Course: 11.4正负分数和小数比较

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正负分数和小数比较

Rahul and his family just moved to California where everyone is talking about the drought. But it's July and he's seen more rain in the past month than he ever did in Arizona where they used to live. He finds a chart that gives the annual rainfall for the past 50 years as a decimal amount above or below the average rainfall. How can he organize the data to see if there is a general trend?
::Rahul和他的家人刚搬到加州,那里每个人都在谈论干旱。但是,去年七月,他看到比过去在亚利桑那(亚利桑那州)更大规模的降雨量。他发现了一张图表,显示过去50年的年降雨量在平均降雨量上下十进制。他如何组织数据,看是否有普遍趋势?

In this concept, you will learn to compare positive and negative fractions and decimals.
::在此概念中,您将学会比较正分数和负分数以及小数。

Comparing Positive and Negative Fractions and Decimals
::比较正和负分数和十进制数

An integer is any positive whole number or its opposite.
::整数是任何正整数或相反的正整数。

A fraction is any number that is written as a ratio of one number to another. $\begin{aligned} \frac{1}{3} a n d \frac{7}{5} \end{aligned}$ are two examples of fractions. The top number in the fraction is the numerator . The bottom number is the denominator .
::一个分数是以一个数字对另一个数字的比率写入的任何数字。 1 3 n d 7 5 是分数的两个例子。分数中的最大数字是分子。底数是分数。

A decimal is another type of number that can be expressed as a ratio. 0.5 and 1.327 are two examples of decimals.
::小数点是另一种可以以比率表示的数字。 0.5和1.327是小数点的两个例子。

Fractions and decimals are examples of . A rational number is any number that can be expressed as a ratio. Integers are rational numbers, but so are many of the numbers between them.
::小数和小数数是 . 的例子。理性数字是可以用比率表示的任何数字。整数是理性数字,但两者之间的数字很多。

In order to compare fractions, you need to give them a common denominator . This allows you to directly compare the numerators. There are many different common denominators for any pair of fractions, but the easiest one to use is usually the Least Common Denominator , that is, the smallest number that both denominators divide into evenly.
::为了比较分数, 您需要给它们一个共同的分母。这样可以直接比较分子。任何一对分数都有许多不同的共同分母, 但最容易使用的通常是最不常见分母, 也就是两个分母均分的最小数。

Here is an example.
::举一个例子。

$\begin{aligned} - \frac{1}{2} > - \frac{3}{4} \end{aligned}$

The denominators are 2 and 4. The smallest number that both divide into here is 4. $\begin{aligned} 4 \div 2 = 2; 4 \div 4 = 1 \end{aligned}$
::分母是 2 和 4 。两者在此分隔的最小数是 4 4 2 = 2 ; 4 4 = 1 。

Next, you need to convert one or both fractions so that they have the common denominator.
::接下来,您需要转换一个或两个分数, 以便它们具有共同的分母。

In this case, the second fraction already has the common denominator, so it can be left alone. But the denominator of the first fraction must be multiplied by 2 in order to make it 4. However, in mathematics you aren't allowed to just multiply the bottom by something because that would change the fraction into a different, non-equivalent, fraction. You are only allowed to multiply the fraction by 1, in fact. Because any number multiplied by 1 is itself. However, you can be creative about what version of 1 you choose. In this example, choose the version of 1= $\begin{aligned} \frac{2}{2} \end{aligned}$
::在此情况下, 第二分数已经具有共同分母, 所以可以单独保留它。但是第一个分分数的分母必须乘以 2 乘以 2 才能成为它 4 。但是, 在数学中, 您不允许仅将底部乘以某种东西, 因为这样会将分数改变为不同的非等值分数。事实上, 您只能将分数乘以 1 。因为任何数字乘以 1 是它本身。但是, 您可以对您选择的 1 的版本有创意。在此示例中, 您选择 1= 2 2 的版本。

Then, multiply it out.
::然后,再乘数出来。

\begin{aligned} - \frac{1}{2} \times \frac{2}{2} = - \frac{2}{4} \end{aligned}

Another way to think of it is that whatever you do to the bottom of the fraction, you also need to do to the top.
::另一种思考方式是无论你对分数底部做什么, 你也需要对顶部做。

After converting, both fractions have the same denominator so their numerators can be compared.
::在转换后,两个分数都有相同的分母,这样可以比较它们的分子。

