Section: 数值表达式求值 | 1.12数值表达式求值

Section outline

Boris is spending the day doing stock inventory at his dad’s fishing and bait store. Boris writes the following expression to represent the fresh bait in the window that day:
::鲍里斯正在他父亲的钓鱼和诱饵商店做股票盘点。鲍里斯写了以下的表达方式,以代表当天窗口中的新诱饵:

$\begin{aligned} 27 + 2 \times 5 - 3 + 8 + 7 = 49 \end{aligned}$

Boris’s dad tells him that all the numbers are correct, except that the total number of packages of fresh bait is only 33, not 49. If Boris has all the right numbers, that must mean he has made a mistake in his . Where can Boris insert " data-term="Parentheses" role="term" tabindex="0"> parentheses to get the right answer and make this a true expression?
::鲍里斯的爸爸告诉他,所有数字都是正确的,除了新诱饵的包件总数只有33个,而不是49个。如果鲍里斯拥有所有正确的号码,这必然意味着他犯了一个错误。鲍里斯在哪里能插入括号来得到正确的答案并把它变成真实的表达方式呢?

In this concept, you will learn how to use order of operations to identify and create numerical expressions that are true.
::在此概念中,您将学会如何使用操作顺序来识别和创建真实的数字表达式。

Evaluating Numerical Expressions
::评价数值表达式

Sometimes you will need to use the order of operations to determine whether or not a numerical expression is true. Sometimes inserting a grouping symbol, like parentheses, in the correct spot can help you to make an expression true.
::有时您需要使用操作顺序来确定数字表达式是否真实。有时,在正确位置插入括号等组合符号可以帮助您实现表达式的真实性。

Let's review the order of operations.
::我们来审查一下行动顺序

Order of Operations
::行动命令令

P - parentheses
::P - 括号括号

E - exponents
::E. 推举

MD - multiplication or division , in order from left to right
::MD - 乘数或除法,按从左到右顺序排列

AS - addition or subtraction , in order from left to right
::AS - 增加或减减,按从左到右顺序排列

Now let's consider an expression that is missing parentheses. You will need to insert parentheses into this expression in order to make it true.
::现在让我们来考虑一个缺少括号的表达式。您需要将括号插入到这个表达式中, 以使它成为事实。

$\begin{aligned} 4 + 2 \times 3 + 7 - 5 + 8^{2} = 76 \end{aligned}$

Since exponents and multiplication and division come before addition and subtraction, it makes sense that you will need to use parentheses to group some part of addition or subtraction so that this operation is completed first.
::由于引言、乘法和除法先于增减,所以有必要使用括号对部分增减进行分组,以便先完成这项工作。

Let's try inserting parentheses around $\begin{aligned} 7 - 5 \end{aligned}$ .
::让我们尝试在7 - 5周围插入括号。

$\begin{aligned} 4 + 2 \times 3 + (7 - 5) + 8^{2} = 76 \end{aligned}$

First, complete the operation in parentheses.
::首先,在括号中填写完成操作。

$\begin{aligned} 4 + 2 \times 3 + (7 - 5) + 8^{2} = & 76 Evaluate (7 - 5) = 2 \\ 4 + 2 \times 3 + 2 + 8^{2} = & 76 \end{aligned}$
Then, evaluate the exponent .

$\begin{aligned} 4 + 2 \times 3 + 2 + 8^{2} = & 76 Evaluate 8^{2} = 64 \\ 4 + 2 \times 3 + 2 + 64 = & 76 \end{aligned}$

Next, multiply.
::下一个,乘数。

$\begin{aligned} 4 + 2 \times 3 + 2 + 64 = & 76 Multiply 2 \times 3 = 6 \\ 4 + 6 + 2 + 64 = & 76 \end{aligned}$

