Section: 比率和百分比 | 1.10 比率和百分比

Section outline

You get a job selling cars. Your salary is determined by a commission or a percentage of the sale of each car. How do you determine how much money you earn per sale?
::你得到卖汽车的工作。你的工资由每辆车销售的佣金或百分比决定。你如何确定每出售一辆车赚多少钱?

In this section, we will cover how to calculate with and percents as we need to do to find this commission.
::在本节中,我们将阐述如何按照我们为找到这个委员会而需要做的计算和百分比。

Percents
::百分比

A percent is a ratio that describes a part of a quantity out of 100. Per represents division and cent is 100, like 100 cents in 1 dollar. A percent is also a part of a whole. For example, we can write 10% as 10 per 100 or
::A% 是一种比例, 它描述在100分中的某一部分, 表示单位为100, 表示单位为100, 表示单位为100, 以1美元计为100美分。 A% 也是整体的一部分。例如, 我们的写法是10%/ 100, 表示单位为10%/ 100, 表示单位为10%/ 100, 表示单位为10%/ 100, 表示单位为10%/ 100, 表示单位为10%/ 100, 表示单位为10%/ 100, 表示单位为100, 表示单位为100美分, 以1美元计为100美分。 A% 也是整体的一部分。例如, 我们的写法是10%/ 100, 或

$\begin{aligned} 10 % = \frac{10}{100} = \frac{1}{10} = 0.10 \end{aligned}$

To Convert from a Percent to a Fraction or a Decimal
::从百分比转换成分数或十进制

To a fraction: Express the percent as part per 100 and reduce if possible.
::分数: 表示百分率为百分数, 并尽可能减少。

To a decimal: Dividing by 100 is the hundredths place value . Just put the number before the percent sign in the hundredths place. For example 237% = 2.37. The 7 is in the hundredths place. You can also think of this as moving the decimal point to the left two spots.
::十进制到小数点 : 将百分位数除以 100 是百位位数。只要在百位数的百分位数前先标出百位数。例如, 237% = 2. 37。 7 是百位数。您也可以将此视为小数点移到左两个点数。

Example 1
::例1

Express $\begin{aligned} \frac{1}{4} \end{aligned}$ as a percent.

::14 以百分比表示。

Solution: Divide 1 by 4 to convert $\begin{aligned} \frac{1}{4} \end{aligned}$ to a decimal.

::解决办法:将1除以4将14转换为小数点。

$\begin{aligned} 0.25 \\ 4 1.00 \\ \underline{- 8} \\ 20 \\ \underline{- 20} \\ 0 \end{aligned}$

Now, express the decimal as a part per 100: $\begin{aligned} 0.25 = \frac{25}{100} = 25 % \end{aligned}$ .

::现在,将小数点表示为每100:0.25=25100=25%。

by Mathispower4u demonstrates how to convert between decimals, percents, and fractions.
::根据 Mathispower4u 演示如何在小数、百分数和分数之间转换。

Example 2
::例2

a. What is 9% of 150?
::a. 150项中的9%是多少?

b. 9% of what number is 13.5?
::b. 数字为13.5的9%?

c. What percent of 150 is 13.5?
::c. 150的13.5%是13.5?

Solution: These three questions are related.
::解决办法:这三个问题是相互关联的。

a. Convert 9% to a decimal and multiply because "of" translates to multiplication: $\begin{aligned} 0.09 \times 150 = 13.5 \end{aligned}$ .
::a. 将9%转换为小数点后乘数,因为“of”会转换成乘法:0.09x150=13.5。

b. First, convert your percent to a decimal, $\begin{aligned} 9 % = 0.09 \end{aligned}$ . Next, translate the statement to an equation: $\begin{aligned} 0.09 n = 13.5 \to n = \frac{13.5}{0.09} = 150 \end{aligned}$ . We found n by dividing both sides of our equation by 0.09. On the left side of the equation, 0.09 canceled, because any number divided by itself is 1. On the right side of the equation, we simply divided 13.5 by 0.09 to obtain 150. We will discuss solving equations like this in the next chapter.
::b. 第一,将百分比转换为小数,90.09。下一步,将语句转换为方程: 0.09n=13.5n=13.50. 50.09=150。我们通过将方程两侧除以 0.09 找到n。在方程的左侧, 0.09 取消, 因为任何数字本身除以是 1 。在方程的右侧, 我们只需将13.5除以 0.09 以获得 150 。我们将在下一章讨论解决方程。

Another technique that we will discuss further in the next chapter is to set up a proportion where we fill in the information that we know from the problem and solve for the information that we do not know.
::我们将在下一章进一步讨论的另一个办法是,确定一个比例,以填充我们从问题中了解到的信息,并解决我们不知道的信息。

In general , we can set up percent proportions as
::总的来说,我们可以将百分比比例设定为:

