Course: 2.12小数乘法的结合律和交换律

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小数乘法的结合律和交换律
Marigold has a lot of tomato plants in her vegetable garden. Marigold is planning to pick the ripe tomatoes and make salsa with them. She looks up a basic recipe. The recipe says that for every cup of tomatoes, she will need 0.5 onion. For every onion, she needs 4 cloves of garlic. How can Marigold determine how many cloves of garlic she needs in terms of the number of cups of tomatoes she picks?
::Marigold的菜园里有很多番茄植物。 Marigold正计划摘熟的番茄, 并和他们一起做番茄酱。她寻找了一个基本的食谱。食谱上说, 每杯番茄, 她都需要0.5洋葱。每杯洋葱, 她需要4个大蒜。玛里戈德如何确定她需要多少大蒜, 以她所摘的番茄数量计算?

In this concept, you will learn to identify and use the commutative and associative properties of multiplication with decimals.
::在此概念中,您将学会识别和使用乘以十进制的通量和关联性。

Commutative and Associative Properties of Multiplication with Decimals
::十进制乘以乘以乘以乘以十进数的通和相交属性

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 $\begin{aligned} b \end{aligned}$ :
::乘法通量表示,在寻找产品时,改变因素的顺序不会改变产品。在符号中,乘法通量表示,对于a和b数字,a和b:

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

Here is an example using simple whole numbers.
::以下是使用简单整数的例子。

Show that $\begin{aligned} 2 \cdot 4 = 4 \cdot 2 \end{aligned}$ .
::显示 2 4 = 4 2 。

First, find $\begin{aligned} 2 \cdot 4 \end{aligned}$ .
::首先,找到2 4 。

$\begin{aligned} 2 \cdot 4 = 8 \end{aligned}$

Next, find $\begin{aligned} 4 \cdot 2 \end{aligned}$ .
::接下来,发现4 2 。

$\begin{aligned} 4 \cdot 2 = 8 \end{aligned}$

Notice that both products are 8.
::注意这两个产品均为8。

The answer is that because both $\begin{aligned} 2 \cdot 4 \end{aligned}$ and $\begin{aligned} 4 \cdot 2 \end{aligned}$ are equal to 8, they are equal to each other.
::答案是,因为2+4和4++2等于8,它们彼此平等。

$\begin{aligned} 2 \cdot 4 = 4 \cdot 2 \end{aligned}$

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 $\begin{aligned} a \end{aligned}$ , $\begin{aligned} b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ :
::乘法联合属性表示,在寻找产品时,改变分类因素的方式不会改变产品。在符号中,乘法联合属性表示,对于a、b和c数,a、b和c:

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

Here is an example using simple whole numbers.
::以下是使用简单整数的例子。

Show that $\begin{aligned} (2 \cdot 5) \cdot 6 = 2 \cdot (5 \cdot 6) \end{aligned}$ .
::显示 (2 □ 5 □ 6 = 2 □ (5 □ 6 ) 。

First find $\begin{aligned} (2 \cdot 5) \cdot 6 \end{aligned}$ . Start by multiplying the numbers in " data-term="Parentheses" role="term" tabindex="0"> parentheses . Then multiply the result with 6.
::第一个发现 (2 + 5 ) = 6 。从括号中乘以数字开始。然后将结果乘以 6 。

$\begin{aligned} \begin{array}{rcl} (2 \cdot 5) \cdot 6 & = & 10 \cdot 6 \\ = & 60 \end{array} \end{aligned}$

Next, find $\begin{aligned} 2 \cdot (5 \cdot 6) \end{aligned}$ . Again, start by multiplying the numbers in parentheses. Then multiply 2 by the result.
::下一步,请查找 2 ( 5 6 ) 。再次, 从括号内的数字乘以数开始。然后乘以 2 乘以结果。

$\begin{aligned} \begin{array}{rcl} 2 \cdot (5 \cdot 6) & = & 2 \cdot 30 \\ = & 60 \end{array} \end{aligned}$

