Section: 分数与带分数比较 | 3.3分数与带分数比较

Section outline

Margaret’s dad is trying to cut back on the salt in his diet. He learned that many packaged foods contain a lot of salt, so he is trying to make some of his favorite foods himself. One of his favorite snacks is salsa, so he is making his own salsa. He found two different recipes. One calls for $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ teaspoons of salt and another calls for $\begin{aligned} 1 \frac{5}{8} \end{aligned}$ teaspoons of salt. How can Margaret help her dad to determine which recipe to use if he wants to use the least amount of salt possible?
::玛格丽特的爸爸正在试图削减饮食中的盐量。他了解到,许多包装食品中含有大量盐量,因此他试图自己做一些他最喜欢的食品。他最喜欢的零食之一是萨尔萨,因此他正在做自己的盐量。他找到两种不同的食谱。其中一种需要112个盐量的茶匙,另一种需要158个盐量的茶量。玛格丽特如何帮助她父亲决定如果他想使用最少的盐量的话,使用哪种配方?

In this concept, you will learn how to compare and order fractions and mixed numbers.
::在这个概念中,你会学习如何比较和排序分数和混合数字。

Comparing Fractions and Mixed Numbers
::比较分数和混合数字

When two fractions have the same denominator , the fraction with the larger numerator will be the bigger fraction. When two fractions do not have the same denominator, comparison is not as easy.
::当两个分数有相同的分母时, 带有较大分子的分数将是更大的分数。当两个分数没有相同的分母时, 比较不那么容易。

One way to compare fractions is by using approximation and benchmarks. Approximate each fraction with one of the three benchmarks: $\begin{aligned} 0, \frac{1}{2} \end{aligned}$ , and $\begin{aligned} 1 \end{aligned}$ . Then, compare the approximations.
::比较分数的方法之一是使用近似值和基准。每种分数与三个基准之一( 0 、 1 2 和 1 ) 相近。然后比较近似值。

Here is an example.
::举一个例子。

Use approximation to order $\begin{aligned} \frac{7}{8}, \frac{2}{5}, 3 \frac{5}{8}, \frac{1}{29} \end{aligned}$ , and $\begin{aligned} \frac{29}{30} \end{aligned}$ greatest to least.
::使用近似于7 8, 2 5, 3 5 8, 1 29, 29 30 至最少。

First, approximate each fraction using the fraction benchmarks.
::首先,使用分数基准,大约每个分数。

$\begin{aligned} \frac{7}{8} : 7 \end{aligned}$ out of 8 is almost 8 out of 8 which would be a whole, so $\begin{aligned} \frac{7}{8} \end{aligned}$ is approximately 1.
::7 8:8人中有7人几乎是8人中有8人将是整体,7人大约是1人。

$\begin{aligned} \frac{2}{5} : 2 \end{aligned}$ is a little less than half of 5, so $\begin{aligned} \frac{2}{5} \end{aligned}$ is approximately $\begin{aligned} \frac{1}{2} \end{aligned}$ .
::2 5: 2 略小于 5 的一半, 所以 2 5 大约是 12 。

$\begin{aligned} 3 \frac{5}{8} \end{aligned}$ is a mixed number greater than 1, so it will automatically be the greatest number in our list.
::3,5,8是超过1的混合数字,因此,它将自动成为我们名单上数量最多的。

$\begin{aligned} \frac{1}{29} \end{aligned}$ the denominator of 29 is much larger than the numerator of 1, so $\begin{aligned} \frac{1}{29} \end{aligned}$ is approximately 0.
::1 29 分母为 29 比 1 的分子大得多, 所以 1 29 大约是 0 。

$\begin{aligned} \frac{29}{30} : 29 \end{aligned}$ out of 30 is almost 30 out of 30 which would be a whole, so $\begin{aligned} \frac{29}{30} \end{aligned}$ is approximately 1.
::29 30:30人中有29人几乎为30人,30人中有30人将是整体,29人30人大约为1人。

