课程： 5.11 使用最小公分母进行分数比较

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选择节使用最小公分母进行分数比较

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使用最小公分母进行分数比较
The 6 ^th grade class is having an ice cream social. Terrence and Emilia estimated that one - third of the class will want to eat vanilla ice cream and four - sevenths of the class will want to eat chocolate ice cream. If these two estimates are accurate, which flavor of ice cream will be the most popular?
::特伦斯和艾蜜莉亚估计该班三分之一的人会想吃香草冰淇淋,七分之四的人会想吃巧克力冰淇淋。如果这两个估计是准确的,那么哪一种冰淇淋最流行?

In this concept, you will learn how to compare fractions using the lowest common denominator .
::在此概念中,您将学会如何使用最低公分母比较分数。

Comparing Fractions Using the Lowest Common Denominator
::使用最低常见分数比较分数

Some fractions have different denominators , the bottom number of a fraction. The numerator refers to the top number of a fraction.
::有些分数有不同的分母, 一个分数的底数。分子指一个分数的顶数。

Here are two fractions with different denominators.
::这里有两个分数,分母不同。

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

Remember that the denominator is the number of parts the whole has been divided into. In the first fraction, one-fourth, the whole has been divided into four parts. The second fraction, two-thirds, has been divided into three parts. In this example, you cannot compare the numerators because the parts of each fraction have different values.
::记住分母是整块被分割的数。在第一个分数, 四分之一, 整块被分为四个部分。第二个分数, 三分之二, 被分为三个部分。在此示例中, 您无法比较数字, 因为每个分数的部位有不同的值。

You use greater than (>), less than (<), or equal to (=) to compare two fractions. It is easy to compare fractions with the same denominator.
::您使用大于( > ) 、小于( < ) 或等于( =) 来比较两个分数。将分数与相同分数比较很容易。

Compare these two fractions.
::比较这两个分数。

$\begin{aligned} \frac{1}{5} \underline{} \frac{3}{5} \end{aligned}$

Both fractions represent a whole that is divided into 5 parts. If the fractions were pizzas that were divided into 5 parts, one-fifth of a pizza would be less than with three-fifths of the same pizza. Therefore, you can compare those fractions like this.
::两个分数代表一个分为五个部分的整块。如果分数是被分为五个部分的比萨饼,那么比萨饼的五分之一将少于同一个比萨饼的五分之三。因此,你可以比较这些分数。

$\begin{aligned} \frac{1}{5} < \frac{3}{5} \end{aligned}$

To compare fractions different denominators, rewrite the fractions so they have a common denominator .
::为了比较分数不同的分母, 请重写分数, 以便它们有一个共同的分母。

Let’s compare the two fractions from earlier.
::让我们比较之前的两部分。

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

Rewrite the denominators by finding the least common multiple of each denominator. Remember that the least common multiple (LCM) is the smallest multiple that two numbers have in common. This LCM becomes the lowest common denominator (LCD).
::通过找到每个分母中最不常见的多个来重写分母。记住, 最小的多分母( LCM) 是两个数字共有的最小的多个。这个 LCM 成为最小的公分母( LCD ) 。

First, list the multiples for 3 and 4 and find the LCM.
::首先,列出3号和4号的倍数并找到LCM

4, 8, 12, 16

3, 6, 9, 12

The LCM for 3 and 4 is 12.
::3和4的LCM为12。

Then, rewrite each fraction in terms of twelfths. Make a fraction equivalent to one-fourth in terms of twelfths, and make a fraction equivalent to two-thirds in terms of twelfths.
::然后以十二分之十二重写每个分数。以十二分之十四重写一个分数, 以十二分之十四重写一个分数, 以十二分之三分之二重写一个分数。

$\begin{aligned} \frac{1}{4} = \frac{}{12} \end{aligned}$

To make , multiply or divide the numerator and the denominator by the same number to create the equal fraction. 4 is multiplied by 3 to get 12. Complete the equivalent fraction by also multiplying the numerator by 3.
::要将分子和分母乘以相同数字来创建相等分数,则要将分子和分母乘以或除以相同数字来创建相等分数。 4 乘以 3 以获得 12 。通过将分子和分母乘以 3 来完成等同分数。

$\begin{aligned} \frac{1}{4} = \frac{3}{12} \end{aligned}$

Now work on rewriting two-thirds in terms of twelfths. 3 is multiplied by 4 to get 12. Multiply the numerator by 4.
::3乘以4 乘以12 乘以4 乘以乘以4 乘以分子乘以4