Notice that both products are 60.
::注意这两种产品均为60。

The answer is that because both $\begin{aligned} (2 \cdot 5) \cdot 6 \end{aligned}$ and $\begin{aligned} 2 \cdot (5 \cdot 6) \end{aligned}$ are equal to 60, they are equal to each other.
::答案是,由于6和2等同于60,它们彼此平等。

$\begin{aligned} (2 \cdot 5) \cdot 6 = 2 \cdot (5 \cdot 6) \end{aligned}$

Both the Commutative Property of Multiplication and the Associative Property of Multiplication can be useful in simplifying expressions. 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.
::举一个例子。

Simplify $\begin{aligned} 29.3 (12.4 x) \end{aligned}$ .
::简化29.3(12.4x)

First, use the Associative Property of Multiplication to regroup the factors.
::首先,使用乘以的共有属性来重新组合各种因素。

$\begin{aligned} 29.3 (12.4 x) \end{aligned}$ is equivalent to $\begin{aligned} (29.3 \cdot 12.4) x \end{aligned}$ .
::29.3 (12.4 x) 等于 (29.3 12.4) x 。

Now, simplify $\begin{aligned} (29.3 \cdot 12.4) x \end{aligned}$ . Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.
::现在, 简化 (29.3 12. 4) x 。乘以括号中的数字。使用您所学到的关于十进制乘法的数值。

$\begin{aligned} 29.3 \\ \underline{\times 12.4} \\ 1172 \\ 5860 \\ \underline{+ 29300} \\ 363.32 \end{aligned}$

$\begin{aligned} (29.3 \cdot 12.4) x \end{aligned}$ simplifies to $\begin{aligned} 363.32 x \end{aligned}$ .
:29.3 12.4) x 简化为363.32 x 。

The answer is that $\begin{aligned} 29.3 (12.4 x) \end{aligned}$ simplifies to $\begin{aligned} 363.32 x \end{aligned}$ .
::答案是29.3(12.4x)简化为363.32x。

Here is another example.
::下面是另一个例子。

Simplify $\begin{aligned} (0.3 x) \cdot 0.4 \end{aligned}$ .
::简化( 0. 3 x ) + + 0.4 。

First, use the Commutative Property of Multiplication to reorder the factors.
::首先,使用乘法的通货性能重新排序系数。

$\begin{aligned} (0.3 x) \cdot 0.4 \end{aligned}$ is equivalent to $\begin{aligned} 0.4 \cdot (0.3 x) \end{aligned}$ .
:0.3x) 0.4等于0.4 (0.3x) 。

Next, use the Associative Property of Multiplication to regroup the factors.
::其次,使用乘以的共有属性来重新组合各种因素。

$\begin{aligned} 0.4 \cdot (0.3 x) \end{aligned}$ is equivalent to $\begin{aligned} (0.4 \cdot 0.3) x \end{aligned}$ .
::0.4 = (0.3x) 等于 (0.4 = 0.3x) x 。

Now, simplify $\begin{aligned} (0.4 \cdot 0.3) x \end{aligned}$ . Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.
::现在, 简化 (0. 4 0. 3) x 。乘以括号中的数字。使用您所学到的关于十进制乘法的知识。

$\begin{aligned} 0.4 \\ \underline{\times 0.3} \\ 0.12 \end{aligned}$

$\begin{aligned} (0.4 \cdot 0.3) x \end{aligned}$ simplifies to $\begin{aligned} 0.12 x \end{aligned}$ .
:0.4 0.3) x 简化为 0.12 x 。

The answer is that $\begin{aligned} (0.3 x) \cdot 0.4 \end{aligned}$ simplifies to $\begin{aligned} 0.12 x \end{aligned}$ .
::答案是 (0.3 x) 0.4 简化为 0. 12 x 。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Marigold, who is planning to make salsa.
::之前,你被问及Marigold的问题, 谁是计划做Salsa的。