Now, write the fractions in a preliminary greatest to least order with the benchmarks in parentheses:
::现在,用括号中的基准, 以初步最大至最小顺序写出分数 :

$\begin{aligned} 3 \frac{5}{8}, \frac{7}{8} (1), \frac{29}{30} (1), \frac{2}{5} (\frac{1}{2}), \frac{1}{29} (0) \end{aligned}$

Notice that the approximation method helped to order most of the numbers, but there are two fractions that are close to 1. You will have to use another method to decide how $\begin{aligned} \frac{7}{8} \end{aligned}$ compares to $\begin{aligned} \frac{29}{30} \end{aligned}$ .
::请注意近似法帮助排列了大部分数字,但有两种分数接近1, 你将不得不使用另一种方法来决定 7 8 与 29 30 之间的比例。

One way to determine which fraction is closest to 1 is to draw two number lines between 0 and 1, arranged so that one number line is above the other. Divide the top number line into eighths and find $\begin{aligned} \frac{7}{8} \end{aligned}$ . Divide the bottom number line into thirtieths and find $\begin{aligned} \frac{29}{30} \end{aligned}$ . Look to see which value is closest to 1.
::确定哪个分数最接近 1 的方法之一是在 0 和 1 之间绘制两个数字行, 并排列一个数字行在另一个数字行之上。将最高数字行除以 8, 并发现 7 8 。将底数行除以 30, 并发现 29 30 。看看哪个值最接近 1 。查看哪个值最接近 1 。

Now you can see that $\begin{aligned} \frac{29}{30} \end{aligned}$ is closer to 1, so it is the greater number.
::现在你可以看到2930接近1 所以数字越大。

The answer is that the numbers ordered from greatest to least are $\begin{aligned} 3 \frac{5}{8}, \frac{29}{30}, \frac{7}{8}, \frac{2}{5}, \frac{1}{29} \end{aligned}$ .
::答案是,从最多到最少订购的号码是3,5,8,29,30,7,8,2,5,1,29。

Another way to compare two fractions with different denominators is by rewriting one or both fractions so that they have the same denominator. To rewrite a fraction, find an equivalent fraction by multiplying both the numerator and the denominator by the same number. Your goal is to choose numbers to multiply by so that the denominators of the will be the same.
::比较两个分数与不同分母的另一种方式是重写一个或两个分数,使其具有相同的分母。要重写一个分数,则通过将分子和分母乘以相同数字来找到相等的分数。您的目标是选择乘以数字,使分母的分母相同。

Here is an example.
::举一个例子。

Compare $\begin{aligned} \frac{2}{3} \end{aligned}$ and $\begin{aligned} \frac{5}{7} \end{aligned}$ .
::比较2 3和5 7

First, notice that the denominators of 3 and 7 are different. You will need to find an equivalent fraction for each given fraction so that their denominators are the same.
::首先, 请注意 3 和 7 的分母是不同的。您需要为每个给定的分母找到一个等效的分母, 以便它们的分母是相同的。

Next, find a common denominator . You are looking for a number that is a multiple of both 3 and 7. The product of 3 and 7 is 21 and in this case that is the least common multiple of 3 and 7.
::接下来,请找到一个共同的分母。您正在寻找一个数字,该数字是3和7的倍数。3和7的产值是21,在此情况下,3和7的最小常见乘数是3和7。

Now, rewrite each fraction as an equivalent fraction with a denominator of 21. Remember to always multiply the numerator and denominator of the fraction by the same number.
::现在, 将每个分数重写为相等的分数, 分母为 21 。记住始终要将分数的分数和分母乘以同一数字。

$\begin{aligned} \begin{matrix} \frac{2}{3} & = & \frac{2 \times 7}{3 \times 7} & = & \frac{14}{21} \\ \frac{5}{7} & = & \frac{5 \times 3}{7 \times 3} & = & \frac{15}{21} \end{matrix} \end{aligned}$

