节: 分数的加法 | 3.4分数的加法

章节大纲

Luke is making cookies to share with his teachers for the holidays. He has decided to make three different types of cookies. For the first type of cookie he needs $\begin{aligned} \frac{3}{4} \end{aligned}$ cups of sugar. For the second type of cookie he needs $\begin{aligned} 1 \frac{1}{3} \end{aligned}$ cups of sugar. For the third type of cookie he needs $\begin{aligned} 2 \frac{1}{2} \end{aligned}$ cups of sugar. His mom just told him that they only have about 5 cups of sugar in the house. How can Luke figure out if he will have enough sugar to make all of his cookies?
::Luke正在做饼干,以便和老师们分享假期。他决定做三种不同的饼干。对于第一种饼干,他需要34杯糖。对于第二种饼干,他需要113杯糖。对于第三种饼干,他需要212杯糖。他的妈妈刚刚告诉他,家里只有5杯糖。Luke怎么弄清楚他是否有足够的糖做他的饼干?

In this concept, you will learn how to add fractions and mixed numbers.
::在这个概念中,你会学会如何增加分数和混合数字。

Adding Fractions and Mixed Numbers
::添加分数和混合数字

A fraction describes a part of a whole number. The number written below the bar in a fraction is the denominator , and it tells you how many parts the whole is divided into. The number above the bar in a fraction is the numerator , and it tells you how many parts of the whole you have. are fractions that describe the same part of a whole.
::一个分数描述整个数字的一部分。分数中写在条下的一个分数是分母, 它会告诉您整个的分母。条上的一个分数是分子, 它会告诉你整体的多少部分。分数是描述整个的同一个部分的分数。

To add fractions:
::添加分数 :

Rewrite each fraction as an equivalent fraction so that the denominators are the same.
::将每一分数重写为相等的分数,以便分母相同。

Add the numerators of the equivalent fractions. Keep the denominator the same.
::添加等效分数的分子。保持分母不变。

Simplify your answer.
::简化您的答案。

Here is an example.
::举一个例子。

Find the sum of $\begin{aligned} \frac{1}{2} \end{aligned}$ and $\begin{aligned} \frac{1}{3} \end{aligned}$ .
::查找12和13之和。

First, find a common denominator . You are looking for a number that is a multiple of both 2 and 3. The product of 2 and 3 is 6 and in this case that is the least common multiple of 2 and 3.
::首先,找到一个共同的分母。您正在寻找一个数字,该数字是2和3的倍数。2和3的产值是6,在此情况下,是2和3的最小公倍数。

Now, rewrite each fraction as an equivalent fraction with a denominator of 6. Remember to always multiply the numerator and denominator of the fraction by the same number.
::现在, 将每个分数重写为等值分数, 分母为 6 。记住始终要将分数的分子和分母乘以相同数字。

$\begin{aligned} \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \\ \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6} \end{aligned}$

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same.
::下一步,添加等效分数。您将添加分子,并将分母保持原样。

$\begin{aligned} \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \end{aligned}$

Finally, check to make sure your answer is in simplest form . $\begin{aligned} \frac{5}{6} \end{aligned}$ is in simplest form because 5 and 6 do not have any common factors besides 1.
::最后,检查以确保您的答复最简单。 5 6 最简单, 因为 5 和 6 除了 1 以外没有任何共同因素。

The answer is $\begin{aligned} \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \end{aligned}$ .
::答案是 1 2+1 3 = 5 6 。

Numbers that consist of a whole number and a fraction are called mixed numbers .
::由整数和分数组成的数字称为混合数。

To add mixed numbers:
::增加混合数字:

Add the fractional parts of the mixed numbers.
::添加混合数字的分数部分。

Add the whole parts of the mixed numbers.
::加上混合数字的整个部分。

Add the sum of the fractional parts to the sum of the whole numbers .
::将分部分的总和加到整个数字的总和中。

Simplify your answer.
::简化您的答案。

Here is an example.
::举一个例子。

Add $\begin{aligned} \frac{3}{4} + 2 \frac{1}{3} \end{aligned}$ .
::加3 4 + 2 1 3 。

