Course: 2.6小数加法的结合律和交换律

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小数加法的结合律和交换律
Jeff’s dad works at an architectural firm in Boston. He has the same commute to work every day. First he drives 3.4 miles to the train station. Then he takes the train for 15.6 miles into the city. He’s lucky because the train station is right next to his office. To get home at the end of the day, Jeff’s dad does the reverse. How many miles does Jeff’s dad travel on his way home from work?
::杰夫的爸爸在波士顿的一家建筑公司工作,他每天有相同的通勤工作。首先,他开车3.4英里到火车站。然后,他坐火车15.6英里到城里。他很幸运,因为火车站就在办公室旁边。到了最后,杰夫的爸爸却反其道而行之。杰夫的爸爸在下班回家的路上走了几英里?

In this concept, you will learn to identify and use the commutative and associative properties of addition with decimals.
::在此概念中,您将学会识别和使用与小数点相加的通量属性和关联属性。

Using Commutative and Associative Properties of Addition with Decimals
::使用与十进制相加的通和相联属性

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 $\begin{aligned} b \end{aligned}$ :
::增加部分的通融性财产规定,在找到一笔金额时,改变增加部分的顺序不会改变其金额。

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

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

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

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

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

Next, find $\begin{aligned} 4 + 2 \end{aligned}$ .
::下一步,发现4+2。

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

Notice that both sums are 6.
::通知两笔金额均为6美元。

The answer is that because both $\begin{aligned} 2 + 4 \end{aligned}$ and $\begin{aligned} 4 + 2 \end{aligned}$ are equal to 6, they are equal to each other. $\begin{aligned} 2 + 4 = 4 + 2 \end{aligned}$ .
::答案是,因为2+4和4+2都等于6,它们彼此平等。 2+4=4+2。

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 $\begin{aligned} a, b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ :
::附加产权协会指出,在寻找一笔金额时,改变添加方的组合方式不会改变其总和。在符号中,添加的关联属性表示,对于数字a、b和c,a、b和c:

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

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

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

First find $\begin{aligned} (2 + 5) + 6 \end{aligned}$ . Start by adding the numbers in " data-term="Parentheses" role="term" tabindex="0"> parentheses . Then add the result with 6.
::第一个查找 (2+ 5) + 6 + 6。开始在括号中添加数字。然后将结果加上 6 。

$\begin{aligned} \begin{array}{rcl} (2 + 5) + 6 & = & 7 + 6 \\ = & 13 \end{array} \end{aligned}$

Next, find $\begin{aligned} 2 + (5 + 6) \end{aligned}$ . Again, start by adding the numbers in parentheses. Then add 2 to the result.
::下一步,请查找 2+ (5+ 6) 。再次,首先在括号中添加数字。然后在结果中添加2。

$\begin{aligned} \begin{array}{rcl} 2 + (5 + 6) & = & 2 + 11 \\ = & 13 \end{array} \end{aligned}$

Notice that both sums are 13.
::通知两笔金额均为13。

The answer is that because both $\begin{aligned} (2 + 5) + 6 \end{aligned}$ and $\begin{aligned} 2 + (5 + 6) \end{aligned}$ are equal to 13, they are equal to each other. $\begin{aligned} (2 + 5) + 6 = 2 + (5 + 6) \end{aligned}$ .
::答案是,由于两个(2+5)+6和2+(5+6)都等于13,它们彼此平等。 (2+5)+6=2+(5+6)

Both the commutative property of addition and the associative property of addition can be useful in solving equations .
::增加的通货财产和增加的连带财产都可用于解决等式问题。

Here is an example.
::举一个例子。

Find the value of $\begin{aligned} x \end{aligned}$ in the following equation .
::在以下方程式中查找 x 值。

$\begin{aligned} 71.321 + 42.29 = x + 71.321 \end{aligned}$
::71.321 + 42.29 = x + 71.321

First, notice that both sides of the equation are the sum of two values. Also, both sides of the equation include the number 71.321 as one of the terms .
::首先,请注意,等式的两面是两个数值的和和。此外,等式的两面包括71.321号,作为条件之一。

By the commutative property of addition, the second term must also be the same on both sides of the equation. $\begin{aligned} x \end{aligned}$ must be equal to 42.29.
::通过增加的通量财产,第二个任期在等式的两侧也必须相同。 x 必须等于42.29。