Next, compare the rewritten fractions. Now that they have the same denominator, you can see that $\begin{aligned} \frac{14}{21} \end{aligned}$ is less than $\begin{aligned} \frac{15}{21} \end{aligned}$ , so $\begin{aligned} \frac{2}{3} \end{aligned}$ is less than $\begin{aligned} \frac{5}{7} \end{aligned}$ .
::接下来,比较重写分数。既然它们具有相同的分母, 你可以看到14 21小于15 21, 所以2 3小于5 7。

The answer is $\begin{aligned} \frac{2}{3} < \frac{5}{7} \end{aligned}$ .
::答案是2 3 < 5 7。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Margaret's dad, who is making salsa.
::早些时候,你被问及玛格丽特的爸爸,他正在做萨尔萨。

He found two recipes. One calls for $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ teaspoons of salt and the other calls for $\begin{aligned} 1 \frac{5}{8} \end{aligned}$ teaspoons of salt. He wants to use the least amount of salt possible.
::他找到了两种食谱,一种是打112个茶匙盐,另一种是打158个盐。他想用最少的盐。

First, Margaret should notice that because both numbers are between 1 and 2, she can compare the fractional parts of the mixed numbers, $\begin{aligned} \frac{1}{2} \end{aligned}$ and $\begin{aligned} \frac{5}{8} \end{aligned}$ , in order to determine which mixed number is greater. Because the denominators are different, she will need to find an equivalent fraction for each given fraction so that their denominators are the same.
::首先,玛格丽特应该注意到,因为这两个数字都介于1到2之间,所以她可以比较混合数字的分数部分,即1、2和5 8,以便确定哪些混合数字较大。由于分母不同,她需要为每个分数找到一个相等的分数,以便它们的分数相同。

Next, she can find a common denominator. She is looking for a number that is a multiple of both 2 and 8. 8 is the least common multiple of 2 and 8.
::接下来,她可以找到一个共同的分母。她正在寻找一个数字,这个数字是2和8的倍数。8是2和8中最不常见的倍数。8是2和8中最不常见的倍数。

Now, she can rewrite each fraction as an equivalent fraction with a denominator of 8. Note that because $\begin{aligned} \frac{5}{8} \end{aligned}$ already has a denominator of 8, she will not need to rewrite that fraction!
::现在,她可以将每一分数重写为相等的分数,分母为8。注意, 因为5 8 分数已经是 8 分数, 她不需要重写这个分数!

$\begin{aligned} \begin{array}{rcl} \frac{1}{2} & = & \frac{1 \times 4}{2 \times 4} = \frac{4}{8} \\ \frac{5}{8} & = & \frac{5}{8} \end{array} \end{aligned}$

Finally, Margaret can compare the rewritten fractions. Now that they have the same denominator, she can see that $\begin{aligned} \frac{4}{8} \end{aligned}$ is less than $\begin{aligned} \frac{5}{8} \end{aligned}$ . This means $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ teaspoons is less than $\begin{aligned} 1 \frac{5}{8} \end{aligned}$ teaspoons.
::最后,玛格丽特可以比较重写部分。既然它们具有相同的分母, 她可以看到4,8小于5,8。这意味着1,12个茶匙小于1,5,8个茶匙。

The answer is that Margaret’s dad should use the recipe that calls for $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ teaspoons of salt.
::答案是玛格丽特的爸爸应该使用一种配方,

Example 2
::例2

In the long jump contest, Peter jumped $\begin{aligned} 5 \frac{3}{8} \end{aligned}$ feet, Sharon jumped $\begin{aligned} 6 \frac{3}{5} \end{aligned}$ feet, and Juan jumped $\begin{aligned} 6 \frac{2}{7} \end{aligned}$ feet. Order their jump distances from greatest to least.
::在漫长的跳跃比赛中,彼得跳了5,3,8英尺,沙龙跳了6,3,5英尺,胡安跳了6,2,7英尺。命令他们从最大跳到最小的跳跃距离。