Then, add and subtract in order from left to right.
::然后,你应当依次增加,然后,你应当依次加以减减减。

$\begin{aligned} 4 + 6 + 2 + 64 = & 76 Add 4 + 6 = 10 \\ 10 + 2 + 64 = & 76 Add 10 + 2 = 12 \\ 12 + 64 = & 76 Add 12 + 64 = 76 \\ 76 = & 76 This is true \end{aligned}$

The simplified equation is 76 = 76, which is true, so inserting parentheses around 7 - 5 makes this a true expression.
::简化的方程式是76=76, 确实如此, 所以在7 - 5前后插入括号, 使这个表达方式成为真实的表达方式。

Look at another expression that is missing parentheses.
::查看另一个缺少括号的表达式。

$\begin{aligned} 5 + 3 \times 2 + 7 - 1 \overset{?}{=} 22 \end{aligned}$

How do you know it's missing parentheses? Because if you evaluate this problem as it is, following order of operations, you get an untrue equation. Here is how that would look:
::你怎么知道它没有括号?因为如果你按操作顺序来评估这个问题, 你会得到一个不真实的方程式。这样看来:

$\begin{aligned} 5 + 3 \times 2 + 7 - 1 \overset{?}{=} & 22 \\ 5 + 6 + 7 - 1 = \\ 11 + 7 - 1 = \\ 18 - 1 \neq & 22 This is an untrue equation \end{aligned}$

Evaluating the operations in a different order will change the outcome, so try putting parentheses around 2 + 7.
::按不同顺序评估业务将改变结果,因此尝试在2+7周围加括号。

$\begin{aligned} 5 + 3 \times (2 + 7) - 1 \overset{?}{=} 22 \end{aligned}$

First, complete the addition in parentheses.
::首先,在括号中填写补充内容。

$\begin{aligned} 5 + 3 \times (2 + 7) - 1 \overset{?}{=} & 22 Evaluate (2 + 7) = 9 \\ 5 + 3 \times 9 - 1 \overset{?}{=} & 22 \end{aligned}$

Then, as there are no exponents, complete the multiplication.
::然后,因为没有表率,所以完成乘法。

$\begin{aligned} 5 + 3 \times 9 - 1 \overset{?}{=} & 22 Multiply 3 \times 9 = 27 \\ 5 + 27 - 1 \overset{?}{=} & 22 \end{aligned}$

Next, complete the addition and subtraction in order from left to right.
::下一步,按照从左到右的顺序完成增减。

$\begin{aligned} 5 + 27 - 1 = & 22 Add 5 + 27 = 32 \\ 32 - 1 = & 22 Subtract 32 - 1 = 31 \\ 31 \neq & 22 This is an untrue statement \end{aligned}$

The simplified equation 31 = 22 is clearly incorrect. Inserting parentheses around $\begin{aligned} 2 + 7 \end{aligned}$ does not make this a true expression.
::简化的等式 31 = 22 明显不正确。在 2 + 7 周围插入括号并不表示这是真实的表达方式。

Try putting the parentheses around 5 + 3.
::试着把括号放在5+3周围

$\begin{aligned} (5 + 3) \times 2 + 7 - 1 = 22 \end{aligned}$

First, complete the addition in parentheses.
::首先,在括号中填写补充内容。

$\begin{aligned} (5 + 3) \times 2 + 7 - 1 = & 22 Evaluate (5 + 3) = 8 \\ 8 \times 2 + 7 - 1 = & 22 \end{aligned}$

Then, there are no exponents, so complete the multiplication.
::那么,就没有推论者,所以完成乘法的乘法。

$\begin{aligned} 8 \times 2 + 7 - 1 = & 22 Multiply 8 \times 2 = 16 \\ 16 + 7 - 1 = & 22 \end{aligned}$

Finally, complete the addition and subtraction in order from left to right.
::最后,按照从左到右的顺序完成增减。

$\begin{aligned} 16 + 7 - 1 = & 22 Add 16 + 7 = 23 \\ 23 - 1 = & 22 Subtract 23 - 1 = 22 \\ 22 = & 22 This is true \end{aligned}$