$\begin{aligned} 71.321 + 42.29 = 42.29 + 71.321 \end{aligned}$

The answer is $\begin{aligned} x = 42.29 \end{aligned}$ .
::答案是x=42.29。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Jeff and his dad’s commute home from work.
::更早之前, 杰夫和他父亲下班后返家时,

To get to work, Jeff’s dad drives for 3.4 miles and then takes the train for 15.6 miles. To get home, he does the reverse. Jeff wants to know how many miles his dad travels to get home.
::为了上班,杰夫的爸爸驾驶了3.4英里,然后坐了15.6英里的火车回家。为了回家,他采取了相反的做法。杰夫想知道他父亲为了回家旅行了几英里。

First, figure out how many miles Jeff’s dad travels to get to work. You know that to get to work Jeff’s dad travels 3.4 miles plus 15.6 more miles.
::首先,找出Jeff的爸爸去上班的路程。你知道Jeff的爸爸去上班的路程是3.4英里加15.6英里。

$\begin{aligned} 3.4 + 15.6 = 19 \end{aligned}$

This means Jeff’s dad travels 19 miles to get to work.
::这表示Jeff的爸爸要走19英里的路去上班。

Now, to get home Jeff’s dad travels in reverse. By the commutative property of addition, you know that changing the order that you add two numbers won’t change the answer.
::现在,为了让杰夫的爸爸回到家,他父亲的旅程会倒转。通过附加的通货性能,你知道改变你添加两个数字的顺序不会改变答案。

$\begin{aligned} 3.4 + 15.6 = 15.6 + 3.4 \end{aligned}$

Therefore , Jeff’s dad must travel 19 miles to get home from work just like he travels 19 miles to get to work.
::因此,Jeff的父亲必须行走19英里才能下班回家,就像他行走19英里才能上班一样。

The answer is Jeff’s dad travels 19 miles to get home from work.
::答案是Jeff的爸爸去19英里的路程下班回家。

Example 2
::例2

Find the value of $\begin{aligned} x \end{aligned}$ in the following equation by using properties of addition.
::使用添加的属性查找以下方程式中的 x 值。

$\begin{aligned} 0.14 + 2.58 = (x + 1.45) + 1.13 \end{aligned}$
::0.14 + 2.58 = (x + 1.45) + 1.13

First, notice that the right side of the equation has two addends grouped together. Use the associative property of addition to rewrite the right side of the equation by changing the way the addends are grouped.
::首先, 请注意方程右侧有两个添加端组合在一起。使用添加的关联属性重写方程右侧, 改变添加端组合的方式。

$\begin{aligned} 0.14 + 2.58 = x + (1.45 + 1.13) \end{aligned}$

Now, combine the terms in parentheses on the right side of the equation.
::现在,将方程右侧括号中的术语合并在一起。

$\begin{aligned} 0.14 + 2.58 = x + 2.58 \end{aligned}$

Next, notice that both sides of the equation now have two terms that are added together. Also, both sides of the equation have a 2.58 as one of the terms. This means the second terms on each side of the equation must be equal. $\begin{aligned} x \end{aligned}$ must be equal to 0.14.
::其次,请注意,等式的两边现在都有两个条件相加。另外, 等式的两边都有2.58个条件之一。这意味着等式两边的第二个条件必须是相等的。 x 必须等于0.14 。

The answer is $\begin{aligned} x = 0.14 \end{aligned}$ .
::答案是x=0.14。

Example 3
::例3

Identify which property is shown below.
::下文列出哪些财产。

$\begin{aligned} 3.4 + 5.6 = 5.6 + 3.4 \end{aligned}$

First, notice that both sides of the equation have the same two addends, but the order has been changed. This is an example of the commutative property of addition.
::首先,请注意方程的两侧都有相同的两个添加端, 但命令已经更改。这是通货性附加属性的一个例子。

The answer is the commutative property of addition.
::答案是增加的通货性能。

Example 4
::例4

Identify which property is shown below.
::下文列出哪些财产。

$\begin{aligned} (3.4 + 3.2) + 5.6 = 3.4 + (3.2 + 5.6) \end{aligned}$

First, notice that both sides of the equation have the same three addends in the same order, but the grouping has been changed. This is an example of the associative property of addition.
::首先,请注意方程的两侧都有相同的三端,顺序相同,但组合已经改变。这是附加连带属性的例子。