First, notice that $\begin{aligned} 5 \frac{3}{8} \end{aligned}$ is less than 6 while $\begin{aligned} 6 \frac{3}{5} \end{aligned}$ and $\begin{aligned} 6 \frac{2}{7} \end{aligned}$ are both greater than 6. That means $\begin{aligned} 5 \frac{3}{8} \end{aligned}$ is the smallest number.
::第一,通知5 3 8小于6,而6 3 5和6 2 7小于6,这意味着5 3 8最小。

Next, compare $\begin{aligned} 6 \frac{3}{5} \end{aligned}$ and $\begin{aligned} 6 \frac{2}{7} \end{aligned}$ . Because both numbers are between 6 and 7, you can compare the fractional parts of the mixed numbers, $\begin{aligned} \frac{3}{5} \end{aligned}$ and $\begin{aligned} \frac{2}{7} \end{aligned}$ , in order to determine which mixed number is greater. Because the denominators are different, you will need to find an equivalent fraction for each given fraction so that their denominators are the same.
::接下来,比较 6 3 5 和 6 6 2 7 。因为这两个数字在 6 和 7 之间, 您可以比较混合数字的分数, 3 5 和 2 7, 以便确定哪个混合数字更大。由于分数不同, 您需要为每个给定分数找到相等的分数, 以便它们的分数相同。

Now, find a common denominator. You are looking for a number that is a multiple of both 5 and 7. 35 is the least common multiple of 5 and 7.
::现在,找到一个共同的分母。你正在寻找一个数字,它是一个5和7的倍数,35是5和7中最不常见的倍数。

Next, rewrite each fraction as an equivalent fraction with a denominator of 35.
::下一步,将每一分数重写为等效分数,分母为35。

$\begin{aligned} \begin{matrix} \frac{3}{5} & = & \frac{3 \times 7}{5 \times 7} & = & \frac{21}{35} \\ \frac{2}{7} & = & \frac{2 \times 5}{7 \times 5} & = & \frac{10}{35} \end{matrix} \end{aligned}$

Finally, compare the rewritten fractions. Now that they have the same denominator, you can see that $\begin{aligned} \frac{21}{35} \end{aligned}$ is greater than $\begin{aligned} \frac{10}{35} \end{aligned}$ . This means $\begin{aligned} 6 \frac{3}{5} \end{aligned}$ is greater than $\begin{aligned} 6 \frac{2}{7} \end{aligned}$ .
::最后,比较重写分数。既然它们具有相同的分母, 你可以看到21 35大于10 35。这意味着 6 3 5 大于6 2 7 。

The answer is that the distances ordered from greatest to least are $\begin{aligned} 6 \frac{3}{5}, 6 \frac{2}{7}, 5 \frac{3}{8} \end{aligned}$ .
::答案是,从最大到最小的距离是6,3,5,5,2,7,5,3,8。

Example 3
::例3

Compare $\begin{aligned} \frac{1}{8} \end{aligned}$ and $\begin{aligned} \frac{5}{6} \end{aligned}$ .
::比较 1 8 和 5 6 。

First, try the approximation method and approximate each fraction with a fraction benchmark.
::首先,尝试近似法,并以一个小数基准来接近每一分数。

$\begin{aligned} \frac{1}{8} : \end{aligned}$ The numerator of 1 is much less than the denominator of 8, so $\begin{aligned} \frac{1}{8} \end{aligned}$ is approximately 0.
::1 8 : 1 的分子数远小于 8 的分母数, 因此18 大约是 0 。

$\begin{aligned} \frac{5}{6} : 5 \end{aligned}$ out of 6 is almost 6 out of 6 which would be a whole, so $\begin{aligned} \frac{5}{6} \end{aligned}$ is approximately 1.
::5 6 :6个中有5个是6个中几乎6个是整体,5个是大约1个。