The simplified equation is 22 = 22. Inserting parentheses around 5 + 3 makes this a true statement.
::简化的等式是22=22,加上括号5+3前后的括号使这一表述成为真实。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Boris and his bait expression.
::早些时候,你被问及鲍里斯和他的诱饵表情

Boris needs to add parentheses somewhere to make a true expression out of:
::鲍里斯需要在某处加括号来表达真实的表达方式:

$\begin{aligned} 27 + 2 \times 5 - 3 + 8 + 7 = 33 \end{aligned}$

First, try inserting parentheses around 27+2.
::首先,尝试在27+2周围插入括号。

$\begin{aligned} (27 + 2) \times 5 - 3 + 8 + 7 = & 33 \\ 29 \times 5 - 3 + 8 + 7 = & 33 \\ 145 - 3 + 8 + 7 = & 33 \\ 142 + 8 + 7 = & 33 \\ 157 \neq & 33 This is an untrue statement \end{aligned}$

Since 157 is not equal to 33, putting parentheses around 27 + 2 did not work.
::由于157不等于33,加括号27+2没有工作。

Next, try inserting parentheses around 3 + 8.
::接下来,请尝试在3+8周围插入括号。

$\begin{aligned} 27 + 2 \times 5 - (3 + 8) + 7 = & 33 \\ 27 + 2 \times 5 - 11 + 7 = & 33 \\ 27 + 10 - 11 + 7 = & 33 \\ 37 - 11 + 7 = & 33 \\ 26 + 7 = & 33 \\ 33 = & 33 This is true. \end{aligned}$

The answer is $\begin{aligned} 27 + 2 \times 5 - (3 + 8) + 7 = 33 \end{aligned}$ .
::答案是27+2×5-(3+8)+7=33。

Boris now has a true expression for the total number of packages of fresh bait in the window.
::鲍里斯现在有一个真实的表达窗口中所有新诱饵的总数。

Example 2
::例2

Determine whether this is a true expression.
::确定此表达式是否为真实表达式。

$\begin{aligned} 7 \times (4 + 1) - 7 \times 2 = 21 \end{aligned}$

First, evaluate the parentheses.
::首先,评价括号。

$\begin{aligned} 7 \times (4 + 1) - 7 \times 2 = & 21 Evaluate (4 + 1) = 5 \\ 7 \times 5 - 7 \times 2 = & 21 \end{aligned}$

Next, there are no exponents to evaluate, so multiply.
::接下来,没有要评估的推手, 所以乘以倍数。

$\begin{aligned} 7 \times 5 - 7 \times 2 = & 21 Multiply 7 \times 5 = 35 \\ 35 - 7 \times 2 = & 21 Multiply 7 \times 2 = 14 \\ 35 - 14 = & 21 \end{aligned}$

Finally, subtract.
::最后,减去。

$\begin{aligned} 35 - 14 = & 21 Subtract 35 - 14 = 21 \\ 21 = & 21 This is true \end{aligned}$

The answer is "The expression is true."
::答案是"表达的确是真实的"

Example 3
::例3

$\begin{aligned} 3^{2} + 4^{2} - 9 + (3 \times 2) = 22 \end{aligned}$
First, evaluate the parentheses.

$\begin{aligned} 3^{2} + 4^{2} - 9 + (3 \times 2) = & 22 Evaluate (3 \times 2) = 6 \\ 3^{2} + 4^{2} - 9 + 6 = & 22 \end{aligned}$
Next, consider the exponents.

