Section: 十进制运算中乘法的性质 | 4.3十进制运算中乘法的性质

Section outline

Joan and Mike are measuring the volume of a box. They can’t remember the exact formula , but they do know that you have to multiply the dimensions of the box. Joan says, “We can multiply the length times the width and then, multiply by the height.” Mike says, “No, we have to multiply the width times the height and then, multiply by the length.” Who is correct?
::琼和麦克正在测量一个盒子的体积。他们不记得确切的公式,但他们确实知道你必须乘以盒子的维度。琼说,“我们可以乘以宽度的倍数,然后乘以高度。” 麦克说 , “ 不,我们必须乘以宽度的倍数,然后乘以长度。 ”谁是正确的?

In this concept, you will learn to apply the multiplication properties in decimal operations .
::在此概念中,您将在小数操作中学习应用乘法属性。

Properties of Multiplying Decimals
::乘以十进制的属性

A property is a rule that makes a statement about the way that numbers interact with each other during certain operations. The key thing to remember about a property is that the statement is true for all numbers.
::财产是一种规则,它可以说明数字在某些操作中相互作用的方式。值得记住的关键是,该声明对所有数字都是真实的。

The commutative property of multiplication states that the order in which you multiply numbers does not matter as the product will be the same.
::乘法的通量属性表示,乘法数的顺序与产品相同无关。

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

When working with decimals and whole numbers, the product will be the same regardless of which number is multiplied first.
::与小数和整数一起工作时,不论先乘哪个数字,产品都是相同的。

$\begin{aligned} 4.5 (7) = 7 (4.5) \end{aligned}$

Check by multiplying.
::通过乘法检查。

$\begin{aligned} \begin{array}{rcl} 4.5 \\ \underline{\times 7} \\ 31.5 \end{array} \end{aligned}$

$\begin{aligned} \begin{array}{rcl} 7 \\ \underline{\times 4.5} \\ 35 \\ \underline{+ 28} \\ 31.5 \end{array} \end{aligned}$

The commutative property of multiplication also applies to multiplication problems with variables. Remember that a variable is a letter used to represent an unknown number. A variable next to a number tells you to multiply.
::乘法的通量属性也适用于变量的乘法问题。请记住,变量是指用于表示未知数字的字母。数字旁边的变量指示您要乘法。

$\begin{aligned} 5.6 a = a 5.6 \end{aligned}$

The product of $\begin{aligned} 5.5 a \end{aligned}$ is the same as the product of $\begin{aligned} a 5.6 \end{aligned}$ . Note that when using variables to express multiplication, the number is written before the variable.
::5.5 a 的产值与 5.6 的产值相同。请注意,当使用变量表示乘法时,数字在变量前面写。

$\begin{aligned} 5 a \end{aligned}$

If $\begin{aligned} a \end{aligned}$ is equal to 3, find the product.
::如果一个等于3, 找到产品。

$\begin{aligned} \begin{array}{rcl} 5.6 \\ \underline{\times 3} \\ 16.8 \end{array} \end{aligned}$

The product of $\begin{aligned} 5 a \end{aligned}$ , if $\begin{aligned} a = 3 \end{aligned}$ , is 16.8.
::5a(如果=3)的产值为16.8。

The associative property of multiplication states that the order in which you group numbers in multiplication does not matter as the product will be the same. Remember that grouping refers to the use of " data-term="Parentheses" role="term" tabindex="0"> parentheses or brackets .
::乘法的连带属性表示,乘法中组合编号的顺序与产品相同无关。请记住,分组是指括号或括号的使用。

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

The same applies when multiplying decimals.
::当乘以十进制时同样适用。

$\begin{aligned} 6 (3.4 \times 2) = (6 \times 3.4) 2 \end{aligned}$

You can change the grouping of the numbers and the product will remain the same. Multiply the numbers in parentheses first.
::您可以更改数字的分组,产品将保持不变。先将括号中的数字乘以。

$\begin{aligned} \begin{array}{rcl} 6 (6.8) & = & 20.4 (2) \\ 40.8 & = & 40.8 \end{array} \end{aligned}$

This is also true with variables.
::在变量方面也是如此。

$\begin{aligned} \begin{array}{rcl} 5 (6 a) & = & (5 \times 6) a \\ 30 a & = & 30 a \end{array} \end{aligned}$

Once again, you can change the grouping of the numbers and variables and the product will remain the same.
::您可以再次改变数字和变量的分组,产品将保持不变。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Joan and Mike trying to calculate volume.
::早些时候,你得到一个问题琼和迈克试图计算体积。

Joan says, “We can multiply the length times the width and then, multiply by the height.” Mike says, “No, we have to multiply the width times the height and then, multiply by the length.” Who is correct? Find the volume of the box.
::琼说,“我们可以乘以宽度的长度倍数,然后乘以高度。” 迈克说,“不,我们必须乘以宽度倍数,然后乘以长度。” 谁是正确的?找到盒子的体积。