$\begin{aligned} \frac{i s}{o f} = \frac{p e r c e n t}{100} \end{aligned}$

::等值=百分率

For this problem, we have
::对于这个问题,我们已:

$\begin{aligned} \frac{13.5}{n} = \frac{9}{100} \end{aligned}$

::13.5n=9100

In this problem, our "of" was the unknown. To solve this proportion, we can use a technique called cross-multiplication where you multiply each of the numerators with the denominators of the other fractions. For this example, we get
::在此问题上, 我们的“ 未知” 是未知的。为了解决这个比例, 我们可以使用一种技术, 叫做交叉乘法, 即您将每个数点乘以其它分数的分数乘以。对于这个例子, 我们得到的是

$\begin{aligned} 1350 = 9 n \end{aligned}$

::1350=9n 1350=9n

To solve for n, we want to divide both sides of our equation by 9. The 9 divided by 9 cancels on the right side of the equation (because any number divided by itself is 1). On the left side of the equation we have 1350 divided by 9. This quotient equals 150.
::为了解决n问题,我们要将我们等式的两边除以9, 9除以9, 等式右侧取消9除以9(因为任何数字本身除以1)。在等式的左侧,我们有1350除以9。这个商数等于150。

c. Again, you can translate to an equations and solve: $\begin{aligned} 150 p = 13.5 ⟶ p = \frac{13.5}{150} = 0.09 = 9 % \end{aligned}$ .
::c再次,您可以转换成方程式和解答:150p=13.5p=13.5150=0.09=9%。

We can also use the proportion technique to solve this one. Here we have
::我们也可以使用比例技术来解决这一难题。

$\begin{aligned} \frac{13.5}{150} = \frac{p}{100} \end{aligned}$

::13.5150=p100

Setting our cross products equal to each other results in the following equation:
::把我们的交叉产品设置为等价,结果产生以下等式:

$\begin{aligned} 1350 = 150 p \end{aligned}$

::1350=150p

Dividing both sides of our equation by 150, results in the final answer of 9. Recall that the 9 is in the percent spot of our proportion, so our answer is 9%.
::我们的等式两边分150分, 最终得出9分的最后答案。回顾9分在我们比例的百分之九, 所以我们的回答是百分之九。

Example 3
::例3

We often want to find a percent of a quantity. For example, the sales tax is a percent of the total purchase . If the cost of your purchase before tax is $49.99 and the sales tax in your state is 5%, what is the total cost of your purchase?

::我们经常想找到一定数量的百分之一。例如,销售税占总购买量的百分之一。如果您在税前的购买成本是49.99美元,而您的州销售税是5%,那么您购买的总成本是多少?

Solution: We need to determine 5% of $49.99. The word "of" indicates multiplication.

::解决方案:我们需要确定49.99的5%。“of”一词表示乘法。

$\begin{aligned} 49.99 \\ \times \underline{0.05} \\ 24995 \end{aligned}$

Since the problem has a total of four decimal places in it, we will move the decimal in our product four places to the left. This results in 2.4995. Since the smallest unit of money is a penny (0.01), we need to round this answer to the nearest hundredth. Therefore, our sales tax is $2.50. The total purchase cost is:
::由于问题总小数点为小数点后四位,我们将将产品小数点中的小数点移到左边四位,结果为2.4995。由于最小的一分钱是一分钱(0.01),我们需要将这一答案四舍五入到最近的一百分之一。因此,我们的销售税是2.50。总购买成本是:

$\begin{aligned} 49.99 \\ \underline{+ 2.50} \\ $ 52.49 \end{aligned}$

(Note: We could also solve this problem by multiplying 49.99 by 1.05. Using the distributive property , this result is precisely the same as the calculations we just performed:
:注:我们也可以通过将49.99乘以1.05来解决这个问题。使用分配性财产,这一结果与我们刚才进行的计算完全相同:

$\begin{aligned} 49.99 (1.05) = 49.99 (1 + 0.05) = (49.99 \times 1) + (49.99 \times 0.05) = 49.99 + 2.50 = 52.49 \end{aligned}$ )

by Mathispower4u explains how to determine the sales tax percent given the amount paid and the sale price.
::Mathispower4u解释如何根据支付的数额和销售价格确定销售税百分比。

Example 4
::例4

Suppose that you get a job selling cars . Your salary is determined by a commission or 2.5% of the selling price of each car that you sell. How much money do you earn for selling a car that costs $27,500?
::假设你找到卖汽车的工作。你的工资是由你卖的每一辆车的佣金或售价的2.5%来决定的。你卖一辆车要花27,500美元,赚了多少钱?

