Course: 5.9 使用百分数来寻找基地 b

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使用百分数来查找基地 b
Alice lives in New Hampshire and by mid-September 35% of the trees lose their leaves. Alice considers the trees in her grandparents’ grove, and sees that approximately 850 trees have lost their leaves. How can Alice know how many trees are there in all?
::爱丽丝生活在新罕布什尔,到9月中旬,35%的树木失去了叶子。爱丽丝认为她祖父母的树林里有树,并且看到大约850棵树失去了叶子。爱丽丝怎么知道有多少棵树?

In this concept, you will learn to use the percent equation to find the base, $\begin{align*}b\end{align*}$ .
::在这个概念中,你会学会使用百分数方程式来找到基数 b。

Finding the Base b
::查找基地b

You can use the proportion $\begin{align*}\frac{a}{b} = \frac{p}{100}\end{align*}$ to solve for a percent. You can also solve by using an equation. In this concept, you will use a proportion to create a different kind of equation that will help you solve percent problems. Sometimes, you will know the percent and a part of the ratio, or part a, but you will need to find the whole or the base, $\begin{align*}b\end{align*}$ .
::您可以使用比例 ab=p100 来解答一个百分比。您也可以使用方程式解答一个百分比。在此概念中,您将使用一个比例来创建一种不同的方程式,帮助您解答%问题。有时,您会知道比例的%和部分,或者部分a,但您需要找到整个或底部,b。

When you solve the proportion $\begin{align*}\frac{a}{b} = \frac{p}{100}\end{align*}$ , you cross multiply to find the missing variable . You can rearrange this formula so that you are solving for just the variable $\begin{align*}a\end{align*}$ .
::当您解析比例 ab=p100 时,要找到缺失的变量,要交叉乘数。您可以重新排列此公式,这样您就可以只解决变量a 。

$\begin{aligned} \begin{array}{rcl} \frac{a}{b} & = & \frac{p}{100} \\ 100 a & = & p b \\ \frac{100 a}{100} & = & \frac{p b}{100} \\ a & = & \frac{p b}{100} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \frac{a}{b} &=& \frac{p}{100} \\ 100a &=& pb \\ \frac{100a}{100} &=& \frac{pb}{100} \\ a &=& \frac{pb}{100} \end{array}\end{align*}$ You could also say that $\begin{align*}a = \frac{pb}{100}\end{align*}$ is equal to $\begin{align*}a = 0.01pb\end{align*}$ . As well, you could convert your percent directly into a decimal and therefore the formula becomes even simpler.
::ab=p100100a=pb100a100=pb100a=pb100a=pb100a=pb100b。您也可以说 a=pb100is 等于 a=0.01pb。同样,您也可以直接将百分比转换为十进制,因此公式变得更加简单。

Let’s look at a problem.
::让我们来看看一个问题。

78 is 65% of what number?
::78是多少数字的65%?

First, write the equation. Remember that $\begin{align*}65\%\end{align*}$ is the same as $\begin{align*}\frac{65}{100}\end{align*}$ .
::首先,写下方程。记住65%与65100相同。

$\begin{aligned} 78 = \frac{65}{100} \times b \end{aligned}$
$\begin{align*}78 = \frac{65}{100} \times b\end{align*}$
::78=65100xb

or
::或

$\begin{aligned} 78 = 0.65 \times b \end{aligned}$
$\begin{align*}78 = 0.65 \times b\end{align*}$
::78=0.65xb

Next, divide both sides of the equation by 0.65 to solve for $\begin{align*}b\end{align*}$ .
::接下来,将方程两边除以0.65,为b解答。

$\begin{aligned} \begin{array}{rcl} 78 & = & 0.65 \times b \\ \frac{78}{0.65} & = & \frac{0.65 \times b}{0.65} \\ b & = & 120 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 78 &=& 0.65 \times b \\ \frac{78}{0.65} &=& \frac{0.65 \times b}{0.65} \\ b &=& 120 \end{array}\end{align*}$
::78=0.65xb780.65=0.65xb0.65b=120

The answer is 120.
::答案是120个

Therefore, 78 is 65% of 120.
::因此,78是120的65%。

Let’s try another example.
::让我们再试一个例子。

11 is 77% of what number?
::11是多少数字的77%?