Her recipe says that for every cup of tomatoes she will need 0.5 onion, and for every onion she will need 4 cloves of garlic. Marigold wants to figure out how many cloves of garlic she will need in terms of the number of cups of tomatoes she picks.
::她的配方说,对于每杯番茄,她需要0.5洋葱,对于每杯洋葱,她需要4个两丁大蒜。 Marigold想弄清楚,从她所选的番茄数量来看,她需要多少大蒜。

First, Marigold should write an expression for this situation. She should start by defining her variable . She doesn’t know how many cups of tomatoes she will have, so that unknown quantity will be her variable.
::首先,Marigold应该为这种情况写一个表达方式。她应该首先定义她的变数。她不知道她会有多少杯西红柿,所以数量不详将是她的变数。

Let $\begin{aligned} x \end{aligned}$ equal the number of cups of tomatoes Marigold picks.
::x 等于西红柿的杯数, Marigold 摘取的杯数。

Now, the problem says that she will need 0.5 onion for every tomato. So the number of onions she needs is $\begin{aligned} 0.5 x \end{aligned}$ .
::问题在于她需要0.5洋葱做每只番茄。所以她需要的洋葱数量是0.5x。

Next, the problem says that she will need 4 cloves of garlic for every onion. Since she will have $\begin{aligned} 0.5 x \end{aligned}$ onions, she will need $\begin{aligned} (0.5 x) \cdot 4 \end{aligned}$ cloves of garlic.
::其次,问题在于她每洋葱需要四丁基大蒜。因为她有0.5 x 洋葱,所以她需要(0.5 x ) + 4 大蒜。

Now, Marigold can simplify the expression.
::Marigold可以简化表达式

First, she can use the Commutative Property of Multiplication to reorder the factors.
::首先,她可以使用乘法的通货性能重新排列因素的顺序。

$\begin{aligned} (0.5 x) \cdot 4 \end{aligned}$ is equivalent to $\begin{aligned} 4 \cdot (0.5 x) \end{aligned}$ .
:0.5x) 4 等于 4 (0.5x) 。

Next, she can use the Associative Property of Multiplication to regroup the factors.
::其次,她可以利用乘法联合属性来重新组合各种因素。

$\begin{aligned} 4 \cdot (0.5 x) \end{aligned}$ is equivalent to $\begin{aligned} (4 \cdot 0.5) x \end{aligned}$ .
::4(0.5x)等于(40.5x)x。

Now, she can simplify $\begin{aligned} (4 \cdot 0.5) x \end{aligned}$ . She can use what she learned about decimal number multiplication to multiply the numbers in parentheses.
::现在,她可以简化( 4 0. 5 ) x 。她可以使用她所学的小数乘法来乘以括号中的数字。

$\begin{aligned} 0.5 \\ \underline{\times 4} \\ 2.0 \end{aligned}$

$\begin{aligned} (4 \cdot 0.5) x \end{aligned}$ simplifies to $\begin{aligned} 2 x \end{aligned}$ .
:4 0.5 ) x 简化为 2 x 。

The answer is that Marigold will need 2 cloves of garlic for every cup of tomatoes she picks.
::答案是Marigold每选一杯咖啡西红柿就需要两丁香蒜

Example 2
::例2

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

$\begin{aligned} 4.5 (9.2 y) \end{aligned}$
::4.5(9.2y)

First, use the Associative Property of Multiplication to regroup the factors.
::首先,使用乘以的共有属性来重新组合各种因素。

$\begin{aligned} 4.5 (9.2 y) \end{aligned}$ is equivalent to $\begin{aligned} (4.5 \cdot 9.2) y \end{aligned}$ .
::4.5(9.2y)等于(4.5 9.2)y。

Now, simplify $\begin{aligned} (4.5 \cdot 9.2) y \end{aligned}$ . Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.
::现在,简化 (4.5 9.2 y) 。乘以括号中的数字。使用您所学到的关于十进制乘法的知识。