Solution: You earn 2.5% of $27,500. We multiply

::解决方案:你挣27,500美元的2.5%。我们乘以

$\begin{aligned} 27, 500 \\ \times \underline{0.025} \\ 137500 \\ \underline{+ 55000} \\ 687500 \end{aligned}$

Since the problem has a total of three decimal places in it, we will move the decimal in our product three places to the left. This results in 687.500. Therefore, our commission would be $687.50.
::由于问题总小数点后有三个位数, 我们将将产品小数点后三个位移到左边。结果是687. 500美元。因此, 我们的佣金是687. 50美元。

Rates

::速率率率率

A rate is a ratio (i.e., fraction) that compares two different quantities and usually those quantities have different units. Suppose you are buying turkey at a supermarket, and you pay $12 for 2 pounds of turkey. That is an example of a rate. The turkey you bought cost $6 per 1 pound of turkey. This is an example of a unit rate , a rate where you compare a quantity to one unit of another quantity.
::一个比率(即分数)比较两个不同的数量,通常数量有不同的单位。假设你在超市买火鸡,你为2磅火鸡支付12美元。这就是一个比率的例子。你购买的火鸡每1磅火鸡费用为6美元。这是一个单位费率的例子,一个单位费率,一个数量与另一个数量的一个单位相比。

Notice the word “per." This word signals to us that we are talking about a rate. Rates show up all the time in our daily lives. We use them when shopping, for example, "$3.99 per pound." We use them when driving, when we say "60 miles per hour." We use them with pricing when we say "$4.00 per yard of material."
::注意“每磅”这个词,这个字告诉我们,我们正在谈论一个利率。利率在我们日常生活中一直出现,我们在购物时使用它,例如“每磅3.99美元”。我们开车时使用它,我们说“每小时60英里”。我们用它来定价,我们说“每院4美元”。

Sometimes, two rates are equivalent or equal. Two rates are equivalent if they show the same relationship between two units of measure. We can use the same strategies to find equivalent rates that we use to see if two fractions are equal.
::有时,两种比率是等的或等的。两种比率是等的,如果它们显示两个计量单位之间的关系相同。我们可以使用同样的策略来找到我们用来看两个分数是否相等的等值率。

Example 5
::例5

Determine if these two rates are equivalent: 40 miles in 2 hours, and 80 miles in 4 hours.
::确定这两个比例是否相等:两小时40英里,四小时80英里。

Solution: First, express each rate as a fraction. Be sure to keep the terms consistent. That is, if the first ratio compares miles to hours, the second ratio should also compare miles to hours.
::解决方案 : 首先, 将每个比例表示为分数。请务必保持条件一致。也就是说, 如果第一个比例将英里与小时比较, 第二个比例也将将英里与小时比较。

$\begin{aligned} 40 miles in 2 hours & = \frac{40 mi}{2 h} = \frac{40}{2} \\ 80 miles in 4 hours & = \frac{80 mi}{4 h} = \frac{80}{4} \end{aligned}$

::2小时40英里=40 MI2 h= 40280 英里= 4小时80 MI4 h= 804

In order to compare them, the denominators need to match. We are going to change the ratio, $\begin{aligned} \frac{40}{2} \end{aligned}$ , to a ratio with 4 as the denominator .
::为了比较它们,分母需要匹配。我们要改变比率,402比4。

Since $\begin{aligned} 2 \times 2 = 4 \end{aligned}$ , multiply both terms of the ratio $\begin{aligned} \frac{40}{2} \end{aligned}$ by 2.
::自2x2=4以来,将402比的两种条件乘以2。

$\begin{aligned} \frac{40}{2} = \frac{40}{2} \times \frac{2}{2} = \frac{80}{4} \end{aligned}$

This shows that the ratio $\begin{aligned} \frac{80}{4} \end{aligned}$ is equivalent to the ratio $\begin{aligned} \frac{40}{2} \end{aligned}$ . This means that the rate 80 miles in 4 hours is equivalent to the rate 40 miles in 2 hours.
::这表明804比率相当于402比率,这意味着4小时80英里的比率相当于2小时40英里的比率。

by Mathispower4u explains how to determine rates and .
::Mathispower4u 解释如何确定费率和.

Example 6
::例6

One brand of dish washing soap costs $2.00 for 18 ounces, and another brand costs $3.00 for 32 ounces. Which is the better buy?
::18盎司的洗碗肥皂要两美元,32盎司的洗碗肥皂要三美元。

Solution: Let's call the the former brand, Brand A, and the latter brand, Brand B. We can compare the rates by finding the unit rate for each brand.

::解决方案:让我们把前品牌, 品牌A, 和后一个品牌B。我们可以通过找到每个品牌的单位费率来比较费率。

Brand A costs $\begin{aligned} $ 2.00 \div 18 = 0. \bar{11} \end{aligned}$ per 1 ounce.

