Course: 3.6分数加法的结合律和交换律

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分数加法的结合律和交换律
Lisa is making bread and weighing the ingredients as she goes. She starts with $\begin{aligned} 16 \frac{3}{4} \end{aligned}$ ounces of flour. Then, she adds water. Finally, she adds $\begin{aligned} \frac{1}{4} \end{aligned}$ ounce of yeast. All together her dough weighs 27 ounces. She forgot to look at the weight of the water after she added it. How can Lisa figure out how much water she added to her dough?
::Lisa正在做面包,并按自己的方式权衡食物成分。她从16 3 4盎司面粉开始, 然后加水。最后, 她加了14盎司的酵母。她的面粉加了14盎司的酵母。她的面粉加了27盎司, 她忘了看水的重量。她加了水之后, 她忘了看水的重量。 Lisa怎么知道她加了多少水呢?

In this concept, you will learn to use the commutative and associative properties of addition with fractions.
::在此概念中,您将学会使用与分数相加的通和和连带特性。

Using Commutative and Associative Properties of Addition with Fractions
::使用带有分数的混合和共同增加属性

Recall that the commutative property of addition states that when finding a sum, changing the order of the addends will not change their sum. In symbols, the commutative property of addition says that for numbers $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ :
::提醒注意,添加的通量财产规定,在寻找一笔金额时,改变添加的顺序不会改变其金额。在符号中,添加的通量财产表示,对于a和b数字,a和b:

$\begin{aligned} a + b = b + a \end{aligned}$

The associative property of addition states that when finding a sum, changing the way addends are grouped will not change their sum. In symbols, the associative property of addition says that for numbers $\begin{aligned} a \end{aligned}$ , $\begin{aligned} b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ :
::附加的连带财产规定,在找到一个总和时,改变添加方的组合方式不会改变其总和。在符号中,加的连带财产表示,对于数字a、b和c,a、b和c:

$\begin{aligned} (a + b) + c = a + (b + c) \end{aligned}$

Both the commutative property of addition and the associative property of addition can be useful in simplifying expressions involving fractions and mixed numbers. The commutative property of addition allows you to reorder terms while the associative property of addition allows you to regroup terms.
::增加的通货财产和增加的连带财产都可用于简化涉及分数和混合数字的表达方式。增加的通货财产允许您重新排序条款,而增加的连带财产允许您重新组合条款。

Here is an example.
::举一个例子。

Simplify the following expression .
::简化以下表达式。

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

First, use the commutative property of addition to reorder the terms.
::首先,使用添加的通货财产重新排列术语的顺序。

$\begin{aligned} 3 \frac{2}{3} + x + \frac{1}{3} \end{aligned}$ is equivalent to $\begin{aligned} 3 \frac{2}{3} + \frac{1}{3} + x \end{aligned}$ .
::3 2 3 3 + x + 1 3 等于 3 2 3 + 1 3 + 3 + x 。

Next, simplify by combining $\begin{aligned} 3 \frac{2}{3} \end{aligned}$ and $\begin{aligned} \frac{1}{3} \end{aligned}$ . Use what you have learned about adding fractions and mixed numbers.
::接下来,将三、二、三、三、三、三、三、三合并,简化。使用你学到的关于添加分数和混合数字的知识。

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

$\begin{aligned} 3 \frac{2}{3} + \frac{1}{3} + x \end{aligned}$ simplifies to $\begin{aligned} 4 + x \end{aligned}$ .
::3 2 3 + 1 3 + x 简化为 4 + x 。

The answer is $\begin{aligned} 3 \frac{2}{3} + x + \frac{1}{3} \end{aligned}$ simplifies to $\begin{aligned} 4 + x \end{aligned}$ .
::答案是 3 2 3 + x + 1 3 简化为 4 + x 。

Here is another example.
::下面是另一个例子。

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} \frac{3}{10} + (\frac{1}{4} + x) \end{aligned}$
::3 10+(1 4+xx)

First, notice that this expression has " data-term="Parentheses" role="term" tabindex="0"> parentheses . That's a clue that you can use the associative property to rewrite it. Use the associative property to regroup the terms.
::首先,请注意此表达式有括号。这是一个线索, 您可以使用关联属性来重写它。使用关联属性来重新组合术语。

$\begin{aligned} \frac{3}{10} + (\frac{1}{4} + x) \end{aligned}$ is equivalent to $\begin{aligned} (\frac{3}{10} + \frac{1}{4}) + x \end{aligned}$ .
::3 10+(1 4+x)等于(3 10+1 4)+x。