First, add the fractional parts: $\begin{aligned} \frac{3}{4} + \frac{1}{3} \end{aligned}$ . Find a common denominator. You are looking for a number that is a multiple of both 4 and 3. 12 is the least common multiple of 4 and 3.
::首先,添加分部分: 3, 4+1, 3。找到一个共同的分母。您正在寻找一个数字, 该数字是 4 和 3. 的倍数, 4 和 3. 12 是 4 和 3 中最不常见的 4 和 3 的倍数。

Now, rewrite each fraction as an equivalent fraction with a denominator of 12.
::现在,将每个分数重写为相等的分数,分母为12。

$\begin{aligned} \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \\ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \end{aligned}$

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same. Rewrite your result as a mixed number .
::下一步,添加相等的分数。您将添加数组,并将分母保持不变。将结果重写为混合数。

$\begin{aligned} \frac{9}{12} + \frac{4}{12} = \frac{13}{12} = 1 \frac{1}{12} \end{aligned}$

Now, add the whole part of the original mixed number to your sum so far.
::现在,把原来混合数字的整部分加入到目前为止的总数中。

$\begin{aligned} 2 + 1 \frac{1}{12} = 3 \frac{1}{12} \end{aligned}$

Finally, check to make sure your answer is in simplest form. $\begin{aligned} 3 \frac{1}{12} \end{aligned}$ is in simplest form because 1 and 12 do not have any common factors besides 1.
::最后,检查以确保你的答复最简单。 3, 1, 12, 最简单, 因为1, 12 除了1, 没有任何共同因素。

The answer is $\begin{aligned} \frac{3}{4} + 2 \frac{1}{3} = 3 \frac{1}{12} \end{aligned}$ .
::答案是3 4 + 2 1 3 = 3 1 12。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Luke and his cookies.
::早些时候,你被问及卢克和他的饼干的问题。

He is making three different types of cookies. For the first type of cookie he needs $\begin{aligned} \frac{3}{4} \end{aligned}$ cups of sugar, for the second type of cookie he needs $\begin{aligned} 1 \frac{1}{3} \end{aligned}$ cups of sugar, and for the third type he needs $\begin{aligned} 2 \frac{1}{2} \end{aligned}$ cups of sugar. His mom told him that they only have about 5 cups of sugar in the house. Luke needs to make sure he has enough sugar to make all of his cookies.
::他正在做三种不同的曲奇。对于第一种曲奇,他需要34杯糖,第二种曲奇需要13杯糖,第三种曲奇需要2,12杯糖。他妈妈告诉他,家里只有5杯糖。卢克需要确保他有足够的糖做他所有的曲奇。

To figure this out, Luke will need to add the three amounts of sugar and see if he needs less than 5 cups.
::卢克需要加三批糖看看他是否需要不到五杯

First, add the fractional parts of the mixed numbers. Find a common denominator. You are looking for a number that is a multiple of 4, 3, and 2. 12 is the least common multiple of 4, 3, and 2.
::首先,添加混合数字的分数部分。找到一个共同的分母。您正在寻找一个数字, 它的乘数是 4 、 3 和 2. 2 。 12 是 4 、 3 和 2 中最不常见的乘数是 4 、 3 和 2 。

Now, rewrite each fraction as an equivalent fraction with a denominator of 12.
::现在,将每个分数重写为相等的分数,分母为12。

$\begin{aligned} \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \\ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \\ \frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} \end{aligned}$

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same. Rewrite your result as a mixed number.
::下一步,添加相等的分数。您将添加数组,并将分母保持不变。将结果重写为混合数。

$\begin{aligned} \frac{9}{12} + \frac{4}{12} + \frac{6}{12} = \frac{19}{12} = 1 \frac{7}{12} \end{aligned}$

Now, add the whole parts of the original mixed numbers.
::现在,加上原来的混合数字的整部分。

$\begin{aligned} 1 + 2 = 3 \end{aligned}$

Next, add the sum of the fractional parts to the sum of the whole numbers.
::下一步,在整数的总和中加入分部分的和。

$\begin{aligned} 3 + 1 \frac{7}{12} = 4 \frac{7}{12} \end{aligned}$

Luke needs $\begin{aligned} 4 \frac{7}{12} \end{aligned}$ cups of sugar.
::卢克需要4 7 12杯糖

The answer is that, because Luke needs $\begin{aligned} 4 \frac{7}{12} \end{aligned}$ cups of sugar and he has 5 cups of sugar, he will have enough sugar to make all three types of cookies.
::答案是,因为卢克需要4 7 12杯糖, 他有5杯糖, 他有足够的糖来做三种饼干。

Example 2
::例2

Donte is making a costume with blue, red, and black fabric. He has $\begin{aligned} 6 \frac{1}{2} \end{aligned}$ yards of blue fabric, $\begin{aligned} 3 \frac{2}{3} \end{aligned}$ yards of red fabric, and $\begin{aligned} 5 \frac{4}{5} \end{aligned}$ yards of black fabric. How many yards of fabric does Donte have all together?
::唐特正在做一件蓝色、红色和黑色布料的服装。他有612码的蓝色布料,323码的红色布料,545码的黑色布料。唐特一共有多少码的布料?