First, write the equation. Remember that 77% is the same as $\begin{align*}\frac{77}{100}\end{align*}$ .
::首先,写下方程。记住,77%与77100相同。

$\begin{aligned} 11 = \frac{77}{100} \times b \end{aligned}$
$\begin{align*}11 = \frac{77}{100} \times b\end{align*}$
::11=77100xb

or
::或

$\begin{aligned} 11 = 0.77 \times b \end{aligned}$
$\begin{align*}11 = 0.77 \times b\end{align*}$
::11=0.77xb

Next, divide both sides of the equation by 0.77 to solve for $\begin{align*}b\end{align*}$ .
::接下来,将方程两边除以0.77,解决 b。

$\begin{aligned} \begin{array}{rcl} 11 & = & 0.77 \times b \\ \frac{11}{0.77} & = & \frac{0.77 \times b}{0.77} \\ b & = & 14.29 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 11 &=& 0.77 \times b \\ \frac{11}{0.77} &=& \frac{0.77 \times b}{0.77} \\ b &=& 14.29 \end{array}\end{align*}$
::11=0.77xb110.77=0.77xb0.77b=14.29

The answer is 14.29.
::答案是14.29。

Therefore, 11 is 77% of 14.29.
::因此,11占14.29的77%。

In this problem, you could round to the nearest hundredths place as you did here. Sometimes, you may be asked to round to the nearest tenths place. In that case, the answer would have been 14.3.
::在此问题上, 您可以像这里一样绕到最近的一百个地方。有时, 您可能会被要求绕到最近的十分之一处。如果是这样的话, 答案应该是14.3 。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Alice’s family grove.
::更早之前,你曾因爱丽丝的家庭园林问题而感到疑惑。

Alice knows that 850 trees in the grove lost their leaves, but doesn’t know the total number of trees in the grove. She also knows that the 850 represents 35% of the total grove.
::爱丽丝知道林中850棵树失去了树叶,但不知道林中树木的总数。她还知道850棵树占林中总面积的35%。

First, write the equation. Remember that 35% is the same as $\begin{align*}\frac{35}{100}\end{align*}$ .

$\begin{aligned} 850 = \frac{35}{100} \times b \end{aligned}$
$\begin{align*}850 = \frac{35}{100} \times b\end{align*}$
::首先,写下方程。记住35%是35100。 850=35100xb

or
::或

$\begin{aligned} 850 = 0.35 \times b \end{aligned}$
$\begin{align*}850 = 0.35 \times b\end{align*}$
::850=0.35xb

Next, divide both sides of the equation by 0.35 to solve for $\begin{align*}b\end{align*}$ .
::接下来,将方程两侧除以0.35,解决 b。

$\begin{aligned} \begin{array}{rcl} 850 & = & 0.35 \times b \\ \frac{850}{0.35} & = & \frac{0.35 \times b}{0.35} \\ b & = & 2428.57 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 850 &=& 0.35 \times b \\ \frac{850}{0.35} &=& \frac{0.35 \times b}{0.35} \\ b &=& 2428.57 \end{array}\end{align*}$
::850=0.35×b8500.35=0.35×b0.35b=2428.57

The answer is 2428.
::答案是2428号

Therefore, 850 is approximately 35% of 2428.
::因此,850人约占2428人的35%。

Example 2
::例2

25 is 60% of what number?
::25是多少数字的60%?

First, write the equation. Remember that 60% is the same as $\begin{align*}\frac{60}{100}\end{align*}$ .
::首先,写下方程。记住60%与60100相同。

$\begin{aligned} 25 = \frac{60}{100} \times b \end{aligned}$
$\begin{align*}25 = \frac{60}{100} \times b\end{align*}$
::25=60100xb

or
::或

$\begin{aligned} 25 = 0.60 \times b \end{aligned}$
$\begin{align*}25 = 0.60 \times b\end{align*}$
::25=0.0.60xb

Next, divide both sides of the equation by 0.60 to solve for $\begin{align*}b\end{align*}$ .
::接下来,将方程的两边除以0.60,解决 b。

$\begin{aligned} \begin{array}{rcl} 25 & = & 0.60 \times b \\ \frac{25}{0.60} & = & \frac{0.60 \times b}{0.60} \\ b & = & 41.67 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 25 &=& 0.60 \times b \\ \frac{25}{0.60} &=& \frac{0.60 \times b}{0.60} \\ b &=& 41.67 \end{array}\end{align*}$ The answer is 41.67.
::25=0.60×b250.60=0.60×b0.60b=41.67 答案是41.67

Therefore, 25 is 60% of 41.67.
::因此,25是41.67的60%。

Example 3
::例3

10 is 50% of what number?
::10%是多少的50%?