Next, simplify by combining $\begin{aligned} \frac{3}{10} \end{aligned}$ and $\begin{aligned} \frac{1}{4} \end{aligned}$ . Use what you have learned about adding fractions. You will need to find a common denominator and rewrite each fraction . Then, add the equivalent fractions.
::接下来,通过合并 3 10 和 14 来简化。使用您学到的添加分数的知识。您需要找到一个共同的分母, 并重写每个分数。然后添加等效的分数。

$\begin{aligned} \begin{array}{rcl} \frac{3}{10} = \frac{3 \times 2}{10 \times 2} & = & \frac{6}{20} \\ \frac{1}{4} = \frac{1 \times 5}{4 \times 5} & = & \frac{5}{20} \\ \frac{6}{20} + \frac{5}{20} & = & \frac{11}{20} \end{array} \end{aligned}$

$\begin{aligned} (\frac{3}{10} + \frac{1}{4}) + x \end{aligned}$ simplifies to $\begin{aligned} \frac{11}{20} + x \end{aligned}$ .
:3 10+1 4)+x简化为11 20+x。

The answer is $\begin{aligned} \frac{3}{10} + (\frac{1}{4} + x) \end{aligned}$ simplifies to $\begin{aligned} \frac{11}{20} + x \end{aligned}$ .
::回答是3 10+(1 4+x)简化为11 20+x。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Lisa and her bread.
::早些时候,你得到一个问题关于丽莎和她的面包。

She started with $\begin{aligned} 16 \frac{3}{4} \end{aligned}$ ounces of flour, then she added water, then she added $\begin{aligned} \frac{1}{4} \end{aligned}$ ounce of yeast. All together her dough weighed 27 ounces. Lisa forgot to weigh the water she added individually. She wants to figure out how much water she added to her dough.
::她先是16,3,4盎司面粉,然后加水,然后加了14盎司酵母。她所有的面粉加了14盎司酵母。Lisa忘记了自己单独加水的重量。她想弄清楚自己加了多少水。

First, Lisa should write an equation to represent this situation. She doesn't know the weight of the water, so let the variable $\begin{aligned} x \end{aligned}$ be equal to the weight of the water. She knows the sum of the flour, water, and yeast is equal to 27.
::首先,丽莎应该写一个方程来代表这种情况。她不知道水的重量, 所以让变量x 等于水的重量。她知道面粉、水和酵母的总和是27。

$\begin{aligned} 16 \frac{3}{4} + x + \frac{1}{4} = 27 \end{aligned}$

Next, Lisa can simplify the left side of her equation. She can use the commutative property to reorder the terms.
::接下来,丽莎可以简化其方程的左侧。她可以使用通货财产重新排序条件。

$\begin{aligned} 16 \frac{3}{4} + x + \frac{1}{4} \end{aligned}$ is equivalent to $\begin{aligned} 16 \frac{3}{4} + \frac{1}{4} + x \end{aligned}$ .
::16 3 4 + x + 1 4 等于 16 3 4 + 1 4 + 4 + x 等于 16 3 4 + 1 4 + x 。

Now, Lisa can simplify by adding the mixed number and fraction.
::现在,丽莎可以通过添加混合数字和分数来简化。

$\begin{aligned} 16 \frac{3}{4} + \frac{1}{4} + x \end{aligned}$ simplifies to $\begin{aligned} 17 + x \end{aligned}$ .
::16 3 4 + 1 4 + x 简化为 17 + x 。

Next, Lisa can rewrite her equation.
::接下来,丽莎可以重写她的方程式

$\begin{aligned} 16 \frac{3}{4} + x + \frac{1}{4} = 27 \end{aligned}$ is equivalent to $\begin{aligned} 17 + x = 27 \end{aligned}$ .
::16 3 4 + x + 1 4 = 27 等于 17 + x = 27。

Finally, Lisa can solve her equation by subtracting 17 from both sides.
::最后,丽莎可以解决她的等式, 从双方减去17。

$\begin{aligned} \begin{array}{rcl} 17 + x - 17 & = & 27 - 17 \\ x & = & 10 \end{array} \end{aligned}$

The answer is Lisa added 10 ounces of water to her bread dough.
::答案是丽莎在面包上加了10盎司的水

Example 2
::例2

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} \frac{3}{7} + y + \frac{2}{7} \end{aligned}$
::3 7 + y + 2 7

First, use the commutative property of addition to reorder the terms.
::首先,使用添加的通货财产重新排列术语的顺序。

$\begin{aligned} \frac{3}{7} + y + \frac{2}{7} \end{aligned}$ is equivalent to $\begin{aligned} \frac{3}{7} + \frac{2}{7} + y \end{aligned}$ .
::3 7 + y + 2 7 等于 3 7 + 2 7 + y 。

Next, simplify by combining $\begin{aligned} \frac{3}{7} \end{aligned}$ and $\begin{aligned} \frac{2}{7} \end{aligned}$ . Use what you have learned about adding fractions.
::接下来,将3 7 和 2 7 合并简化。使用您学到的添加分数的方法。