$\begin{aligned} 4.5 \\ \underline{\times 9.2} \\ 90 \\ \underline{+ 4050} \\ 41.50 \end{aligned}$

$\begin{aligned} (4.5 \cdot 9.2) y \end{aligned}$ simplifies to $\begin{aligned} 41.4 y \end{aligned}$ .
:4.5 9.2)y 简化为41.4 y。

The answer is that $\begin{aligned} 4.5 (9.2 y) \end{aligned}$ simplifies to $\begin{aligned} 41.4 y \end{aligned}$ .
::答案是4.5(9.2y)简化为41.4y。

Example 3
::例3

Simplify $\begin{aligned} 4.8 (3.1 k) \end{aligned}$ .
::简化4.8 (3.1 k).

First, use the Associative Property of Multiplication to regroup the factors.
::首先,使用乘以的共有属性来重新组合各种因素。

$\begin{aligned} 4.8 (3.1 k) \end{aligned}$ is equivalent to $\begin{aligned} (4.8 \cdot 3.1) k \end{aligned}$ .
::4.8 (3.1 k) 等于 (4.8 3.1 ) k 。

Now, simplify $\begin{aligned} (4.8 \cdot 3.1) k \end{aligned}$ . Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.
::现在, 简化 (4. 8 3. 1 ) k。乘以括号中的数字。使用您学到的关于十进制乘法的知识。

$\begin{aligned} 4.8 \\ \underline{\times 3.1} \\ 48 \\ \underline{+ 1440} \\ 14.88 \end{aligned}$

$\begin{aligned} (4.8 \cdot 3.1) k \end{aligned}$ simplifies to $\begin{aligned} 14.88 k \end{aligned}$ .
:4.8 3.1) 简化为14.88 k。

The answer is that $\begin{aligned} 4.8 (3.1 k) \end{aligned}$ simplifies to $\begin{aligned} 14.88 k \end{aligned}$ .
::答案是4.8 (3.1 k) 简化为 14.88 k。

Example 4
::例4

Simplify $\begin{aligned} (3.45 p) \cdot 2.3 \end{aligned}$ .
::简化(3.45页) ________________________________________________________________________________________________________________________________________________________________________________________

First, use the Commutative Property of Multiplication to reorder the factors.
::首先,使用乘法的通货性能重新排序系数。

$\begin{aligned} (3.45 p) \cdot 2.3 \end{aligned}$ is equivalent to $\begin{aligned} 2.3 \cdot (3.45 p) \end{aligned}$ .
::3.45 p) 2.3等于2.3 (3.45 p) 。

Next, use the Associative Property of Multiplication to regroup the factors.
::其次,使用乘以的共有属性来重新组合各种因素。

$\begin{aligned} 2.3 \cdot (3.45 p) \end{aligned}$ is equivalent to $\begin{aligned} (2.3 \cdot 3.45) p \end{aligned}$ .
::2.3 = (3.45 p) 等于 (2.3 = 3.45 ) p.

Now, simplify $\begin{aligned} (2.3 \cdot 3.45) p \end{aligned}$ . Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.
::现在, 简化 (2.3 3. 45) p. 。乘以括号中的数字。使用您所学到的关于十进制乘法的知识。

$\begin{aligned} 3.45 \\ \underline{\times 2.3} \\ 1035 \\ \underline{+ 6900} \\ 7.935 \end{aligned}$

$\begin{aligned} (2.3 \cdot 3.45) p \end{aligned}$ simplifies to $\begin{aligned} 7.935 p \end{aligned}$ .
:2.3) 3.45) 简化到7.935页。

The answer is that $\begin{aligned} (3.45 p) \cdot 2.3 \end{aligned}$ simplifies to $\begin{aligned} 7.935 p \end{aligned}$ .
::答案是: __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Example 5
::例5

