Course: 3.12分数积的结合律和交换律

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分数积的结合律和交换律
Elaine is helping her sister Connie make bracelets to sell at the craft fair. Connie tells Elaine that she will give her $\begin{aligned} \frac{1}{5} \end{aligned}$ of the money she makes at the fair for her help. Elaine decides that whatever money she gets, she will put $\begin{aligned} \frac{1}{2} \end{aligned}$ of it into her savings account at the bank. How can Elaine write an expression to represent the fraction of money that Connie earns that Elaine will be putting into the bank?
::康妮告诉艾琳,她会给她十五分钱。爱琳决定,无论她得到多少钱,她都会把十二分钱放进银行的储蓄账户。伊莲怎么写一个表情来代表康妮赚的钱的一部分,而伊莲会把钱放进银行?

In this concept, you will learn to identify and apply the commutative and associative properties of multiplication with fractions.
::在此概念中,您将学会识别和应用与分数乘法的通量和关联性。

Using Commutative and Associative Properties of Multiplication with Fractions
::使用带有分数的乘法的通和相交属性

The commutative property of multiplication states that when finding a product, changing the order of the factors will not change their product. In symbols, the commutative property of multiplication says that for numbers $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ :
::乘法的通量特性表示,在寻找产品时,改变因素的顺序不会改变产品。在符号中,乘法的通量特性表示,对于a和b数字,a和b:

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

The associative property of multiplication states that when finding a product, changing the way factors are grouped will not change their product. In symbols, the associative property of multiplication says that for numbers $\begin{aligned} a, b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ :
::乘法的连带属性表示,在寻找产品时,改变组合因素的方式不会改变产品。在符号中,乘法的连带属性表示,对于数字a、b和c,a、b和c:

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

The commutative property of multiplication can be useful when multiplying more than two fractions. It allows you to reorder the fractions in order to simplify before multiplying.
::乘法的通量属性在乘法超过两个分数时是有用的。它允许您重新排序分数,以便在乘法之前简化。

Here is an example.
::举一个例子。

Find the product of $\begin{aligned} \frac{6}{8} \cdot \frac{1}{2} \cdot \frac{16}{18} \end{aligned}$ .
::查找6 8 1 2 16 18 的产物。

First, notice that the first fraction and the third fraction have common factors. Use the commutative property to switch the second and third fractions.
::首先,请注意第一个分数和第三个分数有共同因素。使用通量属性转换第二和第三分数。

$\begin{aligned} \frac{6}{8} \cdot \frac{1}{2} \cdot \frac{16}{18} \end{aligned}$ is equivalent to $\begin{aligned} \frac{6}{8} \cdot \frac{16}{18} \cdot \frac{1}{2} \end{aligned}$ .
::6 8 1 2 16 18 等于 6 8 16 18 1 2 。

Now, focus on the first two fractions. Look for common factors along the diagonals in the numerators and the denominators. Notice that 6 and 18 both have a factor of 6. You can divide both the 6 and the 18 by 6.
::现在, 聚焦于前两个部分。寻找在分子和分母的对角线上的常见因素。注意 6 和 18 都有6 的系数。您可以将 6 和 18 除以 6 。

$\begin{aligned} \frac{6}{8} \cdot \frac{16}{18} \cdot \frac{1}{2} = \frac{1}{8} \cdot \frac{16}{3} \cdot \frac{1}{2} \end{aligned}$

Similarly, both the 16 and the 8 have a factor of 8. You can divide both the 16 and the 8 by 8.
::同样,16岁和8岁都有8岁,16岁和8岁都可以除以8。

$\begin{aligned} \frac{1}{8} \cdot \frac{16}{3} \cdot \frac{1}{2} = \frac{1}{1} \cdot \frac{2}{3} \cdot \frac{1}{2} \end{aligned}$

Now you can multiply the fractions. Use what you have learned about fraction multiplication.
::现在您可以乘以分数。使用您所学的分数乘法。

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

Finally, you can simplify your answer.
::最后,你可以简化你的答案。

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

The answer is $\begin{aligned} \frac{6}{8} \cdot \frac{1}{2} \cdot \frac{16}{18} = \frac{1}{3} \end{aligned}$ .
::答案是6 8 □ 1 2 □ 16 18 = 1 3 。