The answer is the associative property of addition.
::答案是附加的连带属性。

Example 5
::例5

Find the value of $\begin{aligned} x \end{aligned}$ in the following equation by using properties of addition.
::使用添加的属性查找以下方程式中的 x 值。

$\begin{aligned} 1.94 + 0.38 = x + 1.94 \end{aligned}$
::1.94 + 0.38 = x + 1.94

First, notice that both sides of the equation are the sum of two values. Also, both sides of the equation include the number 1.94 as one of the terms.
::首先,请注意方程的两面是两个数值的和。此外,方程的两面包括1.94号,作为条件之一。

By the commutative property of addition, the second term must also be the same on both sides of the equation. $\begin{aligned} x \end{aligned}$ must be equal to 0.38.
::通过加法的通量财产,第二个任期在等式的两侧也必须相同。 x 必须等于0.38。

$\begin{aligned} 1.94 + 0.38 = 0.38 + 1.94 \end{aligned}$

The answer is $\begin{aligned} x = 0.38 \end{aligned}$ .
::答案是x=0.38。

Review
::回顾

Identify the property illustrated in each equation.
::查明每个方程式所显示的财产。

$\begin{aligned} 4.5 + (x + y) + 2.6 = (4.5 + x) + y + 2.6 \end{aligned}$
::4.5+4.5+(x+y)+2.6=(4.5+x)+y+2.6

$\begin{aligned} 3.2 + x + y + 5.6 = x + 3.2 + y + 5.6 \end{aligned}$
::3.2+xx+y+y+5.6=x+3.2+y+5.6

$\begin{aligned} 1.5 + (2.3 + y) + 5.6 = (1.5 + 2.3) + y + 5.6 \end{aligned}$
::1.5+(2.3+y)+5.6=(1.5+2.3)+y+5.6

$\begin{aligned} 3.2 + 5.6 + 1.3 + 2.6 = 3.2 + 2.6 + 5.6 + 1.3 \end{aligned}$

$\begin{aligned} 4.5 + 15.6 = 15.6 + 4.5 \end{aligned}$

$\begin{aligned} (x + y) + 5.6 = x + (y + 5.6) \end{aligned}$
:x+y)+5.6=x+(y+5.6)

$\begin{aligned} 17.5 + 18.9 + 2 = 2 + 17.5 + 18.9 \end{aligned}$

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

Find the value of $\begin{aligned} x \end{aligned}$ in the following equations by using properties of addition.
::使用附加属性在以下方程式中查找 x 值。

$\begin{aligned} .8603 + .292 = x + .8603 \end{aligned}$
::8603+.292=x+.8603

$\begin{aligned} (2.65 + x) + 19.35 = 22 + 2.115 \end{aligned}$
:2.65+xx)+19.35=22+2.115

$\begin{aligned} .306 + 1.076 = (.782 + x) + .294 \end{aligned}$
::306+1.076=(782+xx)+294

$\begin{aligned} 6.174 + 76.41 = 76.41 + x \end{aligned}$
::6.174 + 76.41 = 76.41 + x

Solve each problem.
::解决每一个问题。

Lamont measures the amount of water he drinks. His results for four consecutive days were as follows: 0.6 liters, 0.72 liters, 0.84 liters, 0.96 liters. If the pattern continues, how much water will Lamont drink on the fifth day?
::拉蒙特测量他喝的水量,连续四天的结果如下:0.6升,0.72升,0.84升,0.96升。如果这种模式继续下去,第五天拉蒙特将喝多少水?

Barbara is starting a jewelry making enterprise. At the supply store, she spent $19.19 on beads and $6.81 on wire and clasps. If she left the store with $24.50, how much did she start with?
::芭芭拉正在创办一家珠宝制作企业。在供货店,她花了19.19美元用于珠子,6.81美元用于铁丝和扣锁。如果她离开商店时有24.50美元,她从多少开始呢?

A rectangular playground has a length of 18.36 yards and a width of 12.24 yards. What is the perimeter of the playground?
::矩形操场长18.36码,宽12.24码。

Review (Answers)
::回顾(答复)

Section outline

General

小数加法的结合律和交换律

Using Commutative and Associative Properties of Addition with Decimals ::使用与十进制相加的通和相联属性

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 Decimals
::使用与十进制相加的通和相联属性

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)