Simplify $\begin{aligned} 1.98 \cdot (a \cdot 6.4) \end{aligned}$ .
::简化1.98 (a 6.4 ) 。

First, use the Commutative Property of Multiplication to reorder the factors within the parentheses.
::首先,使用乘法的通货性能对括号内的系数进行重新排序。

$\begin{aligned} 1.98 \cdot (a \cdot 6.4) \end{aligned}$ is equivalent to $\begin{aligned} 1.98 \cdot (6.4 \cdot a) \end{aligned}$ .
::1.98 (a 6.4) 等于 1.98 (6.4 a) 。

Next, use the Associative Property of Multiplication to regroup the factors.
::其次,使用乘以的共有属性来重新组合各种因素。

$\begin{aligned} 1.98 \cdot (6.4 \cdot a) \end{aligned}$ is equivalent to $\begin{aligned} (1.98 \cdot 6.4) \cdot a \end{aligned}$
::1.98 (6.4 a) 等于 (1.98 6.4) a

Now, simplify $\begin{aligned} (1.98 \cdot 6.4) \cdot a \end{aligned}$ . Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.
::现在, 简化 (1. 98 6. 4) a 。乘以括号中的数字。使用您所学的关于十进制乘法的数值。

$\begin{aligned} 1.98 \\ \underline{\times 6.4} \\ 792 \\ \underline{+ 11880} \\ 12.672 \end{aligned}$

$\begin{aligned} (1.98 \cdot 6.4) \cdot a \end{aligned}$ simplifies to $\begin{aligned} 12.672 a \end{aligned}$ .
:1.98 6.4 ) 简化为 12.672 a 。

The answer is that $\begin{aligned} 1.98 \cdot (a \cdot 6.4) \end{aligned}$ simplifies to $\begin{aligned} 12.672 a \end{aligned}$ .
::答案是1.98 (a 6.4) 简化为 12.672 a 。

Review
::回顾

Simplify the following expressions.
::简化下列表达式。

$\begin{aligned} (4.21 \times 8.8) \times p \end{aligned}$
:4.21 × 8.8 ) × p

$\begin{aligned} 16.14 \times q \times 6.2 \end{aligned}$
::16.14 × q × 6.2

$\begin{aligned} 3.6 (91.7 x) \end{aligned}$
::3.6 (91.7x)

$\begin{aligned} 5.3 r (2.8) \end{aligned}$
::5.3 r(2.8)

$\begin{aligned} 5.6 x (3.8) \end{aligned}$
::5.6x(3.8)

$\begin{aligned} 2.4 y (2.8) \end{aligned}$
::2.4y(2.8)

$\begin{aligned} 6.7 x (3.1) \end{aligned}$
::6.7x(3.1)

$\begin{aligned} 8.91 r (2.3) \end{aligned}$
::8.91r(2.3)

$\begin{aligned} 5.67 y (2.8) \end{aligned}$
::5.67y(2.8)

$\begin{aligned} 4.53 x (2.2) \end{aligned}$
::4.53x(2.2)

$\begin{aligned} 5.6 (2.8 x) \end{aligned}$
::5.6(2.8x)

$\begin{aligned} 9.2 y (3.2) \end{aligned}$
::9.2y(3.2)

$\begin{aligned} 4.5 x (2.3) \end{aligned}$
::4.5x(2.3)

$\begin{aligned} 15.4 x (12.8) \end{aligned}$
::15.4x( 12.8)

$\begin{aligned} 18.3 y (14.2) \end{aligned}$
::18.3y( 14.2)

Review (Answers)
::回顾(答复)

Section outline

General

小数乘法的结合律和交换律

Commutative and Associative Properties of Multiplication with Decimals ::十进制乘以乘以乘以乘以十进数的通和相交属性

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)

Commutative and Associative Properties of Multiplication with Decimals
::十进制乘以乘以乘以乘以十进数的通和相交属性

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)