Both the commutative property of multiplication and the associative property of multiplication can be useful in simplifying expressions involving fractions. The commutative property of multiplication allows you to reorder factors while the associative property of multiplication allows you to regroup factors.
::乘法的通量属性和乘法的连带属性都可用于简化涉及分数的表达式。乘法的通量属性允许您重新排序因子,而乘法的连带属性允许您重新组合因子。

Here is an example where the commutative property is useful.
::这里的例子就是通融财产是有用的。

Simplify $\begin{aligned} \frac{2}{3} \cdot x \cdot \frac{7}{8} \end{aligned}$ .
::简化2 3 x 7 8 。

First, use the commutative property of multiplication to reorder the factors.
::首先,使用乘法的通量属性重新排序系数。

$\begin{aligned} \frac{2}{3} \cdot x \cdot \frac{7}{8} \end{aligned}$ is equivalent to $\begin{aligned} \frac{2}{3} \cdot \frac{7}{8} \cdot x \end{aligned}$ .
::2 3 × x × 7 8 等于 2 3 × 7 7 8 × 。

Next, simplify $\begin{aligned} \frac{2}{3} \cdot \frac{7}{8} \cdot x \end{aligned}$ . Multiply the fractions using what you have learned about fraction multiplication. Simplify your result.
::下一步, 简化 2 3 7 8 x. 。使用您所学的关于分数乘法的分数乘法乘以分数。简化您的结果。

$\begin{aligned} \frac{2}{3} \cdot \frac{7}{8} = \frac{14}{24} = \frac{7}{12} \end{aligned}$

$\begin{aligned} \frac{2}{3} \cdot \frac{7}{8} \cdot x \end{aligned}$ simplifies to $\begin{aligned} \frac{7}{12} x \end{aligned}$ .
::2 3 7 8 x 简化为 7 12 x 。

The answer is that $\begin{aligned} \frac{2}{3} \cdot x \cdot \frac{7}{8} \end{aligned}$ simplifies to $\begin{aligned} \frac{7}{12} x \end{aligned}$ .
::答案是 2 3 x 7 8 简化为 7 12 x 。

Here is an example where the associative property is useful.
::这里的例子就是连带财产是有用的。

Simplify $\begin{aligned} (x \cdot \frac{1}{2}) \cdot \frac{3}{5} \end{aligned}$ .
::简化 (x 1 2 ) 3 5 。

First, use the associative property of multiplication to regroup the factors.
::首先,利用乘法的共同属性来重新组合各种因素。

$\begin{aligned} (x \cdot \frac{1}{2}) \cdot \frac{3}{5} \end{aligned}$ is equivalent to $\begin{aligned} x \cdot (\frac{1}{2} \cdot \frac{3}{5}) \end{aligned}$ .
::======================================================================================================================================================================================================================x======================================================================================================================================================================================================================================================================================================

Now, simplify $\begin{aligned} x \cdot (\frac{1}{2} \cdot \frac{3}{5}) \end{aligned}$ . Multiply the fractions in the " data-term="Parentheses" role="term" tabindex="0"> parentheses using what you have learned about fraction multiplication.
::现在,简化 x (1、2 3 5 ) 。使用您了解的分数乘法来乘以括号中的分数。

$\begin{aligned} \frac{1}{2} \cdot \frac{3}{5} = \frac{3}{10} \end{aligned}$

$\begin{aligned} x \cdot (\frac{1}{2} \cdot \frac{3}{5}) \end{aligned}$ simplifies to $\begin{aligned} x \cdot \frac{3}{10} \end{aligned}$ .
::x (1、2 3 5 ) 简化为 x 3 10 。

The answer is that $\begin{aligned} (x \cdot \frac{1}{2}) \cdot \frac{3}{5} \end{aligned}$ simplifies to $\begin{aligned} x \cdot \frac{3}{10} \end{aligned}$ or $\begin{aligned} \frac{3}{10} x \end{aligned}$ .
::答案是 (x ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Elaine and Connie and their bracelets.
::早些时候,你被问及伊莱恩和康妮还有他们的手镯

Connie is going to give Elaine $\begin{aligned} \frac{1}{5} \end{aligned}$ of the money she makes. Elaine is going to put $\begin{aligned} \frac{1}{2} \end{aligned}$ of that money into a savings account at the bank. Elaine wants to know how she could write an expression to represent the fraction of money that Connie earns that Elaine will be putting into the bank.
::康妮要给艾琳15分钱。艾琳要把12分钱放进银行的储蓄账户。艾琳想知道她如何写一个表达方式来代表康妮赚的钱的一部分,而艾琳会把钱放进银行。

