节: 使用比例来解决百分比问题 | 5.6 使用比例来解决百分比问题

章节大纲

The student council decided to conduct a survey to see how many middle school students attend football games on Friday nights. They asked the students when they entered the stadium what grade they were in and then recorded the results. The student council members did this for three weeks to get a good estimate on how many middle school students attend the games. They found that 42% of the 850 students who attended were from the middle school. How many students does this represent?
::学生会决定进行一次调查,以了解星期五晚上有多少中学生参加足球比赛,他们问学生进入体育场时是哪一年级,然后记录结果。学生会成员做了三个星期,以便很好地估计有多少中学生参加比赛。他们发现,参加比赛的850名学生中有42%来自中学。这代表了多少学生?

In this concept, you will learn to use proportions to solve .
::在这一概念中,你将学会用比例来解决问题。

Proportions
::比例比例

Proportions can be used to compare any equal quantity. Remember that proportions are created when two ratios are equal. Because of this, proportions can be used to find any missing piece. If you know a percent, you can also find a part of the whole.
::比例可以用来比较任何相等的数量。记住当两个比例相等时, 比例会形成。因此, 比例可以用来查找任何缺失的部分。如果您知道百分比, 您也可以找到整体的一部分。

Let’s look at an example.
::让我们举个例子。

At Big Town Middle School, approximately 37% of the students participate in athletic programs. If there are 968 students at the school, how many uniforms do they need to purchase?
::在大城中学,大约37%的学生参加体育课程,如果学校有968名学生,他们需要购买多少套校服?

Although 37% is a useful number, it does not tell you how many students need uniforms. You know that for every 100 students, 37 will need a uniform. But that is still not enough information to make a purchase.
::虽然37%是一个有用的数字,但它并没有告诉你有多少学生需要校服。你知道,每100名学生中,有37名学生需要制服。但这还不足以提供购买所需的信息。

First, if there are 968 students at the school, use the proportion $\begin{align*}\frac{a}{b}= \frac{p}{100}\end{align*}$ to find the missing value. In this case you are looking for $\begin{align*}a\end{align*}$ .
::首先,如果学校有968名学生,请使用 ab=p100 的比例来查找缺失值。在此情况下,您正在寻找一个。

$\begin{aligned} \begin{array}{rcl} \frac{a}{b} & = & \frac{p}{100} \\ \frac{a}{968} & = & \frac{37}{100} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{a}{b} &=& \frac{p}{100}\\ \frac{a}{968} &=& \frac{37}{100} \end{array}\end{align*}$
::ab=p100a968=37100

Next, use cross products to help isolate $\begin{align*}a\end{align*}$ . In other words, cross multiply.
::接下来,使用交叉产品帮助隔离 a。换句话说,交叉乘法。

$\begin{aligned} \begin{array}{rcl} \frac{a}{968} & = & \frac{37}{100} \\ 100 a & = & 968 \times 37 \\ 100 a & = & 35816 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{a}{968} &=& \frac{37}{100}\\ 100a &=& 968 \times 37\\ 100a &=& 35816 \end{array}\end{align*}$
::a968=37100100a=968x37100a=35816

Then, divide by 100 to find the number of uniforms to purchase.
::然后除以100 找到需要购买的制服数量

$\begin{aligned} \begin{array}{rcl} 100 a & = & 35816 \\ \frac{100 a}{100} & = & \frac{35816}{100} \\ a & = & 358.16 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 100a &=& 35816\\ \frac{100a}{100} &=& \frac{35816}{100}\\ a &=& 358.16 \end{array}\end{align*}$
::100a=35816100a100=35816100a=35816100a=358。

The answer is 358.16.
::答案是358.16。

Therefore , the team needs to purchase 358 uniforms.
::因此,小组需要购买358套制服。

Since you can use a proportion to solve for any missing variable , you can also find the whole if you know a percent and its corresponding part.
::由于您可以使用比例来解决任何缺失变量,如果您知道一个百分比及其相应部分,您也可以找到整个变量。

Let’s look at another example.
::让我们再看看另一个例子。

At Big Town High School, approximately 58% participate in athletic programs. If 1670 students are involved in an athletics program, then how many students are there in the school?
::在大城镇高中,大约58%的学生参加运动计划。如果有1 670名学生参加运动计划,那么学校里有多少学生?