First, write the equation. Remember that 50% is the same as $\begin{align*}\frac{50}{100}\end{align*}$ .
::首先,写下方程,记住50%等于50100

$\begin{aligned} 10 = \frac{50}{100} \times b \end{aligned}$
$\begin{align*}10 = \frac{50}{100} \times b\end{align*}$
::10=50100xb

or
::或

$\begin{aligned} 10 = 0.50 \times b \end{aligned}$
$\begin{align*}10 = 0.50 \times b\end{align*}$
::10=0.0.50xb

Next, divide both sides of the equation by 0.50 to solve for $\begin{align*}b\end{align*}$ .
::接下来,将方程两边除以0.50,解决 b。

$\begin{aligned} \begin{array}{rcl} 10 & = & 0.50 \times b \\ \frac{10}{0.50} & = & \frac{0.50 \times b}{0.50} \\ b & = & 20 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 10 &=& 0.50 \times b \\ \frac{10}{0.50} &=& \frac{0.50 \times b}{0.50} \\ b &=& 20 \end{array}\end{align*}$
::10=0.50×b100.50=0.50×b0.50b=20

The answer is 20.
::答案是20岁

Therefore, 10 is 50% of 20.
::因此,10是20的50%。

Example 4
::例4

45 is 20% of what number?
::45是什么数字的20%?

First, write the equation. Remember that 20% is the same as $\begin{align*}\frac{20}{100}\end{align*}$ .
::首先,写下方程。记住20%与20100年相同。

$\begin{aligned} 45 = \frac{20}{100} \times b \end{aligned}$
$\begin{align*}45 = \frac{20}{100} \times b\end{align*}$
::45=20100xb

or
::或

$\begin{aligned} 45 = 0.20 \times b \end{aligned}$
$\begin{align*}45 = 0.20 \times b\end{align*}$
::45=0.20xb

Next, divide both sides of the equation by 0.20 to solve for $\begin{align*}b\end{align*}$ .
::接下来,将方程的两边除以0.20,解决 b。

$\begin{aligned} \begin{array}{rcl} 45 & = & 0.20 \times b \\ \frac{45}{0.20} & = & \frac{0.20 \times b}{0.20} \\ b & = & 225 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 45 &=& 0.20 \times b \\ \frac{45}{0.20} &=& \frac{0.20 \times b}{0.20} \\ b &=& 225 \end{array}\end{align*}$
::45=0.20xb4.50.20=0.20xb0.20b=225

The answer is 225.
::答案是225

Therefore, 45 is 20% of 225.
::因此,45是225的20%。

Example 5
::例5

68 is 40% of what number?
::68是多少数字的40%?

First, write the equation. Remember that 40% is the same as $\begin{align*}\frac{40}{100}\end{align*}$ .
::首先,写下方程,记住40%和40100相同

$\begin{aligned} 68 = \frac{40}{100} \times b \end{aligned}$
$\begin{align*}68 = \frac{40}{100} \times b\end{align*}$
::68=40100xb

or
::或

$\begin{aligned} 68 = 0.40 \times b \end{aligned}$
$\begin{align*}68 = 0.40 \times b\end{align*}$
::68=0.40xb

Next, divide both sides of the equation by 0.40 to solve for $\begin{align*}b\end{align*}$ .
::接下来,将方程的两边除以0.40,解决 b。

$\begin{aligned} \begin{array}{rcl} 68 & = & 0.40 \times b \\ \frac{68}{0.40} & = & \frac{0.40 \times b}{0.40} \\ b & = & 170 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 68 &=& 0.40 \times b \\ \frac{68}{0.40} &=& \frac{0.40 \times b}{0.40} \\ b &=& 170 \end{array}\end{align*}$ The answer is 170.
::68=0.40xb680.40=0.40xb0.40b=170 答案是170。

Therefore, 68 is 40% of 170.
::因此,68是170的40%。

Review
::回顾

Solve each percent problem. You may round your answers to the nearest tenth when necessary.
::解决每个百分率问题。必要时您可以将答案绕到最近的十分之一。

23 is 9% of what number?
::23是多少的9%?

10 is 35% of what number?
::10是多少数字的35%?

580 is 82% of what number?
::580是多少数字的82%?

58 is 8% of what number?
::58是什么数字的8%?

58 is 80% of what number?
::58是多少数字的80%?

11 is 82% of what number?
::11是多少数字的82%?

33 is 2% of what number?
::33是什么数字的2%?

14 is 9% of what number?
::14是多少数字的9%?

50 is 67% of what number?
::50是多少数字的67%?

33 is 45% of what number?
::33是多少数字的45%?

40 is 80% of what number?
::40是多少数字的80%?

68 is 99% of what number?
::68是多少数字的99%?

78 is 55% of what number?
::78是多少数字的55%?

16 is 12% of what number?
::16是哪个数字的12%?

1450 is 80% of what number?
::1450是多少数字的80%?

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

Section outline

General

使用百分数来查找基地 b

Finding the Base b ::查找基地b

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

Finding the Base b
::查找基地b

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