Next, compare the two numbers. $\begin{aligned} \frac{1}{8} \end{aligned}$ is approximately 0 and $\begin{aligned} \frac{5}{6} \end{aligned}$ is approximately 1. That means $\begin{aligned} \frac{1}{8} \end{aligned}$ is definitely less than $\begin{aligned} \frac{5}{6} \end{aligned}$ .
::接下来,比较这两个数字。 18个大约是0个,5个6个大约是1个,这意味着18个绝对少于5个6个。

The answer is $\begin{aligned} \frac{1}{8} < \frac{5}{6} \end{aligned}$ .
::答案是18 < 5 6。

Example 4
::例4

Compare $\begin{aligned} \frac{4}{9} \end{aligned}$ and $\begin{aligned} \frac{7}{15} \end{aligned}$ .
::比较:4 9和7 15

First, notice each fraction is approximately $\begin{aligned} \frac{1}{2} \end{aligned}$ , so the approximation method for comparison won’t work this time. Because the denominators are different, you will need to find an equivalent fraction for each given fraction so that their denominators are the same.
::首先,通知每个分数大约是12分,因此近似比较方法这次不会起作用。由于分母不同,你需要为每个分数找到一个等值分数,以便它们的分母相同。

Next, find a common denominator. You are looking for a number that is a multiple of both 9 and 15. 45 is the least common multiple of 9 and 15, though any common multiple of 9 and 15 would work.
::接下来,找到一个共同的分母。您正在寻找一个数字,这个数字是9和15的倍数,45是9和15的最小常见的倍数,尽管任何9和15的常见倍数都会有效。

Now, rewrite each fraction as an equivalent fraction with a denominator of 45.
::现在,将每一分数重写为等效分数,分母为45。

$\begin{aligned} \begin{matrix} \frac{4}{9} & = & \frac{4 \times 5}{9 \times 5} & = & \frac{20}{45} \\ \frac{7}{15} & = & \frac{7 \times 3}{15 \times 3} & = & \frac{21}{45} \end{matrix} \end{aligned}$

Next, compare the rewritten fractions. Now that they have the same denominator, you can see that $\begin{aligned} \frac{20}{45} \end{aligned}$ is less than $\begin{aligned} \frac{21}{45} \end{aligned}$ , so $\begin{aligned} \frac{4}{9} \end{aligned}$ is less than $\begin{aligned} \frac{7}{15} \end{aligned}$ .
::接下来,比较重写分数。既然它们具有相同的分母, 你可以看到20 45小于21 45, 所以4 9小于7 15。

The answer is $\begin{aligned} \frac{4}{9} < \frac{7}{15} \end{aligned}$ .
::答案是4 9 < 7 15。

Example 5
::例5

Compare $\begin{aligned} \frac{8}{9} \end{aligned}$ and $\begin{aligned} \frac{3}{4} \end{aligned}$ .
::比较 8 9 和 3 4 。

First, notice each fraction is approximately 1, so the approximation method for comparison won’t work this time. Because the denominators are different, you will need to find an equivalent fraction for each given fraction so that their denominators are the same.
::首先,通知每个分数大约为 1 , 因此这一次比较的近似方法不会起作用。因为分母不同, 您需要为每个分数找到一个等值分数, 以便它们的分母相同。

Next, find a common denominator. You are looking for a number that is a multiple of both 9 and 4. 36 is the least common multiple of 9 and 4.
::下一步,找到一个共同的分母。您正在寻找一个数字,它是一个9和4的倍数,36是9和4的最小的常见的倍数。

Now, rewrite each fraction as an equivalent fraction with a denominator of 36.
::现在,将每一分数重写为等效分数,分母为36。

$\begin{aligned} \begin{matrix} \frac{8}{9} & = & \frac{8 \times 4}{9 \times 4} & = & \frac{32}{36} \\ \frac{3}{4} & = & \frac{3 \times 9}{4 \times 9} & = & \frac{27}{36} \end{matrix} \end{aligned}$