First, if there are 1670 students at the school, use the proportion $\begin{align*}\frac{a}{b}= \frac{p}{100}\end{align*}$ to find the missing value. In this case you are looking for $\begin{align*}b\end{align*}$ .
::首先,如果学校有1 670名学生,请使用 ab=p100 的比例来查找缺失值。在此情况下,您正在寻找 b。

$\begin{aligned} \begin{array}{rcl} \frac{a}{b} & = & \frac{p}{100} \\ \frac{1670}{b} & = & \frac{58}{100} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{a}{b} &=& \frac{p}{100}\\ \frac{1670}{b} &=& \frac{58}{100} \end{array}\end{align*}$
::ab=p1001670b=58100

Next, use cross products to help isolate $\begin{align*}b\end{align*}$ . In other words, cross multiply.
::接下来,使用交叉产品帮助分离 b. 换句话说,交叉乘法。

$\begin{aligned} \begin{array}{rcl} \frac{1670}{b} & = & \frac{58}{100} \\ 58 b & = & 1670 \times 100 \\ 58 b & = & 167000 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{1670}{b} &=& \frac{58}{100}\\ 58b &=& 1670 \times 100\\ 58b &=& 167000 \end{array}\end{align*}$
::1670b=5810058b=1670x10058b=167000

Then, divide by 58 to find the number of students in the school.
::然后除以58 以寻找在校学生人数。

$\begin{aligned} \begin{array}{rcl} 58 b & = & 167000 \\ \frac{58 b}{58} & = & \frac{167000}{58} \\ b & = & 2879.3 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 58b &=& 167000\\ \frac{58b}{58} &=& \frac{167000}{58}\\ b &=& 2879.3 \end{array}\end{align*}$
::58b=16700058b58=16700058b=2879.3

The answer is 2879.3.
::答案是2879.3。

Therefore there are 2879 students in the school.
::因此,学校有2 879名学生。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the student football crowd.
::早些时候,你遇到一个学生足球队的问题

First, you know that 42% of the students at the games are in middle school. Write the percent as a ratio.
::首先,你知道,42%的游戏学生是中学学生。把这个比例写成一个比例。

$\begin{aligned} 42 % = \frac{42}{100} \end{aligned}$
$\begin{align*}42\%= \frac{42}{100}\end{align*}$

Next, there are 850 students attending the games. Write the proportion to start solving the problem.
::接下来,有850名学生参加游戏。写下比例来开始解决问题。

$\begin{aligned} \begin{array}{rcl} \frac{a}{850} & = & \frac{42}{100} \\ 100 a & = & 42 \times 850 \\ 100 a & = & 35700 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{a}{850} &=& \frac{42}{100}\\ 100a &=& 42 \times 850\\ 100a &=& 35700 \end{array}\end{align*}$
::a850=42100100a=42x850100a=35700

Then, divide by 100 to solve for $\begin{align*}a\end{align*}$ .
::然后,除以100 解决一个。

$\begin{aligned} \begin{array}{rcl} 100 a & = & 35700 \\ \frac{100 a}{100} & = & \frac{35700}{100} \\ a & = & 357 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 100a &=& 35700\\ \frac{100a}{100} &=& \frac{35700}{100}\\ a &=& 357 \end{array}\end{align*}$
::100a=35700100a100=35700100a=35700100a=357

The answer is $\begin{align*}a = 357\end{align*}$ .
::答案是a=357。

Therefore, 357 middle school students attend the game.
::因此,357名中学生参加了比赛。

Example 2
::例2

A forest range discovered that 25% of the trees in his area were infected with a parasite. If there are 3060 trees in his area, how many trees are infected?
::一个森林范围发现,他所在地区的25%的树木感染了寄生虫,如果他所在地区有3 060棵树,有多少棵树受到感染?