::品牌A每1盎司$2.00$18=0.11

Brand B costs $\begin{aligned} $ 3.00 \div 32 = 0.09375 \end{aligned}$ per 1 ounce.

::品牌B每1盎司费用为3.00 32=0.09375美元。

Brand B is the better buy.

::品牌B是最好的买家

Summary
::摘要

To convert between percents and decimals, we can just move the decimal point two units right or left.
::要在百分率和小数点之间转换,我们可以将小数点点两单位向右或向左移动。

To convert between percents and fractions, we can create fractions that have a denominator of 100 (per cent).
::为了在百分数和分数之间转换,我们可以产生分数,分数分母为100%(%)。

A rate is a ratio that compares two quantities with different units.
::a 比率是两个数量和不同单位的比较比率。

A unit rate is a rate where there is one unit of the quantity in the denominator.
::单位费率是指分母中数量有一个单位的费率。

Review
::回顾

Convert the following percents to fractions and decimals.
::将以下百分比转换为分数和小数。

50%

12%

$\begin{aligned} 33 \frac{1}{3} \end{aligned}$ %?

Convert the following fractions to decimals and percents.
::将下列分数转换为小数点和百分数。

$\begin{aligned} \frac{3}{4} \end{aligned}$

$\begin{aligned} \frac{5}{8} \end{aligned}$

$\begin{aligned} \frac{5}{4} \end{aligned}$

Answer the following.
::答复如下。

What percent of 150 is 15?
::150是15是多少?

What is 110% of 200?
::200的110%是多少?

Determine whether each is equivalent or not.
::确定每项是否相等。

60 units in 3 minutes and 80 units in 4 minutes
::60个单位每3分钟60个单位每3分钟,80个单位每4分钟

$16 for 8 pounds and $22 for 10 pounds
::8磅16块,10磅22块22块

50 kilometers in 2 hours and 75 kilometers in 3 hours
::2小时50公里 3小时75公里

Explore More
::探索更多

1. A copy machine can produce 310 copies in 5 minutes. How many copies can the machine produce per minute?
::1. 复印机5分钟内可制作310份,每分钟可印制多少份?

2. Nadia needs to buy some cheddar cheese. An unce package of cheddar cheese costs $2.40. A 12-ounce package of cheddar cheese costs $3.36.
::2. Nadia需要购买一些切达乳酪,一套8盎司的切达乳酪包价2.40美元,一套12盎司的切达乳酪包价3.36美元。

a. Find the unit price for the unce package.
::a. 寻找8盎司包件的单价。

b. Find the unit price for the 12-ounce package.
::b. 寻找12盎司包件的单价。

c. Which is the better buy? What is the difference in the unit price of the unce and the 12-ounce packages of cheese ?
:c) 哪种购买更好?8盎司和12盎司奶酪包的单价有什么差别?

3. For parts a-c, a hair stylist charges $70 for a haircut. Depending on how much you tip, what will be the total cost of the haircut?
::3. 对于A-C部分,理发师理发费为70美元,视你小费多少,理发的总费用是多少?

a . You tip 15%. b . You tip 20%. c . You tip 25%.
::a. 提示15%. b. 提示20%. c. 提示25%.

4. 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?
::4. 商店的雇员目前每小时9.50美元的工资,如果全年工作,她得到12%的加薪,在加薪之后她新的小时工资率会是多少?

5. Stan is late on his car payment. The finance company charges 3% interest per month if it is late. His monthly payment is $300.00. What is the total amount he will owe if he pays the August first bill October first? (Assuming he was able to make his September bill on time and the interest was not compounded.)
::5. Stan的汽车付款迟到,金融公司每月收取3%的利息,如果迟到,则每月支付300.00美元,如果他支付8月10月第一笔帐单,他将拖欠的总额是多少? (假设他能够按时提出9月份的帐单,利息没有增加。 )

6. The pie chart shows the College of Arts and Sciences majors chosen by incoming freshman at a college. What percent of students chose each of the four majors?
::6. 派图显示大学新生选择的艺术和科学学院主要学科,四大学科中学生各占多少?

Answers to Review and Explore More Problems
::对审查和探讨更多问题的答复

Please see the Appendix.

::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section:
::尝试这一互动,强化本节所探讨的概念:

Section outline

Percents ::百分比

To Convert from a Percent to a Fraction or a Decimal ::从百分比转换成分数或十进制

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Rates ::速率率率率

Example 5 ::例5

Example 6 ::例6

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers to Review and Explore More Problems ::对审查和探讨更多问题的答复

PLIX ::PLIX

Percents
::百分比

To Convert from a Percent to a Fraction or a Decimal
::从百分比转换成分数或十进制

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Rates

::速率率率率

Example 5
::例5

Example 6
::例6

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers to Review and Explore More Problems
::对审查和探讨更多问题的答复

PLIX
::PLIX