Then, check the signs. In this case, both numbers are negative. When comparing negative numbers , the larger number is further from zero and therefore less.
::然后,请检查迹象。在这个例子中,这两个数字都是负数。比较负数时,较大数字比零数字要远,因此更少。

In this case, that means that -3<-2.
::在这种情况下,这意味着 -3 < 2。

That means $\begin{aligned} - \frac{1}{2} > - \frac{3}{4} \end{aligned}$
::这意味着 - 1 2 > - 3 4

The answer is >. (The sideways V always opens in the direction of the larger number.)
::答案是 > 。 (第五侧线总是朝较大数字的方向打开。 )

Rational numbers with decimals are easier to compare because they don't have to be converted. Just look to see which number is bigger. Remember, a positive number is always greater than a negative number, and when comparing negative numbers, the one farther from zero is less.
::小数点的理性数字比较容易,因为它们不需要转换。只需看看哪个数字更大。记住, 正数总是大于负数, 比较负数时, 更远于零的数字会更少。

Here is an example.
::举一个例子。

-.29 ____ -.56

First, check the signs.
::首先,检查标志。

In this case, they are both negative, so the bigger number is less.
::在这种情况下,两者都是负数,因此数字越大越少。

Then compare them to see which is bigger.
::然后比较它们看哪个更大。

In this case, .56 is bigger than .29, so it is further from zero.
::在这种情况下, . 56 大于 29, 所以它比零更远。

Therefore, -.29 is greater than -.56: -.29>-.56.
::因此,-29大于-56:-29>-56。

The answer is >.
::答案是 > 。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Rahul and his question about the California drought.
::早些时候,你得到一个问题关于Rahul和他的问题关于加利福尼亚干旱。

He wants to write the past fifty years in order of rainfall to see if he can tell if California is actually getting drier. Here is his data:
::他想写过去五十年的书以降雨量为序看看他能否看清加州是否真正变干燥。

Season (July 1-June 30) (Year given represents end of season) ::季节(7月1日至6月30日)	Inches Above/Below (+/-) 135 Year Average ::高于/低于(+/-)英寸(+/-)135年平均数
2012	-6.29
2011	+5.22
2010	+1.38
2009	-5.90
2008	-1.45
2007	-11.77
2006	-1.79
2005	+22.98
2004	-5.73
2003	+1.44
2002	-10.56
2001	+2.96
2000	-3.41
1999	-5.89
1998	+16.03
1997	-2.58
1996	-2.54
1995	+9.37
1994	-6.87
1993	+12.38
1992	+6.02
1991	-2.99
1990	-7.63
1989	-6.90
1988	-2.50
1987	-7.32
1986	+2.88
1985	-2.16
1984	-4.55
1983	+16.3
1982	-4.27
1981	-6.02
1980	+12.00
1979	+4.69
1978	+18.46
1977	-2.68
1976	-7.77
1975	-0.63
1974	-0.06
1973	+6.28
1972	-7.81
1971	-2.66
1970	-7.24
1969	+12.49
1968	+1.60
1967	+7.02
1966	+5.46
1965	-1.30
1964	-7.05
1963	-6.60

In order to arrange the years from least rainfall to most, first Rahul looks for the year with the least rainfall. He knows this will be a negative value. He looks for the biggest negative whole numbers first, ignoring the decimals.
::为了安排从降雨最少到降雨最多的年份, 首先Rahul在寻找一年时降雨最少。他知道这将是负值。他首先寻找负数最大的整数, 忽略小数数。

He finds that 2007 has a -11. That is the biggest negative whole number, so 2007 has the least.
::他发现2007年的负数最大,因此2007年的负数最少。

Next, he goes through the rest of the years looking for the next biggest negative whole numbers.
::接下来,他花了几年时间寻找下一个最大的负数

2002 has a -10, so it's next.
::2002年有一个 -10, 所以这是下一个。

Then, he hits a snag because there are 6 years with a -7. 1990=-7.63; 1987=-7.32; 1976=-7.77; 1972=-7.81; 1970=-7.24; 1964=-7.05. In order to order these, he looks at the decimals. The biggest decimal is the most negative. The decimals thus ordered are: .81, .77, .63, .32, .24, .05. Of the -7 years, the least rainfall is 1972 and the most is1964.
::然后,他打中了一个小孔,因为有6年,有-7.1990=7.63;1987=7.32;1976=7.77;1976=7.77;1972=7.81;1970=7.24;1964年=7.05。为了排列这些小孔,他查看了小数小数,最大的小数是负数。因此,小数是:.81,.77,.63,32,24,05。在7年中,降雨量最少的是1972年,最多的是1964年。