First, if there are 968 students at the school, use the proportion $\begin{align*}\frac{a}{b}= \frac{p}{100}\end{align*}$ to find the missing value. In this case you are looking for $\begin{align*}a\end{align*}$ .
::首先,如果学校有968名学生,请使用 ab=p100 的比例来查找缺失值。在此情况下,您正在寻找一个。

$\begin{aligned} \begin{array}{rcl} \frac{a}{b} & = & \frac{p}{100} \\ \frac{a}{3060} & = & \frac{25}{100} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{a}{b} &=& \frac{p}{100}\\ \frac{a}{3060} &=& \frac{25}{100} \end{array}\end{align*}$ Next, use cross products to help isolate $\begin{align*}a\end{align*}$ .
::ab=p100a3060=25100 下一个,使用交叉产品帮助分离 a。

$\begin{aligned} \begin{array}{rcl} \frac{a}{3060} & = & \frac{25}{100} \\ 100 a & = & 3060 \times 25 \\ 100 a & = & 76500 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{a}{3060} &=& \frac{25}{100}\\ 100a &=& 3060 \times 25\\ 100a &=& 76500 \end{array}\end{align*}$
::a3060=25100100a=3060x25100a=76500

Then, divide by 100 to find the number of trees infected.
::然后除以100 以找到感染的树木数量

$\begin{aligned} \begin{array}{rcl} 100 a & = & 76500 \\ \frac{100 a}{100} & = & \frac{76500}{100} \\ a & = & 765 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 100a &=& 76500\\ \frac{100a}{100} &=& \frac{76500}{100}\\ a &=& 765 \end{array}\end{align*}$
::100a=76500100a100=76500100a=765

The answer is 765.
::答案是765

Therefore, there are 765 of the 3060 trees infected.
::因此,3 060棵树中有765棵被感染。

Example 3
::例3

Solve for $\begin{align*}b\end{align*}$ in the proportion: $\begin{align*}\frac{9}{b} = \frac{20}{100}\end{align*}$
::比例为 b 的解决比例: 9b=20100

First, use cross products to help isolate $\begin{align*}b\end{align*}$ .
::首先,使用交叉产品帮助分离b。

$\begin{aligned} \begin{array}{rcl} \frac{9}{b} & = & \frac{20}{100} \\ 20 a & = & 9 \times 100 \\ 20 a & = & 900 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{9}{b} &=& \frac{20}{100}\\ 20a &=& 9 \times 100\\ 20a &=& 900 \end{array}\end{align*}$
::9b=2010020a=9x10020a=900

Next, divide by 20 to solve for $\begin{align*}b\end{align*}$ .
::接下来,除以 20 来解决b。

$\begin{aligned} \begin{array}{rcl} 20 b & = & 900 \\ \frac{20 b}{20} & = & \frac{900}{20} \\ b & = & 45 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 20b &=& 900\\ \frac{20b}{20} &=& \frac{900}{20}\\ b &=& 45 \end{array}\end{align*}$
::20b=90020b20=90020b=45

The answer is $\begin{align*}b = 45\end{align*}$ .
::答案是b=45

Example 4
::例4

Solve for $\begin{align*}a\end{align*}$ in the proportion: $\begin{align*}\frac{a}{10}=\frac{40}{100}\end{align*}$
::解决一个比例比例: a10=40100

First, use cross products to help isolate $\begin{align*}a\end{align*}$ .
::首先,使用交叉产品帮助隔离a。

$\begin{aligned} \begin{array}{rcl} \frac{a}{10} & = & \frac{40}{100} \\ 100 a & = & 40 \times 10 \\ 100 a & = & 400 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{a}{10} &=& \frac{40}{100}\\ 100a &=& 40 \times 10\\ 100a &=& 400 \end{array}\end{align*}$
::a10=40100100a=40x10100a=400

Next, divide by 100 to solve for $\begin{align*}a\end{align*}$ .
::下一个,除以100 解决一个。

$\begin{aligned} \begin{array}{rcl} 100 a & = & 400 \\ \frac{100 a}{100} & = & \frac{400}{100} \\ a & = & 4 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 100a &=& 400\\ \frac{100a}{100} &=& \frac{400}{100}\\ a &=& 4 \end{array}\end{align*}$
::100a=400100a100=400100a=4

The answer is $\begin{align*}a = 45\end{align*}$ .
::答案是a=45。

Example 5
::例5

Solve for $\begin{align*}a\end{align*}$ in the proportion: $\begin{align*}\frac{a}{3}= \frac{18}{100}\end{align*}$
::解决比例比例: a3=18100