First, Elaine can write an expression and then she can simplify it. She doesn't know how much money Connie will earn, so that unknown will be her variable . Let $\begin{aligned} x \end{aligned}$ be the amount of money Connie earns.
::首先,Elaine可以写一个表达方式,然后她可以简化它。她不知道Connie会挣多少钱,所以未知的将是她的变数。让x是Connie挣的钱。

Elaine will be getting $\begin{aligned} \frac{1}{5} \end{aligned}$ of the money Connie earns, so Elaine will be getting $\begin{aligned} \frac{1}{5} \cdot x \end{aligned}$ . Elaine is going to put $\begin{aligned} \frac{1}{2} \end{aligned}$ of that money in the bank. So the money Elaine will be putting in the bank can be represented by the expression.
::Elaine将会得到Connie收入的15美元,所以Elaine将会得到15美元。Elaine将会把这笔钱的12美元放入银行。所以Elaine将会把这笔钱放入银行,Elaine会把这笔钱放在银行里。

$\begin{aligned} \frac{1}{2} \cdot (\frac{1}{5} \cdot x) \end{aligned}$

Now Elaine can simplify the expression. She can use the associative property of multiplication to regroup the factors.
::现在伊莱恩可以简化表达方式。她可以使用乘法的关联属性来重新组合各种因素。

$\begin{aligned} \frac{1}{2} \cdot (\frac{1}{5} \cdot x) \end{aligned}$ is equivalent to $\begin{aligned} (\frac{1}{2} \cdot \frac{1}{5}) \cdot x \end{aligned}$ .
::1 2 □ (1 5 □ x ) 等于 (1 2 □ 1 (5) ) x 。

Next, she can simplify $\begin{aligned} (\frac{1}{2} \cdot \frac{1}{5}) \cdot x \end{aligned}$ . She can multiply the fractions in the parentheses using what she has learned about fraction multiplication.
::其次,她可以简化( 1 2 1 5 ) x 。她可以使用她了解的分数乘法来乘以括号中的分数。

$\begin{aligned} \frac{1}{2} \cdot \frac{1}{5} = \frac{1}{10} \end{aligned}$

$\begin{aligned} (\frac{1}{2} \cdot \frac{1}{5}) \cdot x \end{aligned}$ simplifies to $\begin{aligned} \frac{1}{10} x \end{aligned}$ .
:1 2 1 5 ) x 简化为 1 10 x 。

The answer is that Elaine will be putting $\begin{aligned} \frac{1}{10} x \end{aligned}$ dollars in the bank, where $\begin{aligned} x \end{aligned}$ is the amount of money Connie earns selling bracelets.
::答案是Elaine会在银行里放10美元康妮卖手镯的钱是X

Example 2
::例2

Multiply $\begin{aligned} \frac{9}{16} \cdot \frac{1}{2} \cdot \frac{8}{15} \end{aligned}$ .
::乘以 9 16 1 2 8 15 。

First, notice that the first fraction and the third fraction have common factors. Use the commutative property to switch the second and third fractions.
::首先,请注意第一个分数和第三个分数有共同因素。使用通量属性转换第二和第三分数。

$\begin{aligned} \frac{9}{16} \cdot \frac{1}{2} \cdot \frac{8}{15} \end{aligned}$ is equivalent to $\begin{aligned} \frac{9}{16} \cdot \frac{8}{15} \cdot \frac{1}{2} \end{aligned}$ .
::9 16 1 2 8 15 等于 9 16 8 15 1 2 。

Now, focus on the first two fractions. Look for common factors along the diagonals in the numerators and the denominators. Notice that 9 and 15 both have a factor of 3. You can divide both the 9 and the 15 by 3.
::现在, 聚焦于前两个部分。寻找在分子数和分母的对角线上的常见因素。注意 9 和 15 都有3 的系数。您可以将 9 和 15 除以 3 。

$\begin{aligned} \frac{9}{16} \cdot \frac{8}{15} \cdot \frac{1}{2} = \frac{3}{16} \cdot \frac{8}{5} \cdot \frac{1}{2} \end{aligned}$

Similarly, both the 8 and the 16 have a factor of 8. You can divide both the 8 and the 16 by 8.
::同样,8和16的系数为8,8和16的系数为8。 8和16的系数可除以8。