$\begin{aligned} 3^{2} + 4^{2} - 9 + 6 = & 22 Evaluate 3^{2} = 9 \\ 9 + 4^{2} - 9 + 6 = & 22 Evaluate 4^{2} = 16 \\ 9 + 16 - 9 + 6 = & 22 \end{aligned}$

Finally, as there is no multiplication or division, complete addition and subtraction from left to right.
::最后,由于没有乘法或除法,完全加法和从左向右减法。

$\begin{aligned} 9 + 16 - 9 + 6 = & 22 Add 9 + 16 = 25 \\ 25 - 9 + 6 = & 22 Subtract 25 - 9 = 16 \\ 16 + 6 = & 22 Add 16 + 6 = 22 \\ 22 = & 22 This is true \end{aligned}$

The answer is "The expression is true."
::答案是"表达的确是真实的"

Example 4
::例4

Determine if the following is a true expression.
::确定以下是否为真实表达式。

$\begin{aligned} 6 + 3 \times 2 - 5 + (7 - 1) = 19 \end{aligned}$

First, evaluate the parentheses.
::首先,评价括号。

$\begin{aligned} 6 + 3 \times 2 - 5 + (7 - 1) = & 19 Evaluate (7 - 1) = 6 \\ 6 + 3 \times 2 - 5 + 6 = & 19 \end{aligned}$

Next, there are no exponents to evaluate, so multiply.
::接下来,没有要评估的推手, 所以乘以倍数。

$\begin{aligned} 6 + 3 \times 2 - 5 + 6 = & 19 Multiply 3 \times 2 = 6 \\ 6 + 6 - 5 + 6 = & 19 \end{aligned}$

Then, complete the addition and subtraction in order from left to right.
::然后,完成增量和减量,以便从左到右。

$\begin{aligned} 6 + 6 - 5 + 6 = & 19 Add 6 + 6 = 12 \\ 12 - 5 + 6 = & 19 Subtract 12 - 5 = 7 \\ 7 + 6 = & 19 Add 7 + 6 = 13 \\ 13 \neq & 19 This is not a true statement \end{aligned}$

The answer is "This expression is not true."
::答案是"这个表达方式不是真的"

Review
::回顾

Check each answer using order of operations. Write whether the answer is true or false.
::使用操作顺序检查每个答案。写入答案是真实的还是虚假的。

$\begin{aligned} 4 + 5 \times 2 + 8 - 7 = 15 \end{aligned}$

$\begin{aligned} 4 + 3 \times 9 + 6 - 10 = 104 \end{aligned}$

$\begin{aligned} 6 + 2^{2} \times 4 + 3 \times 6 = 150 \end{aligned}$

$\begin{aligned} 3 + 6 \times 3 + 9 \times 7 - 18 = 66 \end{aligned}$

$\begin{aligned} 7 \times 2^{3} + 4 - 9 \times 3 - 8 = 25 \end{aligned}$

$\begin{aligned} 2 \times 3^{3} + 7 \times 3 = 183 \end{aligned}$

$\begin{aligned} 4 \times 3 + 4^{2} - 9 + 8 = 25 \end{aligned}$

$\begin{aligned} 3^{2} \times 2^{3} + 14 - 9 = 77 \end{aligned}$

$\begin{aligned} 6 \times 3^{3} - 24 + 19 \times 2 - 4 = 310 \end{aligned}$

$\begin{aligned} 5 \times 2 + 5^{2} - 10 \times 2 - 7 = 8 \end{aligned}$

Insert parentheses to make each expression true.
::插入括号以使每个表达式都变为真实。

$\begin{aligned} 4 + 5 - 2 + 3 - 2 = 8 \end{aligned}$

$\begin{aligned} 2 + 3 \times 2 - 4 = 6 \end{aligned}$

$\begin{aligned} 1 + 9 \times 4 \times 3 + 2 - 1 = 180 \end{aligned}$

$\begin{aligned} 7 + 4 \times 3 - 5 \times 2 = 23 \end{aligned}$

$\begin{aligned} 2^{2} + 5 \times 8 - 3 + 4 = 33 \end{aligned}$

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Resources
::资源

Section outline

Evaluating Numerical Expressions ::评价数值表达式

Order of Operations ::行动命令令

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Evaluating Numerical Expressions
::评价数值表达式

Order of Operations
::行动命令令

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源