$\begin{aligned} \frac{3}{7} + \frac{2}{7} = \frac{5}{7} \end{aligned}$

$\begin{aligned} \frac{3}{7} + \frac{2}{7} + y \end{aligned}$ simplifies to $\begin{aligned} \frac{5}{7} + y \end{aligned}$ .
::3 7 + 2 7 + y 简化为 5 7 + y 。

The answer is $\begin{aligned} \frac{3}{7} + y + \frac{2}{7} \end{aligned}$ simplifies to $\begin{aligned} \frac{5}{7} + y \end{aligned}$ .
::答案是 3 7 + y + 2 7 简化为 5 7 + y 。

Example 3
::例3

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} \frac{2}{3} + y + \frac{1}{5} \end{aligned}$
::2 3 + y + + 1 5

First, use the commutative property of addition to reorder the terms.
::首先,使用添加的通货财产重新排列术语的顺序。

$\begin{aligned} \frac{2}{3} + y + \frac{1}{5} \end{aligned}$ is equivalent to $\begin{aligned} \frac{2}{3} + \frac{1}{5} + y \end{aligned}$ .
::2 3 + y + 1 5 等于 2 3 + 1 5 + y 。

Next, simplify by combining $\begin{aligned} \frac{2}{3} \end{aligned}$ and $\begin{aligned} \frac{1}{5} \end{aligned}$ . Use what you have learned about adding fractions. You will need to find a common denominator and rewrite each fraction. Then, add the equivalent fractions.
::接下来,通过合并 2 3 和 15 来简化。使用您学到的添加分数的知识。您需要找到一个共同的分母, 并重写每个分数。然后, 添加相等的分数。

$\begin{aligned} \begin{array}{rcl} \frac{2}{3} = \frac{2 \times 5}{3 \times 5} & = & \frac{10}{15} \\ \frac{1}{5} = \frac{1 \times 3}{5 \times 3} & = & \frac{3}{15} \\ \frac{10}{15} + \frac{3}{15} & = & \frac{13}{15} \end{array} \end{aligned}$

$\begin{aligned} \frac{2}{3} + \frac{1}{5} + y \end{aligned}$ simplifies to $\begin{aligned} \frac{13}{15} + y \end{aligned}$ .
::2 3 + 1 5 + y 简化为 13 15 + y 。

The answer is $\begin{aligned} \frac{2}{3} + y + \frac{1}{5} \end{aligned}$ simplifies to $\begin{aligned} \frac{13}{15} + y \end{aligned}$ .
::答案是 2 3 + y + 1 5 简化为 13 15 + y 。

Example 4
::例4

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} \frac{1}{2} + (\frac{1}{2} + x) \end{aligned}$
::1 2 + (1 2 + x )

First, use the associative property to regroup the terms.
::首先,使用连带财产来重新组合术语。

$\begin{aligned} \frac{1}{2} + (\frac{1}{2} + x) \end{aligned}$ is equivalent to $\begin{aligned} (\frac{1}{2} + \frac{1}{2}) + x \end{aligned}$ .
::1 2+ (1、2+x) 等于 (1、2+1 2) + x 。

Next, simplify by combining $\begin{aligned} \frac{1}{2} \end{aligned}$ and $\begin{aligned} \frac{1}{2} \end{aligned}$ . Use what you have learned about adding fractions.
::下一步, 将 1 2 和 1 2 合并简化。使用您学到的添加分数的方法。

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

$\begin{aligned} (\frac{1}{2} + \frac{1}{2}) + x \end{aligned}$ simplifies to $\begin{aligned} 1 + x \end{aligned}$ .
:1、2 + 1 2 ) + x 简化为 1 + x 。

The answer is $\begin{aligned} \frac{1}{2} + (\frac{1}{2} + x) \end{aligned}$ simplifies to $\begin{aligned} 1 + x \end{aligned}$ .
::答案是 1 2 + ( 1 2 + x ) , 简化为 1 + x 。

Example 5
::例5

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} (x + \frac{4}{9}) + \frac{2}{9} \end{aligned}$
:xx+4+4 9)+2 9)

First, use the associative property to regroup the terms.

