Section: 涉及科学标记的解决百分率问题 | 5.16 解决涉及科学名词的百分率问题

Section outline

On certain dates, Mars is about $\begin{aligned} 4.9 \times 10^{7} \end{aligned}$ miles from Earth. If a spacecraft is headed toward Mars and has traveled 30% of the distance, how many miles has it gone?
::在某些日期,火星距离地球约4.9×107英里。如果航天器向火星飞去,飞行了30%的距离,它走了多少英里?

In this concept, you will learn to solve percent problems involving scientific notation.
::在这个概念中,你将学会解决与科学符号有关的百分率问题。

Scientific Notation
::科学符号

Scientific notation is a useful mathematical tool that allows you to work with very large or very small numbers. Scientific notation is when a number is written as a rational number and a power of 10. This means that you are using to represent the power of 10. Scientific notation has the general form: $\begin{aligned} a \times 10^{b} \end{aligned}$
::科学标记是一个有用的数学工具, 使您能够使用非常大或非常小的数字工作。科学标记是指当数字被写成一个理性数字和10的功率时。这意味着您用来代表10的功率。科学标记具有一般形式: ax10b。

where ‘ $\begin{aligned} a \end{aligned}$ ’ is any number between 1 and 10, and ‘ $\begin{aligned} b \end{aligned}$ ’ is the exponent of 10.
::其中`a ' 是1至10之间的任何数字,`b ' 是10的指数。

When you conduct operations involving percent with numbers in scientific notation, you can use any operation with the ‘ $\begin{aligned} a \end{aligned}$ ’ value as you have seen in these sections. Then you will be sure to write your answer in scientific notation. You may need to adjust the ‘ $\begin{aligned} a \end{aligned}$ ’ and ‘ $\begin{aligned} b \end{aligned}$ ’ values.
::当您在科学符号中用数字进行占%的操作时,您可以使用与这些章节中看到的`a ' 值有关的任何操作。然后您可以将答案写在科学符号中。您可能需要调整“a”和“b”值。

Let’s look at an example.
::让我们举个例子。

Find 25% of $\begin{aligned} 3 \times 10^{12} \end{aligned}$ .
::查找 3x1012 的25%。

First, multiply the 25% by 3.
::首先,乘以25%乘以3

$\begin{aligned} \begin{array}{rcl} 0.25 \times (3 \times 10^{12}) & = & (0.25 \times 3) \times 10^{12} \\ = & 0.75 \times 10^{12} \end{array} \end{aligned}$

Next, since the ‘ $\begin{aligned} a \end{aligned}$ ’ value is not between 1 and 10, you must move the decimal place over one place. In other words, 0.75 must change to 7.5. Since you are moving the decimal one place to the right, your exponent $\begin{aligned} b \end{aligned}$ will go down by 1.
::其次,由于“a”值不介于1到10之间,您必须将小数点位移到一个地方。换句话说,0.75必须改为7.5。既然您将小数点位移到右边,您的指数b将下降1。

$\begin{aligned} 0.75 \times 10^{12} = 7.5 \times 10^{11} \end{aligned}$

The answer is $\begin{aligned} 7.5 \times 10^{11} \end{aligned}$ .
::答案是7.5x1011。

Very large and very small numbers are not always written in scientific notation. Some numbers remain expanded or in standard notation . You can still work with numbers that are very large but again must be most careful of decimal places.
::非常大和非常小的数字并不总是在科学符号中写成。有些数字仍然在扩展或标准符号中。您仍然可以使用非常大的数字工作,但必须十分小心小数位数。

Let’s look at an example.
::让我们举个例子。

In the year 2000, the United States had a population of about 280,000,000 people. By 2010, the population was about 308,000,000. What will the percent increase have been in those 10 years?
::2000年,美国人口约为280 000 000人,到2010年,人口约为308 000 000人,这十年增长的百分比是多少?

First, find the difference in the populations.
::首先,找出人口的差别。

$\begin{aligned} 308, 000, 000 - 280, 000, 000 = 28, 000, 000 \end{aligned}$

Next, use the percent increase formula to find the answer.
::接下来,使用百分比增长公式来找到答案。

$\begin{aligned} \begin{array}{rcl} percent of increase & = & \frac{amount of increase}{original amount} \times 100 \\ percent of increase & = & \frac{28, 000, 000}{280, 000, 000} \times 100 \\ percent of increase & = & 0.10 \times 100 \\ percent of increase & = & 10 % \end{array} \end{aligned}$