In order to solve this problem, you need to add the three given mixed numbers.
::为了解决这个问题,您需要添加三个给定的混合数字。

First, add the fractional parts of the mixed numbers. Find a common denominator. You are looking for a number that is a multiple of 2, 3, and 5. 30 is the least common multiple of 2, 3, and 5.
::首先,添加混合数字的分数部分。找到一个共同的分母。您正在寻找一个数字, 数字的乘数是 2, 3, 5, 5, 30 是 2, 3, 5 中最不常见的乘数是 2, 3, 5 。

Now, rewrite each fraction as an equivalent fraction with a denominator of 30.
::现在,将每一分数重写为30分母的等同分数。

$\begin{aligned} \frac{1}{2} = \frac{1 \times 15}{2 \times 15} = \frac{15}{30} \\ \frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \\ \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30} \end{aligned}$

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same. Rewrite your result as a mixed number.
::下一步,添加相等的分数。您将添加数组,并将分母保持不变。将结果重写为混合数。

$\begin{aligned} \frac{15}{30} + \frac{20}{30} + \frac{24}{30} = \frac{59}{30} = 1 \frac{29}{30} \end{aligned}$

Now, add the whole parts of the original mixed numbers.
::现在,加上原来的混合数字的整部分。

$\begin{aligned} 6 + 3 + 5 = 14 \end{aligned}$

Next, add the sum of the fractional parts to the sum of the whole numbers.
::下一步,在整数的总和中加入分部分的和。

$\begin{aligned} 14 + 1 \frac{29}{30} = 15 \frac{29}{30} \end{aligned}$

Finally, check to make sure your answer is in simplest form. $\begin{aligned} 15 \frac{29}{30} \end{aligned}$ is in simplest form because 29 and 30 do not have any common factors besides 1.
::最后,检查以确保您的答复最简单。 15 29 30 最简单, 因为 29 和 30 除了 1 以外没有任何共同因素。

The answer is that Donte has $\begin{aligned} 15 \frac{29}{30} \end{aligned}$ yards of fabric all together.
::答案是唐特有152930码的布料

Example 3
::例3

Add $\begin{aligned} 9 \frac{1}{2} + 22 \frac{1}{4} \end{aligned}$ .
::加9 1 2 + 22 1 4 。

First, add the fractional parts of the mixed numbers. Find a common denominator. You are looking for a number that is a multiple of both 2 and 4. 4 is the least common multiple of 2 and 4.
::首先,添加混合数字的分部分。找到一个共同的分母。您正在寻找一个数字, 数字是2和4的倍数, 4是2和4中最不常见的倍数。

Now, rewrite each fraction as an equivalent fraction with a denominator of 4.
::现在,将每一分数重写为等效分数,分母为4。

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

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same.
::下一步,添加等效分数。您将添加分子,并将分母保持原样。

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

Now, add the whole parts of the original mixed numbers.
::现在,加上原来的混合数字的整部分。

$\begin{aligned} 9 + 22 = 31 \end{aligned}$

Next, add the sum of the fractional parts to the sum of the whole numbers.
::下一步,在整数的总和中加入分部分的和。

$\begin{aligned} 31 + \frac{3}{4} = 31 \frac{3}{4} \end{aligned}$

Finally, check to make sure your answer is in simplest form. $\begin{aligned} 31 \frac{3}{4} \end{aligned}$ is in simplest form because 3 and 4 do not have any common factors besides 1.
::最后,检查以确保您的答复最简单。 31 3 4 最简单, 因为 3 和 4 除了1 以外没有任何共同因素。

The answer is $\begin{aligned} 9 \frac{1}{2} + 22 \frac{1}{4} = 31 \frac{3}{4} \end{aligned}$ .
::答案是 9 1 2 + 22 1 4 = 31 3 4 。