$\begin{aligned} \frac{3}{16} \cdot \frac{8}{5} \cdot \frac{1}{2} = \frac{3}{2} \cdot \frac{1}{5} \cdot \frac{1}{2} \end{aligned}$

Now, you can multiply the fractions. Use what you have learned about fraction multiplication.
::现在,您可以乘以分数。使用您学到的关于分数乘法的知识。

$\begin{aligned} \frac{3}{2} \cdot \frac{1}{5} \cdot \frac{1}{2} = \frac{3}{20} \end{aligned}$

The answer is $\begin{aligned} \frac{9}{16} \cdot \frac{1}{2} \cdot \frac{8}{15} = \frac{3}{20} \end{aligned}$ .
::答案是 9 16 1 2 8 15 = 3 20 。

Example 3
::例3

Simplify $\begin{aligned} (x \cdot \frac{4}{5}) \cdot \frac{1}{2} \end{aligned}$ .
::简化 (x 4 5 ) 1 2 。

First, use the associative property of multiplication to regroup the factors.
::首先,利用乘法的共同属性来重新组合各种因素。

$\begin{aligned} (x \cdot \frac{4}{5}) \cdot \frac{1}{2} \end{aligned}$ is equivalent to $\begin{aligned} x \cdot (\frac{4}{5} \cdot \frac{1}{2}) \end{aligned}$ .
::=============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

Now, simplify $\begin{aligned} x \cdot (\frac{4}{5} \cdot \frac{1}{2}) \end{aligned}$ . Multiply the fractions in the parentheses using what you have learned about fraction multiplication.
::现在,简化 x ( 4 5 1 2 ) 。使用您了解的分数乘法来乘以括号中的分数。

$\begin{aligned} \frac{4}{5} \cdot \frac{1}{2} = \frac{4}{10} = \frac{2}{5} \end{aligned}$

$\begin{aligned} x \cdot (\frac{4}{5} \cdot \frac{1}{2}) \end{aligned}$ simplifies to $\begin{aligned} x \cdot \frac{2}{5} \end{aligned}$ .
::x (4 5 1 2 ) 简化为 x 2 5 。

The answer is that $\begin{aligned} (x \cdot \frac{4}{5}) \cdot \frac{1}{2} \end{aligned}$ simplifies to $\begin{aligned} x \cdot \frac{2}{5} 5 \end{aligned}$ or $\begin{aligned} \frac{2}{5} x \end{aligned}$ .
::答案是 (x 4 5 ) 1 2 简化为 x 2 5 5 或 2 5 x 。

Example 4
::例4

Simplify $\begin{aligned} \frac{6}{7} \cdot x \cdot \frac{1}{3} \end{aligned}$ .
::简化 6 7 × 1 3 。

First, use the commutative property of multiplication to reorder the factors.
::首先,使用乘法的通量属性重新排序系数。

$\begin{aligned} \frac{6}{7} \cdot x \cdot \frac{1}{3} \end{aligned}$ is equivalent to $\begin{aligned} \frac{6}{7} \cdot \frac{1}{3} \cdot x \end{aligned}$ .
::6 7 × x × 1 3 等于 6 7 × 1 3 x 。

Next, simplify $\begin{aligned} \frac{6}{7} \cdot \frac{1}{3} \cdot x \end{aligned}$ . Multiply the fractions using what you have learned about fraction multiplication. Simplify your result.
::下一步, 简化 6 7 1 3 x 。使用您所学到的关于分数乘法的分数乘以分数。简化您的结果。

$\begin{aligned} \frac{6}{7} \cdot \frac{1}{3} = \frac{6}{21} = \frac{2}{7} \end{aligned}$

$\begin{aligned} \frac{6}{7} \cdot \frac{1}{3} \cdot x \end{aligned}$ simplifies to $\begin{aligned} \frac{2}{7} x \end{aligned}$ .
::6 7 1 3 x x 简化为 2 7 x 。

The answer is that $\begin{aligned} \frac{6}{7} \cdot x \cdot \frac{1}{3} \end{aligned}$ simplifies to $\begin{aligned} \frac{2}{7} x \end{aligned}$ .
::答案是 6 7 x 13 简化为 2 7 x 。

Example 5
::例5

Simplify $\begin{aligned} \frac{2}{3} \cdot x \cdot \frac{4}{9} \end{aligned}$ .
::简化2 3 x 4 9 。