::增加的百分率=0.10×增加的100% 增加的原数×100% = 28 000280 000 000×100% 增加的0.10×100% 增加的10% 增加的0.10×100% 增加的10%

The answer is 10%.
::答案是10%。

The population increased 10% in 10 years.
::人口在10年内增长了10%。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the trip to Mars.
::早些时候,有人给了你一个火星之旅的问题

You want to find the number of miles a spacecraft has gone if it is 30% of the way to Mars, which at its nearest is about $\begin{aligned} 4.9 \times 10^{7} \end{aligned}$ miles from Earth.
::您想要找到一个航天器去的英里数, 如果它到达火星的路程是30%, 最近的火星距离地球大约是4.9×107英里。

First, change 30% into a decimal.
::首先,将30%改为小数点。

$\begin{aligned} \begin{array}{rcl} 30 % & = & \frac{30}{100} \\ = & 0.30 \end{array} \end{aligned}$

Next, multiply 0.30 by $\begin{aligned} 4.9 \times 10^{7} \end{aligned}$
::下一个乘以0.30乘4.9×107

$\begin{aligned} \begin{array}{rcl} 0.30 \times (4.9 \times 10^{7}) & = & (0.30 \times 4.9) \times 10^{7} \\ = & 1.47 \times 10^{7} \end{array} \end{aligned}$

The answer is $\begin{aligned} 1.47 \times 10^{7} \end{aligned}$ .
::答案是1.47×107。

The spacecraft has traveled $\begin{aligned} 1.47 \times 10^{7} \end{aligned}$ miles towards Mars if it is 30% of the way.
::该航天器已经向火星飞行1.47×107英里如果它行驶到30%的路程的话

Example 2
::例2

Find 32% of 0.00000054.
::发现百分之三十二的万万五十四。

First, change 32% into a decimal. Remember that percent means that the denominator is 100.
::首先,将32%修改为小数点。记住这个百分比意味着分母为100。

$\begin{aligned} \begin{array}{rcl} 32 % & = & \frac{32}{100} \\ = & 0.32 \end{array} \end{aligned}$

Next, multiply 0.32 by 0.00000054
::下一个乘以0.32乘以0.000054

$\begin{aligned} 0.32 \times 0.00000054 = 0.0000001728 \end{aligned}$

Then, change to scientific notation. Notice that the number is smaller than 1 so the exponent $\begin{aligned} (b) \end{aligned}$ will be negative. Move the decimal place until it is after the 1. This means you are moving the decimal 7 places to the right; your exponent $\begin{aligned} (b) \end{aligned}$ is -7.
::然后, 更改为科学符号。请注意, 数字小于 1 , 引号(b) 将是负数。移动小数位数, 到 1 之后。这意味着您要将小数位数 7 位数移到右边; 您的引号(b) 是 - 7 。

$\begin{aligned} 0.0000001728 = 1.728 \times 10^{- 7} \end{aligned}$

The answer is $\begin{aligned} 1.728 \times 10^{- 7} \end{aligned}$ .
::答案是1.728×10-7。

Example 3
::例3

Find 30% of .000567
::发现0.00567占0.00567的30%

First, change 30% into a decimal. Remember that percent means that the denominator is 100.
::首先,将30%修改为小数点。记住这个百分比意味着分母为100。

$\begin{aligned} \begin{array}{rcl} 30 % & = & \frac{30}{100} \\ = & 0.30 \end{array} \end{aligned}$

Next, multiply 0.30 by 0.000567
::下一个乘以0.30乘以0.0567

$\begin{aligned} 0.30 \times 0.000567 = 0.0001701 \end{aligned}$

Then, change to scientific notation. Notice that the number is smaller than 1 so the exponent $\begin{aligned} (b) \end{aligned}$ will be negative. Move the decimal place until it is after the 1. This means you are moving the decimal 4 places to the right; your exponent $\begin{aligned} (b) \end{aligned}$ is -4.
::然后, 更改为科学符号。请注意, 数字小于 1 , 引号(b) 将是负数。移动小数位数, 到 1 之后。这意味着您要将小数点 4 移到右边; 您的引号(b) 是 - 4 。

$\begin{aligned} 0.0001701 = 1.701 \times 10^{- 4} \end{aligned}$

The answer is $\begin{aligned} 1.701 \times 10^{- 4} \end{aligned}$ .
::答案是 1.701×10-4。

Example 4
::例4

Find 10% of 123,000,000
::发现123 000 000 000 的10%

First, change 10% into a decimal.
::首先,将10%改为十进制。

$\begin{aligned} \begin{array}{rcl} 10 % & = & \frac{10}{100} \\ = & 0.10 \end{array} \end{aligned}$