$\begin{aligned} \frac{2}{3} = \frac{8}{12} \end{aligned}$

Next, compare the fractions now that both fractions have been written in terms of twelfths.
::接下来,比较分数,因为这两个分数都是以十二分写成的。

$\begin{aligned} \frac{3}{12} < \frac{8}{12} \end{aligned}$

so
::so so, so, so, so, so, so, so, so, so,

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

$\begin{aligned} \frac{1}{4} \end{aligned}$ is less than $\begin{aligned} \frac{2}{3} \end{aligned}$ .
::1 4小于2 3。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the ice cream social.
::之前,你曾遇到冰淇淋社交问题

Terrence and Emilia estimated that one - third of the class will want to eat vanilla ice cream and four - sevenths of the class will want to eat chocolate ice cream. Compare the fractions to see which flavor will be more popular.
::Terrence和Emilia估计,三分之一的班级将想吃香草冰淇淋,七分之四的班级将想吃巧克力冰淇淋。比较分数,看看哪一种口味更受欢迎。

$\begin{aligned} \frac{1}{3} \underline{} \frac{4}{7} \end{aligned}$

First, find the LCM of 3 and 7. The lowest common denominator will be 21.
::首先,找到3和7的LCM, 最低共同标准为21。

Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of $\begin{aligned} \frac{1}{3} \end{aligned}$ by 7. Multiply the numerator and denominator of $\begin{aligned} \frac{4}{7} \end{aligned}$ by 3.
::然后用最小的公分母重写每一分数。乘以 13 乘以 13 乘以 7 乘以 4 乘以 7 乘以 4 乘以 7 乘以 3 。

$\begin{aligned} \frac{1}{3} = \frac{7}{21} \end{aligned}$

$\begin{aligned} \frac{4}{7} = \frac{12}{21} \end{aligned}$

Next, compare the equivalent fractions.
::接下来,比较相等的分数。

$\begin{aligned} \frac{7}{21} < \frac{12}{21} \end{aligned}$

Chocolate will be more popular.
::巧克力会更受欢迎

Example 2
::例2

Rewrite each fraction with the lowest common denominator and compare using <, >, or =.
::以最低公分母重写每个分数,并使用 < 、 > 或 = 比较。

$\begin{aligned} \frac{6}{9} \underline{} \frac{3}{4} \end{aligned}$

First, find the LCM of 9 and 4. The lowest common denominator will be 36.
::首先,找到9和4的LCM。最低共同标准为36。

9 = 9, 18, 27, 36

4 = 4, 12, 16, 20, 24, 28, 32, 36

Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of $\begin{aligned} \frac{6}{9} \end{aligned}$ by 4. Multiply the numerator and denominator of $\begin{aligned} \frac{3}{4} \end{aligned}$ by 9.
::然后用最小的公分母重写每一分数。乘以 6 9 乘以 4 乘以 3 乘以 4 乘以 3 乘以 4 。

$\begin{aligned} \frac{6}{9} = \frac{24}{36} \end{aligned}$

$\begin{aligned} \frac{3}{4} = \frac{27}{36} \end{aligned}$

Next, compare the equivalent fractions.
::接下来,比较相等的分数。

$\begin{aligned} \frac{24}{36} < \frac{27}{36} \end{aligned}$

$\begin{aligned} \frac{6}{9} \end{aligned}$ is less than $\begin{aligned} \frac{3}{4} \end{aligned}$ .
::6 9 小于3 4。

Example 3
::例3

Compare the fractions.
::比较分数。

$\begin{aligned} \frac{2}{5} \underline{} \frac{6}{10} \end{aligned}$

First, find the LCM of 5 and 10. The lowest common denominator will be 10.
::首先,找到5和10的LCM, 最低共同标准为10。

Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of $\begin{aligned} \frac{2}{5} \end{aligned}$ by 2. $\begin{aligned} \frac{6}{10} \end{aligned}$ is already a fraction of tenths.
::然后以最小的公分母重写每个分数。乘以 2 5 乘以 2 乘以 2 乘以 2 乘以 6 10 的分子和分数已经是十分之一。

$\begin{aligned} \frac{2}{5} = \frac{4}{10} \end{aligned}$

Next, compare the equivalent fractions.
::接下来,比较相等的分数。

$\begin{aligned} \frac{4}{10} < \frac{6}{10} \end{aligned}$

$\begin{aligned} \frac{2}{5} \end{aligned}$ is less than $\begin{aligned} \frac{6}{10} \end{aligned}$ .
::2 5小于6 10。