Example 4
::例4

Add $\begin{aligned} 2 \frac{1}{3} + 8 \frac{2}{3} \end{aligned}$ .
::加2 1 3 3 + 8 2 3 。

First, add the fractional parts of the mixed numbers. You will add the numerators and keep the denominator the same. Rewrite your result as a whole number.
::首先,添加混合数字的分数部分。您将添加分子,并保持分母不变。将结果重写为整数。

$\begin{aligned} \frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1 \end{aligned}$

Now, add the whole parts of the original mixed numbers.
::现在,加上原来的混合数字的整部分。

$\begin{aligned} 2 + 8 = 10 \end{aligned}$

Next, add the sum of the fractional parts to the sum of the whole numbers.
::下一步,在整数的总和中加入分部分的和。

$\begin{aligned} 10 + 1 = 11 \end{aligned}$

The answer is $\begin{aligned} 2 \frac{1}{3} + 8 \frac{2}{3} = 11 \end{aligned}$ .
::答案是 2 1 3 + 8 2 3 = 11 。

Example 5
::例5

Add $\begin{aligned} 5 \frac{1}{3} + \frac{1}{7} \end{aligned}$ .
::加5 1 3 + 1 7 。

First, add the fractional parts of the mixed numbers. Find a common denominator. You are looking for a number that is a multiple of both 3 and 7. 21 is the least common multiple of 3 and 7.
::首先,添加混合数字的分部分。找到一个共同的分母。您正在寻找一个3和7的倍数, 21是3和7中最不常见的3和7的倍数。

Now, rewrite each fraction as an equivalent fraction with a denominator of 21.
::现在,将每一分数重写为等效分数,分母为21。

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

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same.
::下一步,添加等效分数。您将添加分子,并将分母保持原样。

$\begin{aligned} \frac{7}{21} + \frac{3}{21} = \frac{10}{21} \end{aligned}$

Now, add the sum of the fractional parts to the original whole number.
::现在,在原整数中加入分部分的总和。

$\begin{aligned} 5 + \frac{10}{21} = 5 \frac{10}{21} \end{aligned}$

Finally, check to make sure your answer is in simplest form. $\begin{aligned} 5 \frac{10}{21} \end{aligned}$ is in simplest form because 10 and 21 do not have any common factors besides 1.
::最后,检查以确保你的答复最简单。 5 10 21 最简单, 因为 10 和 21 除了1 以外没有任何共同因素。

The answer is $\begin{aligned} 5 \frac{1}{3} + \frac{1}{7} = 5 \frac{10}{21} \end{aligned}$ .
::答案是5 1 3+1 7 = 5 10 21。

Review
::回顾

Add the following fractions and mixed numbers. Simplify your answers.
::添加以下分数和混合数字。简化您的答案。

$\begin{aligned} \frac{3}{7} + \frac{1}{14} \end{aligned}$

$\begin{aligned} \frac{3}{4} + \frac{1}{3} \end{aligned}$

$\begin{aligned} \frac{2}{5} + \frac{3}{10} \end{aligned}$

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

$\begin{aligned} 2 \frac{3}{5} + \frac{17}{20} \end{aligned}$

$\begin{aligned} 8 \frac{5}{12} + 2 \frac{1}{6} \end{aligned}$

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

$\begin{aligned} 2 \frac{1}{5} + 4 \frac{14}{15} \end{aligned}$

$\begin{aligned} 2 \frac{1}{5} + 4 \frac{2}{15} \end{aligned}$

$\begin{aligned} 12 \frac{1}{5} + 4 \frac{6}{15} \end{aligned}$

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

$\begin{aligned} 3 \frac{4}{10} + 6 \frac{1}{9} \end{aligned}$

$\begin{aligned} 2 \frac{1}{30} + \frac{12}{15} \end{aligned}$

$\begin{aligned} 4 \frac{1}{2} + 14 \frac{4}{5} \end{aligned}$

$\begin{aligned} 16 \frac{8}{9} + 14 \frac{9}{12} \end{aligned}$

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。

Resources
::资源

章节大纲

Adding Fractions and Mixed Numbers ::添加分数和混合数字

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)

Resources ::资源

Adding Fractions and Mixed Numbers
::添加分数和混合数字

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)

Resources
::资源