Next, multiply 0.10 by 123,000,000
::下一个乘以0.10乘123 000 000

$\begin{aligned} 0.10 \times 123, 000, 000 = 12, 300, 000 \end{aligned}$

Then, change to scientific notation. Notice that the number is larger than 1 so the exponent $\begin{aligned} (b) \end{aligned}$ will be positive . Move the decimal place until it is after the 1. This means you are moving the decimal 7 places to the left; your exponent $\begin{aligned} (b) \end{aligned}$ is 7.
::然后, 更改为科学符号。请注意, 数字大于 1 , 指数(b) 将是正数。移动小数点的位置, 到 1 之后。这意味着您要将小数点 7 点移到左边; 您的指数(b) 是 7 点。

$\begin{aligned} 12, 300, 000 = 1.23 \times 10^{7} \end{aligned}$

The answer is $\begin{aligned} 1.23 \times 10^{7} \end{aligned}$ .
::答案是1.23×107

Example 5
::例5

Find 25% of 0.0000987.
::发现0.000987的25%。

First, change 25% into a decimal.
::首先,将25%改为小数点后。

$\begin{aligned} \begin{array}{rcl} 25 % & = & \frac{25}{100} \\ = & 0.25 \end{array} \end{aligned}$

Next, multiply 0.25 by 0.0000987
::下一个乘以0.25乘以0.000987

$\begin{aligned} 0.25 \times 0.0000987 = 0.000024675 \end{aligned}$

Then, change to scientific notation. Notice that the number is smaller than 1 so the exponent $\begin{aligned} (b) \end{aligned}$ will be negative. Move the decimal place until it is after the 1. This means you are moving the decimal 5 places to the right; your exponent $\begin{aligned} (b) \end{aligned}$ is -5.
::然后, 更改为科学符号。请注意, 数字小于 1 , 引号(b) 将是负数。移动小数点的位置, 到 1 之后。这意味着您要将小数点 5 位移到右边; 您的引号(b) 是 - 5 。

$\begin{aligned} 0.000024675 = 2.4675 \times 10^{- 5} \end{aligned}$

The answer is $\begin{aligned} 2.47 \times 10^{- 5} \end{aligned}$ .
::答案是2.47×10-5。

Review
::回顾

Round the ‘ $\begin{aligned} a \end{aligned}$ ’ values to the nearest hundredth and place your answers in scientific notation.
::将“a”值圈到最近的第一百个值,并将你的答案放在科学符号中。

1. Find 62% of $\begin{aligned} 3.5 \times 10^{9} \end{aligned}$ .
::1. 发现3.5×109的62%。

2. Find 5% of $\begin{aligned} 9.1 \times 10^{13} \end{aligned}$ .
::2. 发现9.1x1013的5%。

3. Find 180% of $\begin{aligned} 6.3 \times 10^{- 17} \end{aligned}$ .
::3. 发现6.3×10-17的180%。

4. Find 12% of $\begin{aligned} .18 \times 10^{3} \end{aligned}$ .
::4. 发现18×103的12%。

5. Find 22% of $\begin{aligned} 56.4 \times 10^{- 2} \end{aligned}$ .
::5. 发现56.4x10-2的22%。

6. Find 14% of $\begin{aligned} 56.4 \times 10^{- 2} \end{aligned}$ .
::6. 发现56.4x10-2的14%。

7. Find 30% of $\begin{aligned} 9.99 \times 10^{11} . \end{aligned}$
::7. 发现9.99×1011的30%。

Answer each question and leave your answers in standard form. Round your answers up to the seven decimal places.
::回答每个问题并按标准格式留下答案。将答案四舍五入到小数点后七位。

8. What percent of 8,570,000 is 152?
::8. 8,570,000的百分之八是152?

9. Find 230% of .00000488.
::9. 发现0.000488的230%。

10. The number .00036 is 45% of what number?
::10.00036是多少的45%?

11. Find 23% of 98.78.
::11. 发现98.78的23%。

12. Find 150% of .0000866.
::12. 发现1000866的150%。

13. Find 210% of .002368.
::13. 发现0.002368的210%。

14. Find 30% of .000009.
::14. 发现0.00009的30%。

15. A light year is about 5,880,000,000,000 miles. In one month, it travels about 8.3% of that distance. About how far does it travel in one month?
::15. 光年约5 880 000 000万英里,一个月内,其飞行距离约为8.3%,一个月内,其飞行距离有多远?

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

Scientific Notation ::科学符号

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

Scientific Notation
::科学符号

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