First, use cross products to help isolate $\begin{align*}a\end{align*}$ .
::首先,使用交叉产品帮助隔离a。

$\begin{aligned} \begin{array}{rcl} \frac{a}{3} & = & \frac{18}{100} \\ 100 a & = & 3 \times 18 \\ 100 a & = & 54 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{a}{3} &=& \frac{18}{100}\\ 100a &=& 3 \times 18\\ 100a &=& 54 \end{array}\end{align*}$
::a3=18100100a=3x18100a=54

Next, divide by 100 to solve for $\begin{align*}a\end{align*}$ .
::下一个,除以100 解决一个。

$\begin{aligned} \begin{array}{rcl} 100 a & = & 54 \\ \frac{100 a}{100} & = & \frac{54}{100} \\ a & = & 0.54 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 100a &=& 54\\ \frac{100a}{100} &=& \frac{54}{100}\\ a &=& 0.54 \end{array}\end{align*}$
::100a=54100a100=54100a=54100a=0.544

The answer is $\begin{align*}a = 0.54\end{align*}$ .
::答案是=0.54

Review
::回顾

Solve the following proportions for the $\begin{align*}a\end{align*}$ value.
1. $\begin{align*}\frac{a}{23} = \frac{85}{100}\end{align*}$
::解决一个值的以下比例值。 1. a23=85100

2. $\begin{align*}\frac{a}{1500} = \frac{7}{100}\end{align*}$
::2. 1500=7100

3. $\begin{align*}\frac{a}{5} = \frac{61}{100}\end{align*}$
::3.5=6100

4. $\begin{align*}\frac{a}{4} = \frac{75}{100}\end{align*}$
::4.4=75100

5. $\begin{align*}\frac{a}{5} = \frac{40}{100}\end{align*}$
::5.5=40100

6. A small car company sold 65,000 cars last year. Ninety-five percent of those cars had airbags. How many cars had airbags?
::6. 去年,一家小汽车公司出售了65 000辆汽车,其中95%的汽车有气袋,有多少汽车有气袋?

Solve the following proportions for the $\begin{align*}b\end{align*}$ value. Round to the nearest tenths place.
::解决 b 值的以下比例。圆到最近的十分位。

7. $\begin{align*}\frac{88}{b} = \frac{22}{100}\end{align*}$
::7. 88b=22100

8. $\begin{align*}\frac{600}{b} = \frac{74}{100}\end{align*}$
::8. 600b=74100

9. $\begin{align*}\frac{1}{b} = \frac{85}{100}\end{align*}$
::9. 1b=85100

10. $\begin{align*}\frac{2}{b} = \frac{66}{100}\end{align*}$
::10. 2b=66100

11. $\begin{align*}\frac{3}{b} = \frac{75}{100}\end{align*}$
::11. 3b=75100

12. A recent government survey shows that in New City, people spend 35% of their monthly income on rent. If average rent is $780, what is the average income? Round your answer to the nearest cent.
::12. 最近的一项政府调查显示,在新城,人们将月收入的35%花在房租上,如果平均租金为780美元,平均收入是多少?

Use the proportion $\begin{align*}\frac{a}{b} = \frac{p}{100}\end{align*}$ to solve the following problems.
::使用比例 ab=p100 来解决下列问题。

13 A fluorescent light bulb uses 35% as much energy as an incandescent bulb. If an incandescent uses 75 watts, how much does the fluorescent use? Round your answer to the nearest hundredths place.
::13 荧光灯泡的能量与白光灯泡的能量一样大,为35%。如果白光灯泡的能量为75瓦,那么荧光灯灯泡的用量是多少?答案是最近的100个地方。

14. An average human weighs 160 pounds. An elephant weights 1500% more. How much does the elephant weigh?
::14. 人的平均体重为160磅,大象的重量增加了1500磅。大象的重量是多少?

15. An average elephant eats about 350lbs of food per day. Using your calculation from problems #14, what percent of its own weight does it consume each day? Round your answer to the nearest hundredths place.
::15. 平均大象每天食用350磅左右的食物,使用你对问题14的计算,它每天消耗的重量的多少百分比?

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

章节大纲

Proportions ::比例比例

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Proportions
::比例比例

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源