Rations=55722222250.230, 或23%。

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  23% of USDA minority employees in 2004 were neither African-American nor Hispanic.
  ::2004年,美国农业部23%的少数族裔雇员既不是非裔美国人,也不是西班牙裔美国人。

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  Examples
  ::实例

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  In 1995 New York had 18136000 residents. There were 827025 reported crimes, of which 152683 were violent. By 2005 the population was 19254630 and there were 85839 violent crimes out of a total of 491829 reported crimes. (Source: New York Law Enforcement Agency Uniform Crime Reports.) Calculate the percentage change from 1995 to 2005 in:
  ::1995年,纽约共有居民18136 000人,报告的犯罪为827025人,其中暴力犯罪为152683人,2005年,人口为19254630人,暴力犯罪为85839人,而报告的犯罪总数为491829人。 (资料来源:纽约执法机构统一犯罪报告。 )

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  This is a percentage change problem. Remember the formula for percentage change:
  ::这是一个百分比变化问题。 记住百分比变化的公式 :

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  $$\begin{array}{r}\text{Percent change}=\frac{\text{final amount - original amount}}{\text{original amount}}\times 100\mathrm{\%}\end{array}$$

  
  ::百分比变化=最终数额 -- -- 原始数额x100%

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  In these problems, the final amount is the 2005 statistic, and the initial amount is the 1995 statistic.
  ::在这些问题上,最后数额是2005年的统计数字,最初数额是1995年的统计数字。

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  Example 1
  ::例1

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  Calculate the percentage change from 1995 to 2005 in the population of New York 
  ::计算1995年至2005年纽约人口百分比变化

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  Population:
  ::人口:

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  $$\begin{array}{rl}\text{Percent change}& =\frac{19254630-18136000}{18136000}\times 100\mathrm{\%}\\ & =\frac{1118630}{18136000}\times 100\mathrm{\%}\\ & \approx 0.0617\times 100\mathrm{\%}\\ & =6.17\mathrm{\%}\end{array}$$

  
  ::百分率变化=19254630-1813600018136000x100 111863018136000x100 111863018136000x100 0.0617×100}6.17%

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  The population grew by 6.17%.
  ::人口增长了6.17%。

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  Example 2
  ::例2

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  Calculate the percentage change from 1995 to 2005 in the total reported crimes. 
  ::计算1995年至2005年所报告犯罪总数的百分比变化。

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  Total reported crimes:
  ::已报告犯罪总数:

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  $$\begin{array}{rl}\text{Percent change}& =\frac{491829-827025}{827025}\times 100\mathrm{\%}\\ & =\frac{-335196}{827025}\times 100\mathrm{\%}\\ & \approx -0.4053\times 100\mathrm{\%}\\ & =-40.53\mathrm{\%}\end{array}$$

  
  ::百分比变化=491829-827025827025x100}3351968-27025×100 0.4053×10040.53%

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  The total number of reported crimes fell by 40.53%.
  ::报告的犯罪总数下降了40.53%。

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  Example 3
  ::例3

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  Calculate the percentage change from 1995 to 2005 in violent crimes. 
  ::计算1995年至2005年暴力犯罪的百分比变化。

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  Violent crimes:
  ::暴力犯罪:

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  $$\begin{array}{rl}\text{Percent change}& =\frac{85839-152683}{152683}\times 100\mathrm{\%}\\ & =\frac{-66844}{152683}\times 100\mathrm{\%}\\ & \approx -0.4377\times 100\mathrm{\%}\\ & =-43.77\mathrm{\%}\end{array}$$

  
  ::百分比变化=85839-152683152683×10066844152683×1000.4377×10043.77%

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  The total number of violent crimes fell by 43.77%.
  ::暴力犯罪总数下降了43.77%。

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  Review 
  ::回顾

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  For questions 1-3, a hair stylist charges $70 for a haircut. Depending on how much you tip, what will be the total cost of the haircut?
  ::对于问题1-3,理发师的理发费为70美元。根据你的小费多少,理发的总成本是多少?

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  1. You tip 15%.
     ::你出15%的钱
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  2. You tip 20%.
     ::你给20%。
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  3. You tip 25%.
     ::你给25%。
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  4. 250 is what percentage of 195?
     ::195人中250人的百分比是多少?
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  5. 0.0032 is what percentage of 0.045?
     ::0.0032是0.045的多少百分比?
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  6. An employee at a store is currently paid $9.50 per hour. If she works a full year she gets a 12% pay raise. What will her new hourly rate be after the raise?
     ::商店的雇员目前每小时9.50美元的工资,如果工作满一年,她就会得到12%的工资提高。 在加薪之后,她新的小时工资率会是多少?
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  7. A TV is advertised on sale. It is 35% off and now costs $195. What was the pre-sale price?
     ::电视广告是销售广告,现在售出35%,现在成本为195美元。 预售价格是多少?
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  8. A TV was advertised on sale. If you saved $40, and bought it for $160, what percentage off was it?
     ::如果你存了40美元,然后以160美元买下它,那它的百分比是多少?
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  9. Another TV is advertised on sale. If this TV is also $40 cheaper than the pre-sale price, was it also the same percentage off as the TV in the question above? Explain!
     ::如果这台电视比预售价还便宜40美元,
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  10. Store $\begin{array}{r}A\end{array}$ and Store $\begin{array}{r}B\end{array}$ both sell bikes, and both buy bikes from the same supplier at the same prices. Store $\begin{array}{r}A\end{array}$ has a 40% mark-up for their prices, while store $\begin{array}{r}B\end{array}$ has a 250% mark-up. Store $\begin{array}{r}B\end{array}$ has a permanent sale and will always sell at 60% off the marked-up prices. Which store offers the better deal?
      ::商店A和商店B都出售自行车,并且都以相同价格从同一供应商购买自行车。 商店A的价格上涨40%,商店B加价250 % 。 商店B有长期销售,总以60%的标价出售。 哪家店提供更好的交易?

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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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  Texas Instruments Resources
  ::得克萨斯州工具资源

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  In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See .
  ::在CK-12得克萨斯州仪器代数I FlexBook资源中,有图表计算活动,旨在补充本章某些经验教训的目标。