So far, his list of rainfall from least to greatest is: 2007, 2002, 1972, 1976, 1990, 1987, 1970, 1964.
::迄今为止,他的降雨量最少到最多清单是:2007年、2002年、1972年、1976年、1990年、1987年、1970年、1964年。

Carrying out this process, he makes the following list of years under the average rainfall:
::在开展这一进程时,他列出了平均降雨量下年数清单如下:

2007, 2002, 1972, 1976, 1990, 1987, 1970, 1964, 1989, 1994, 2012, 1963, 1981, 2009, 1999, 2004, 1984, 1982, 2000, 1999, 1977, 1971, 1997, 1996, 1988, 1985, 2006, 2008, 1965, 1975, 1974.

At this point, he's getting tired of sorting data. He looks at the list so far to see if he can see any trends in it. He sees that 6 out of the last 10 years are under the average rainfall. 13 of the last 20 years are below the average. 16 of the last 25 years are.
::此时,他厌倦了整理数据。他看列表, 看是否能看到任何趋势。他看到过去10年中有6年的平均降雨量低于平均降雨量。过去20年中有13年低于平均降雨量。过去25年中有16年低于平均降雨量。

He writes those as fractions:
::他写作为分数:

\begin{aligned} \frac{6}{10} \frac{13}{20} \frac{16}{25} \end{aligned}

Because he's still confused, and he really wants to figure out what everyone is talking about, he converts these fractions to a common denominator to compare them.
::因为他仍然困惑, 而且他真的想弄清楚每个人都在谈论什么, 他把这些分数转换成一个共同的分母来比较它们。

First, he decides on a common denominator.
::首先,他决定了一个共同点。

In this case, 100 works.
::在这种情况下,有100个作品。

Then he figures out that the first one must be multiplied by 10/10, the second one by 5/5, and the third one by 4/4 to get them to have the common denominator.
::然后他发现第一个必须乘以10/10,第二个必须乘以5/5,第三个必须乘以4/4,才能获得共同的分母。

When he does this, he gets these fractions:
::当他这样做时,他得到这些分数:

\begin{aligned} \frac{60}{100} \frac{65}{100} \frac{64}{100} \end{aligned}

He concludes that more than half of the driest years in the past 50 years have been in the more recent 25 years.
::他的结论是,过去50年中最干燥的一年中有一半以上是最近25年。

In the following examples, compare the two fractions.
::在以下例子中,比较两个部分。

Example 2
::例2

$\begin{aligned} - \frac{2}{5} \end{aligned}$ _____ $\begin{aligned} - \frac{6}{7} \end{aligned}$

First, create a common denominator by multiplying the two denominators together.
::首先,通过将两个分母相乘而形成一个共同分母。

In this case, $\begin{aligned} 5 \times 7 = 35 \end{aligned}$ , so the common denominator is 35.
::在这种情况下,5×7=35,共同标准是35。

Next, decide what version of "one" each fraction must be multiplied by to make its denominator 35.
::其次,决定每一分数的“一”版本必须乘以多少版本才能达到35分母。

The first must be multiplied by $\begin{aligned} \frac{7}{7} \end{aligned}$ .
::第一个必须乘以 7 7。

The second must be multiplied by $\begin{aligned} \frac{5}{5} \end{aligned}$ .
::第二组必须乘以5 5。

Then, convert each fraction accordingly so they can be compared.
::然后,将每一分数依次转换,以便进行比较。

\begin{aligned} - \frac{2}{5} \times \frac{7}{7} = - \frac{14}{35} \end{aligned}

\begin{aligned} - \frac{6}{7} \times \frac{5}{5} = - \frac{30}{35} \end{aligned}

Then, re-write the initial problem.
::然后重写最初的问题。

$\begin{aligned} - \frac{14}{35} \end{aligned}$ ____ $\begin{aligned} - \frac{30}{35} \end{aligned}$