First, write the equation to find the volume using Joan’s method and Mike’s method.
::首先,用琼的方法和麦克的方法写出方程式,以寻找音量。

$\begin{aligned} \begin{array}{rcl} Joan Mike \\ Volume = (length \times width) (height) Volume = (width \times height) (length) \\ Volume = (12.5 i n . \times 4 i n .) (5 i n .) Volume = (4 i n . \times 5 i n .) (12.5 i n .) \end{array} \end{aligned}$

Then, look at each of their equations. Remember that the commutative and associative properties of multiplication tell you that the order and grouping of numbers do not matter in multiplication. The products for both Joan and Mike will be equal.
::然后,看看他们的方程。记住乘法的通和和关联性能告诉大家数字的顺序和组合在乘法中并不重要。琼和麦克的产品是平等的。

$\begin{aligned} (12.5 i n . \times 4 i n .) (5 i n .) = (4 i n . \times 5 i n .) (12.5 i n .) \end{aligned}$

Joan and Mike are both correct.
::琼和麦克都是对的

Next, multiply the three numbers to find the volume of the box.
::下一步,乘以三个数字以找到盒子的体积。

$\begin{aligned} \begin{array}{rcl} Volume & = & 12.5 i n . \times 4 i n . \times 5 i n . \\ Volume & = & 250 i n .^{3} \end{array} \end{aligned}$

The volume of the box is 250 cubic inches.
::盒子的体积是250立方英寸

Example 2
::例2

Identify the property.
::说明财产。

$\begin{aligned} 4 (3.67) = 3.67 (4) \end{aligned}$

The only thing that changed in this problem is the order of the values being multiplied.
::这个问题唯一改变的就是数值乘以的顺序。

This demonstrates the commutative property of multiplication.
::这表明乘法的通量特性。

Example 3
::例3

Identify the property.
::说明财产。

$\begin{aligned} 4.5 (5 a) = (4.5 \times 5) a \end{aligned}$

The order of the numbers is the same, but the groupings are different.
::数字的顺序相同,但分组不同。

This demonstrates the associative property of multiplication.
::这表明乘法的共同属性。

Example 4
::例4

Identify the property.
::说明财产。

$\begin{aligned} 6.7 (4) = 4 (6.7) \end{aligned}$

The order of the numbers being multiplied has changed.
::乘数的顺序已改变。

This demonstrates the commutative property of multiplication.
::这表明乘法的通量特性。

Example 5
::例5

Identify the property.
::说明财产。

$\begin{aligned} 5.4 a = a 5.4 \end{aligned}$

The order of the numbers being multiplied has changed.
::乘数的顺序已改变。

This demonstrates the commutative property of multiplication.
::这表明乘法的通量特性。

Review
::回顾

Identify the property illustrated in each problem.
::查明每个问题所说明的财产。

$\begin{aligned} 4.6 a = a 4.6 \end{aligned}$
::4.6 a = 4.6

$\begin{aligned} (4 a) (b) = 4 (a b) \end{aligned}$
:4 a) (b) = 4 (a) = 4 (b)

$\begin{aligned} (5.5 a) (c) = 5.5 (a c) \end{aligned}$
:5.5a) (c) = 5.5(a) = (c)

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

$\begin{aligned} 6 a b = a b (6) \end{aligned}$
::6 a b = a b (6 )

$\begin{aligned} 6 \times 4 = 4 \times 6 \end{aligned}$

$\begin{aligned} 5 (a b) = (5 a) \times b \end{aligned}$
::5 (a b) = (5 a) × b

$\begin{aligned} 7 (8 x) = (7 \times 8) x \end{aligned}$
::7 (8x) = (7x8) x

$\begin{aligned} 2 x y = 2 y x \end{aligned}$
::2xy=2yxx 2yxx

$\begin{aligned} 3 (4 a) = (3 \times 4) a \end{aligned}$
::3 (4 a) = (3 × 4) a

$\begin{aligned} 6 \times 7 \times 4 = 4 \times 7 \times 6 \end{aligned}$

$\begin{aligned} a b c = c a b \end{aligned}$
::a b c = c a b

$\begin{aligned} x y (a z) = x (y a z) \end{aligned}$
::x y( a z ) = x ( y a z )

$\begin{aligned} a b c d = d c a b \end{aligned}$
::a b b c d = d c a b

$\begin{aligned} 2 a (b c) = (2 a) b c \end{aligned}$
::2(a)(b)(c)=(2a)(b)c

Review (Answers)
::回顾(答复)

Section outline

Properties of Multiplying Decimals ::乘以 十进制的属性

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)

Properties of Multiplying Decimals
::乘以十进制的属性

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)