Next, compare the rewritten fractions. Now that they have the same denominator, you can see that $\begin{aligned} \frac{32}{36} \end{aligned}$ is greater than $\begin{aligned} \frac{27}{36} \end{aligned}$ , so $\begin{aligned} \frac{8}{9} \end{aligned}$ is greater than $\begin{aligned} \frac{3}{4} \end{aligned}$ .
::接下来,比较重写分数。既然它们具有相同的分母, 你可以看到 32 36 大于 27 36, 所以 8 9 大于 3 4 。

The answer is $\begin{aligned} \frac{8}{9} > \frac{3}{4} \end{aligned}$ .
::答案是8 9 > 3 4。

Review
::回顾

Compare each pair of fractions or mixed numbers using an inequality symbol or equals sign.
::使用不平等符号或等号比较每对分数或混合数字。

$\begin{aligned} \frac{2}{5} and \frac{3}{5} \end{aligned}$
::2、5和3 5 5

$\begin{aligned} \frac{2}{6} and \frac{1}{6} \end{aligned}$
::2、6和1 6 6

$\begin{aligned} \frac{4}{5} and \frac{3}{5} \end{aligned}$
::4 5 和 3 5 5

$\begin{aligned} \frac{2}{15} and \frac{13}{15} \end{aligned}$
::2、15和13 15

$\begin{aligned} \frac{2}{5} and \frac{3}{7} \end{aligned}$
::2、5和3 7

$\begin{aligned} \frac{1}{5} and \frac{1}{7} \end{aligned}$
::1、5和1 7

$\begin{aligned} \frac{12}{15} and \frac{13}{30} \end{aligned}$
::12、15和13 30

$\begin{aligned} \frac{4}{5} and \frac{7}{9} \end{aligned}$
::4、5和7 9

$\begin{aligned} \frac{3}{8} and \frac{4}{7} \end{aligned}$
::8、8和4 8和4 7

$\begin{aligned} \frac{1}{2} and \frac{3}{6} \end{aligned}$
::1 2和3 6

Write each set in order from least to greatest.
::每一组都按从最少到最大的顺序写下来。

$\begin{aligned} \frac{5}{6}, \frac{1}{3}, \frac{4}{9} \end{aligned}$

$\begin{aligned} \frac{6}{7}, \frac{1}{4}, \frac{2}{3} \end{aligned}$

$\begin{aligned} \frac{6}{6}, \frac{4}{5}, \frac{2}{3} \end{aligned}$

$\begin{aligned} \frac{1}{2}, \frac{3}{5}, \frac{2}{3} \end{aligned}$

$\begin{aligned} \frac{2}{7}, \frac{1}{4}, \frac{3}{6} \end{aligned}$

$\begin{aligned} \frac{1}{6}, \frac{2}{9}, \frac{2}{5} \end{aligned}$

Brantley is making an asparagus soufflé which calls for $\begin{aligned} 3 \frac{3}{7} \end{aligned}$ cups of cheese, $\begin{aligned} 3 \frac{2}{3} \end{aligned}$ cups of asparagus, and $\begin{aligned} 2 \frac{2}{5} \end{aligned}$ cups of parsley. Using approximation, order the ingredients from largest amount used to least amount used.
::Brantley正在做一个香肠酸奶,需要3,3,7杯奶酪、3,2,3杯香肠和2,2,5杯柑橘。使用近似值,将用过的成分从最大量订购到用量最小。

Geraldine is putting in a pool table in her living room. She wants to put it against the longest wall of the room. Wall A is $\begin{aligned} 12 \frac{4}{9} \end{aligned}$ feet and wall B is $\begin{aligned} 12 \frac{2}{5} \end{aligned}$ feet. Against which wall will Geraldine put her pool table?
::她想把它放在房间最长的墙壁上。墙A是12,4,9英尺,墙B是12,2,5英尺。墙B是12,2,5英尺。

Review (Answers)
::回顾(答复)

Section outline

Comparing Fractions and Mixed Numbers ::比较分数和混合数字

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)

Comparing Fractions and Mixed Numbers
::比较分数和混合数字

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)