Finally, compare the two.
::最后,比较一下两者。

Both are between 0 and -1. But $\begin{aligned} - \frac{30}{35} \end{aligned}$ is closer to -1 which means that it is more negative. And more negative is smaller. Therefore $\begin{aligned} - \frac{6}{7} < - \frac{2}{5} \end{aligned}$ .
::两者均介于0和-1之间。但- 30 35比-1更接近- 1,这意味着其负值更大。而负值则较小。因此- 6 7 < - 2 5。

The answer is $\begin{aligned} - \frac{2}{5} > - \frac{6}{7} \end{aligned}$ .
::答案是 - 2 5 > - 6 7。

Example 3
::例3

-.98 ____ -.88

First, notice the signs of the two decimals.
::首先,注意小数点后两个小数点的符号。

Because they are both negative, the one that is the bigger number is less than the other.
::因为两者都是负数, 数字越大, 数字越小。

Next, determine which number is bigger.
::下一步,决定哪个数字更大。

.98 is bigger than .88.
::98比88还大

Therefore, -.98 < -.88
::因此,98 < -.88

Example 4
::例4

$\begin{aligned} - \frac{1}{4} \underline{} - \frac{1}{2} \end{aligned}$

First, determine the common denominator.
::首先,确定共同的分母。

In this case, 4 works.
::在这种情况下,有4件作品。

Next, determine which version of "one" each fraction needs to be multiplied by so that it has the common denominator.
::下一步,决定每一分数的“一”的哪个版本需要乘以使其具有共同的分母。

The first fraction already has 4 in the denominator, so it can be left alone. 2x2=4, so the second fraction must be multiplied by $\begin{aligned} \frac{2}{2} \end{aligned}$ .
::第一个分数在分母中已经有 4 个, 所以它可以单独留下。 2x2=4, 所以第二个分数必须乘以 2 2 。

Then, re-write the second fraction with the common denominator.
::然后用共同分母重写第二个分数。

\begin{aligned} - \frac{1}{2} \times \frac{2}{2} = - \frac{2}{4} \end{aligned}

Then, re-write the initial problem with the new fraction.
::然后用新分数重写初始问题。

$\begin{aligned} - \frac{1}{4} \end{aligned}$ ____ $\begin{aligned} - \frac{2}{4} \end{aligned}$

Then, note the signs.
::然后,注意迹象。

Since they are both negative, the bigger number is less.
::由于两者都是负数,因此数字越大越少。

-2 is bigger than -1 so the answer is $\begin{aligned} - \frac{1}{4} > - \frac{1}{2} \end{aligned}$
::-2大于 -1 所以答案是 -1 4 > -1 2

Example 5
::例5

.67 ____ -.67

First, note the signs.
::首先,注意这些迹象。

.67 is positive. -.67 is negative.
::67是正数 67是负数

Then, remember that a positive number is always greater than a negative.
::那么,记住一个正数总是大于负数。

Therefore, the answer is .67>-.67

::因此,答案是.67>-.67

Review
::回顾

Compare each pair of values using <, > or =.
::比较使用 < 、 > 或 = 的每对数值。

-0.18 ____ -0.27
-0.23 ____ -0.98
-9 ____ -11
-18 ____ -29
-67 ____ -89
$\begin{aligned} - \frac{1}{4} \underline{} - \frac{4}{5} \end{aligned}$
$\begin{aligned} - \frac{3}{4} \underline{} - \frac{1}{3} \end{aligned}$
$\begin{aligned} - \frac{5}{10} \underline{} - \frac{1}{2} \end{aligned}$
$\begin{aligned} - \frac{3}{4} \underline{} - 0.75 \end{aligned}$
$\begin{aligned} - \frac{1}{4} \underline{} - 0.25 \end{aligned}$
$\begin{aligned} - .25 \underline{} - \frac{3}{4} \end{aligned}$
$\begin{aligned} - \frac{18}{20} \underline{} - \frac{1}{2} \end{aligned}$

Write the following values in order from least to greatest.
::写下以下值, 以从最小到最大顺序排列。

-4, -12, -19, -8, 0, -2, -1
5, 7, 23, 8, -9, -11
$\begin{aligned} \frac{- 1}{2}, \frac{- 1}{4}, \frac{- 5}{6}, \frac{- 3}{4} \end{aligned}$

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。

Section outline

General

正负分数和小数比较

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