Example 4
::例4

Compare the fractions.
::比较分数。

$\begin{aligned} \frac{2}{3} \underline{} \frac{1}{9} \end{aligned}$

First, find the LCM of 3 and 9. The lowest common denominator will be 9.
::首先,找到3和9的LCM,最低共同标准为9。

Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of $\begin{aligned} \frac{2}{3} \end{aligned}$ by 3. $\begin{aligned} \frac{1}{9} \end{aligned}$ does not change.
::然后, 重写每个分数的最小公分母。乘以 2 3 乘以 3. 1 9 的分子和分母不会改变。

$\begin{aligned} \frac{2}{3} = \frac{6}{9} \end{aligned}$

Next, compare the equivalent fractions.
::接下来,比较相等的分数。

$\begin{aligned} \frac{6}{9} > \frac{1}{9} \end{aligned}$

$\begin{aligned} \frac{2}{3} \end{aligned}$ is greater than $\begin{aligned} \frac{1}{9} \end{aligned}$ .
::2 3 大于 19 。

Example 5
::例5

Compare the fractions.
::比较分数。

$\begin{aligned} \frac{3}{4} \underline{} \frac{6}{8} \end{aligned}$

First, find the LCM of 4 and 8. The lowest common denominator will be 8.
::首先,找到4和8的LCM,最低共同标准为8。

Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of $\begin{aligned} \frac{3}{4} \end{aligned}$ by 2. $\begin{aligned} \frac{6}{8} \end{aligned}$ does not change.
::然后, 重写每个分数的最小公分母。乘以 3 4 乘以 2. 6 8 的分子和分母不会改变。

$\begin{aligned} \frac{3}{4} = \frac{6}{8} \end{aligned}$

Next, compare the equivalent fractions.
::接下来,比较相等的分数。

$\begin{aligned} \frac{6}{8} = \frac{6}{8} \end{aligned}$

$\begin{aligned} \frac{3}{4} \end{aligned}$ is equal to $\begin{aligned} \frac{6}{8} \end{aligned}$ .
::3等于6 8。

Review
::回顾

Rename each in terms of tenths.
::以十分数重命名。

1. $\begin{aligned} \frac{1}{5} \end{aligned}$

2. $\begin{aligned} \frac{3}{5} \end{aligned}$

3. $\begin{aligned} \frac{1}{2} \end{aligned}$

4. $\begin{aligned} \frac{4}{5} \end{aligned}$

Complete each equal fraction.
::完成每个相等的分数。

5. $\begin{aligned} \frac{1}{3} = \frac{}{9} \end{aligned}$

6. $\begin{aligned} \frac{2}{3} = \frac{}{18} \end{aligned}$

7. $\begin{aligned} \frac{5}{6} = \frac{}{18} \end{aligned}$

8. $\begin{aligned} \frac{2}{7} = \frac{}{14} \end{aligned}$

9. $\begin{aligned} \frac{4}{9} = \frac{}{36} \end{aligned}$

10. $\begin{aligned} \frac{3}{4} = \frac{}{48} \end{aligned}$

Identify the lowest common multiple for each pair of numbers.
::确定每对数字的最低公数倍数。

11. 3 and 6
::11. 3和6

12. 4 and 10
::12. 4和10

13. 5 and 3
::13. 5和3

14. 7 and 2
::14. 7和2

15. 8 and 4
::15. 8和4

16. 6 and 4
::16. 6和4

17. 8 and 5
::17. 8和5

18. 12 and 5
::18. 12和5

19. 9 and 2
::19. 9和2

20. 6 and 7
::20. 6和7

Compare the following fractions using <, >, or =
::比较以下使用 <, > 或 = 的分数

21. $\begin{aligned} \frac{1}{2} \underline{} \frac{1}{3} \end{aligned}$

22. $\begin{aligned} \frac{2}{3} \underline{} \frac{3}{9} \end{aligned}$

23. $\begin{aligned} \frac{4}{6} \underline{} \frac{2}{3} \end{aligned}$

24. $\begin{aligned} \frac{6}{10} \underline{} \frac{4}{5} \end{aligned}$

25. $\begin{aligned} \frac{9}{18} \underline{} \frac{3}{6} \end{aligned}$

Review (Answers)
::回顾(答复)

章节大纲

常规

使用最小公分母进行分数比较

Comparing Fractions Using the Lowest Common Denominator ::使用最低常见分数比较分数

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 Using the Lowest Common Denominator
::使用最低常见分数比较分数

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)