First, use the commutative property of multiplication to reorder the factors.
::首先,使用乘法的通量属性重新排序系数。

$\begin{aligned} \frac{2}{3} \cdot x \cdot \frac{4}{9} \end{aligned}$ is equivalent to $\begin{aligned} \frac{2}{3} \cdot \frac{4}{9} \cdot x \end{aligned}$ .
::2 3 × x × 4 9 等于 2 3 × 4 9 × x 。

Next, simplify $\begin{aligned} \frac{2}{3} \cdot \frac{4}{9} \cdot x \end{aligned}$ . Multiply the fractions using what you have learned about fraction multiplication. Simplify your result.
::下一步, 简化 2 3 4 9 x 。使用您所学到的关于分数乘法的分数乘以分数。简化您的结果。

$\begin{aligned} \frac{2}{3} \cdot \frac{4}{9} = \frac{8}{27} \end{aligned}$

$\begin{aligned} \frac{2}{3} \cdot \frac{4}{9} \cdot x \end{aligned}$ simplifies to $\begin{aligned} \frac{8}{27} x \end{aligned}$ .
::2 3 4 9 x 简化为 8 27 x 。

The answer is that $\begin{aligned} \frac{2}{3} \cdot x \cdot \frac{4}{9} \end{aligned}$ simplifies to $\begin{aligned} \frac{8}{27} x \end{aligned}$ .
::答案是 2 3 x 4 9 简化为 8 27 x 。

Review
::回顾

Multiply.
::乘数

$\begin{aligned} \frac{2}{3} \cdot \frac{9}{12} \cdot \frac{6}{7} \end{aligned}$

$\begin{aligned} \frac{1}{2} \cdot 1 \frac{4}{5} \cdot \frac{3}{4} \end{aligned}$

$\begin{aligned} (\frac{4}{9} \cdot \frac{5}{8}) \cdot \frac{3}{7} \end{aligned}$

$\begin{aligned} \frac{10}{12} \cdot (3 \frac{1}{5} \cdot \frac{7}{10}) \end{aligned}$

$\begin{aligned} \frac{1}{3} \cdot \frac{9}{18} \cdot \frac{6}{7} \end{aligned}$

$\begin{aligned} \frac{4}{5} \cdot \frac{9}{20} \cdot \frac{2}{9} \end{aligned}$

$\begin{aligned} \frac{1}{3} \cdot \frac{4}{12} \cdot \frac{2}{9} \end{aligned}$

Simplify each expression.
::简化每个表达式。

$\begin{aligned} \frac{7}{8} \cdot x \cdot \frac{4}{5} \end{aligned}$
::7 8 x x 4 5

$\begin{aligned} x \cdot 2 \frac{2}{3} \cdot \frac{5}{6} \end{aligned}$
::x 2 2 2 2 3 □ 5 6

$\begin{aligned} \frac{5}{8} \cdot (1 \frac{2}{3} \cdot x) \end{aligned}$
::5 8 _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} \frac{6}{8} \cdot x \cdot \frac{2}{3} \end{aligned}$
::6 8 × 8 × × 2 3

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

$\begin{aligned} \frac{3}{10} \cdot x \cdot \frac{4}{5} \end{aligned}$
::3 10 x 4 5

Solve each problem.
::解决每一个问题。

Crazy Sal's is having a Delirious Discount Sale. He is selling everything in his store for $\begin{aligned} \frac{3}{8} \end{aligned}$ of the marked price. Rowena finds a t-shirt that is marked at $36. How much will she pay for the shirt at the discounted price?
::Rowena找到一件T恤衫,印价36美元。她要花多少钱买这件衬衫?

Dan is cutting plywood for his science fair project. He cuts a board that is $\begin{aligned} 3 \frac{1}{4} \end{aligned}$ feet long. After he cuts it, he realizes that he really needs a piece about $\begin{aligned} \frac{2}{3} \end{aligned}$ of this length. How long will the new piece of wood that Dan cuts be?
::丹为他的科学博览会项目剪切胶合板。他剪切板的长度是34英尺。在切切之后,他意识到他真的需要这一长度的大约23英尺。丹切开的新木板需要多长时间?

Review (Answers)
::回顾(答复)

Section outline

General

分数积的结合律和交换律

Using Commutative and Associative Properties of Multiplication with Fractions ::使用带有分数的乘法的通和相交属性

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 Multiplication with Fractions
::使用带有分数的乘法的通和相交属性

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)