$\begin{aligned} (x + \frac{4}{9}) + \frac{2}{9} \end{aligned}$ is equivalent to $\begin{aligned} x + (\frac{4}{9} + \frac{2}{9}) \end{aligned}$ .
::首先,使用连带财产重新组合术语。 (x+4 9)+2 9等于x+(4 9+2 9)

Next, simplify by combining $\begin{aligned} \frac{4}{9} \end{aligned}$ and $\begin{aligned} \frac{2}{9} \end{aligned}$ . Use what you have learned about adding fractions. Rewrite your result in simplest form.
::下一步, 将 4 9 和 2 9 合并简化。使用您所学到的添加分数的知识。以最简单的形式重写您的结果。

$\begin{aligned} \frac{4}{9} + \frac{2}{9} = \frac{6}{9} = \frac{2}{3} \end{aligned}$

$\begin{aligned} x + (\frac{4}{9} + \frac{2}{9}) \end{aligned}$ simplifies to $\begin{aligned} x + \frac{2}{3} \end{aligned}$ .
::x + (4 9 + 2 9) 简化为 x + 2 3 。

The answer is $\begin{aligned} (x + \frac{4}{9}) + \frac{2}{9} \end{aligned}$ simplifies to $\begin{aligned} x + \frac{2}{3} \end{aligned}$ .
::答案是 (x+4 9) + 2 9 简化为 x + 2 3 。

Review
::回顾

Simplify each expression using the commutative and associative properties of addition.
::使用添加的通量和连带性来简化每个表达式。

$\begin{aligned} \frac{1}{6} + y + \frac{2}{6} \end{aligned}$
::1 6 + y + 2 6 6

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

$\begin{aligned} \frac{2}{9} + y + \frac{5}{9} \end{aligned}$
::2 9 + 9 + y + 5 9

$\begin{aligned} 2 \frac{7}{8} + x + 1 \frac{1}{8} \end{aligned}$
::2 7 8 + x + 1 1 8

$\begin{aligned} (x + 3 \frac{2}{3}) + 5 \frac{1}{6} \end{aligned}$
:xx+3+3 2 3)+5 1 6

$\begin{aligned} \frac{1}{4} + x + \frac{5}{8} \end{aligned}$
::1 4 + x + 5 8

$\begin{aligned} (\frac{1}{9} + x) + \frac{2}{9} \end{aligned}$
:1 9+xx)+2 9

$\begin{aligned} 2 \frac{1}{14} + (x + 3 \frac{5}{7}) \end{aligned}$
::2 1 14+(x+3 5 7)

$\begin{aligned} 3 \frac{1}{4} + (x + 1 \frac{2}{3}) \end{aligned}$
::3 1 4+(x+1 2 3)

$\begin{aligned} 2 \frac{1}{10} + (x + 3 \frac{1}{3}) \end{aligned}$
::2 1 10+(x+3 1 3)

$\begin{aligned} 4 \frac{1}{2} + (x + 2 \frac{1}{6}) \end{aligned}$
::4 1 2 + (x+ 2 1 6 )

$\begin{aligned} 3 \frac{1}{9} + (x + 2 \frac{2}{18}) \end{aligned}$
::3 1 9+(x+2 2 2 18)

One-third of the CDs in Joseph’s CD collection are classical music CDs. Two-sevenths of the CDs are hip-hop CDs. What fraction of Joseph’s collection is classical and hip-hop?
::约瑟夫的CD收藏中,三分之一的CD是古典音乐CD。 77%的CD是嘻哈CD。约瑟的CD收藏中,哪一部分是古典和嘻哈?

Naira is making pinecone stew. First, she mixes $\begin{aligned} 3 \frac{1}{5} \end{aligned}$ cups of chopped pinecones with $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ cups of mud. For the snail sauce on top, she uses another $\begin{aligned} 1 \frac{3}{8} \end{aligned}$ cups of chopped pinecones. How much chopped pinecone does Naira use for her recipe?
::首先,她混合了35杯松松饼和12杯泥。对于上面的蜗牛酱,她又使用了138杯松饼。Naira用多少松松饼做配方?

Jennifer is trying to determine if the cheerleading squad has enough ribbon for the pep rally on Friday. Kurt contributes $\begin{aligned} 9 \frac{1}{6} \end{aligned}$ feet of gold ribbon. Estelle contributes $\begin{aligned} \frac{3}{4} \end{aligned}$ foot of red ribbon. Aaron brings in $\begin{aligned} 5 \frac{2}{7} \end{aligned}$ feet of gold ribbon at the last minute. How much ribbon does the cheerleading squad have?
::Jennifer正试图确定啦啦队队是否有足够的丝带来参加周五的节日集会。 Kurt贡献了96英尺的金丝带。 Estelle贡献了34英尺的红丝带。Aaron最后一分钟带来了527英尺的金丝带。啦啦队有多少丝带?

Review (Answers)
::回顾(答复)

Section outline

General

分数加法的结合律和交换律

Using Commutative and Associative Properties of Addition with Fractions ::使用带有分数的混合和共同增加属性

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)

Using Commutative and Associative Properties of Addition with Fractions
::使用带有分数的混合和共同增加属性

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)