Course: 1.6 模式和表达式

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模式和表达式

Patterns and Equations
::模式和等价

In mathematics, and especially in algebra, we look for patterns in the numbers we see. The tools of algebra help us describe these patterns with words and with equations (formulas or functions). An equation is a mathematical recipe that gives the value of one variable in terms of another.
::在数学中,特别是在代数中,我们寻找我们所看到的数字模式。代数工具帮助我们用文字和方程(公式或函数)来描述这些模式。方程式是一种数学配方,从另一个变量的角度给出一个变量的价值。

For example, if a theme park charges $12 admission, then the number of people who enter the park every day and the amount of money taken in by the ticket office are related mathematically, and we can write a rule to find the amount of money taken in by the ticket office.
::例如,如果一个主题公园收取12美元的入场费,那么每天进入公园的人数和入场券办公室所收钱的数额,在数学上是相关的,我们可以写一条规则,找到入场券办公室所收钱的数额。

In words, we might say “The amount of money taken in is equal to twelve times the number of people who enter the park.”
::换句话说,我们可以说,“所收钱的金额相当于进入公园人数的12倍。”

We could also make a table. The following table relates the number of people who visit the park and the total money taken in by the ticket office.
::下表列出参观公园的人数和票务办公室收取的钱总额。

\begin{aligned} Number of visitors 1 2 3 4 5 6 7 \\ Money taken in ($) 12 24 36 48 60 72 84 \end{aligned}

::12 24 36 48 60 72 84

Clearly, we would need a big table to cope with a busy day in the middle of a school vacation!
::显然,我们需要一张大桌子来应付一个忙碌的一天在学校假期中间!

A third way we might relate the two quantities (visitors and money) is with a graph. If we plot the money taken in on the vertical axis and the number of visitors on the horizontal axis , then we would have a graph that looks like the one shown below. Note that this graph shows a smooth line that includes non-whole number values of $\begin{aligned} x \end{aligned}$ (e.g. $\begin{aligned} x = 2.5 \end{aligned}$ ). In real life this would not make sense, because fractions of people can’t visit a park. This is an issue of domain and range , something we will talk about later.
::第三,我们可以将这两个数量(访客和金钱)与图表联系起来。如果我们在垂直轴和水平轴上访问者人数上图出钱,那么我们就会有一个类似于下文所示的图表。请注意,这个图表显示了一条包括非全数x值(如x=2.5)的平滑线。在现实生活中,这没有意义,因为人们无法访问公园的分数。这是一个域和范围问题,我们以后会讨论这个问题。

The method we will examine in detail in this lesson is closer to the first way we chose to describe the relationship. In words we said that “The amount of money taken in is twelve times the number of people who enter the park.” In mathematical terms we can describe this sort of relationship with variables . A variable is a letter used to represent an unknown quantity. We can see the beginning of a mathematical formula in the words:
::我们将在这一教训中详细研究的方法更接近于我们首选描述这种关系的方法。用我们的话说,“钱的金额是进入公园人数的12倍。”用数学术语,我们可以描述这种与变量的关系。变量是用来代表未知数量的字母。我们可以看到数学公式的开头语是:

The amount of money taken in is twelve times the number of people who enter the park.
::收入是进入公园人数的12倍

This can be translated to:
::这可以翻译为:

\begin{aligned} The amount of money taken in = 12 \times (the number of people who enter the park) \end{aligned}

::12x(进入公园的人数)

We can now see which quantities can be assigned to letters. First we must state which letters (or variables ) relate to which quantities. We call this defining the variables :
::我们现在可以看到哪些数量可以指定给字母。首先我们必须说明哪些字母( 或变量) 与哪些数量相关。我们称之为定义变量 :

Let $\begin{aligned} x = \end{aligned}$ the number of people who enter the theme park.
::x=进入主题公园的人数。

Let $\begin{aligned} y = \end{aligned}$ the total amount of money taken in at the ticket office.
::买票办公室收的钱总额吧

We now have a fourth way to describe the relationship: with an algebraic equation .
::我们现在有第四种方式来描述这种关系:代数方程式。

\begin{aligned} y = 12 x \end{aligned}

::y=12x y=12x

Writing a mathematical equation using variables is very convenient. You can perform all of the operations necessary to solve this problem without having to write out the known and unknown quantities over and over again. At the end of the problem, you just need to remember which quantities $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ represent.
::使用变量写入数学方程式非常方便。您可以执行解决这个问题的所有必要操作, 而不必一次又一次地写出已知和未知的数量。在问题结束时, 您只需要记住哪些数量 x 和 y 代表。

Wr ite an Equation
::写一个方程式

'Equation' is a term used to describe a collection of numbers and variables related through mathematical operators . An algebraic equation will contain letters that represent real quantities. For example, if we wanted to use the algebraic equation in the example above to find the money taken in for a certain number of visitors, we would substitute that number for $\begin{aligned} x \end{aligned}$ and then solve the resulting equation for $\begin{aligned} y \end{aligned}$ .
::“ 等式” 是一个术语, 用来描述通过数学运算符收集的数字和变量。代数方程式将包含代表真实数量的字母。例如, 如果我们想使用上述示例中的代数方程式来找到一定数量的访问者所收钱, 我们可以用这个数字来代替 x, 然后解答 y 的等式。

Writing and Solving an Equation
::书写和解决方程式

A theme park charges $12 entry to visitors. Find the money taken in if 1296 people visit the park.
::一个主题公园向参观者收取12美元的入境费。如果有1296人参观公园,请查查所收的钱。

Let’s break the solution to this problem down into steps. This will be a useful strategy for all the problems in this lesson.
::让我们把这一问题的解决方案化为步骤。这将是解决这一教训中所有问题的有用策略。

Step 1: Extract the important information.
::第1步:提取重要信息。

\begin{aligned} (number of dollars taken in) = 12 \times (number of visitors) \\ (number of visitors) = 1296 \end{aligned}

::=12xx(游客人数(游客人数)=1296

Step 2: Translate into a mathematical equation. To do this, we pick variables to stand for the numbers.
::步骤2: 转换成数学等式。要做到这一点, 我们选择变量来表示数字。

\begin{aligned} Let y = (number of dollars taken in) . \\ Let x = (number of visitors) . \end{aligned}

::Let y = (美元数取自) Let x = (访问者数) 。 Let x = (访问者数) 。

\begin{aligned} (number of dollars taken in) & = 12 \times (number of visitors) \\ y & = 12 \times x \end{aligned}

以美元计数)=12x(来访者人数)y=12xx

Step 3: Substitute in any known values for the variables.
::第3步:替代变量的任何已知值。

\begin{aligned} y & = 12 \times x \\ x & = 1296 \\ y & = 12 \times 1296 \end{aligned}

::y=12xxxxx=1296y=12x1296

Step 4: Solve the equation.
::步骤4:解决方程式问题。

\begin{aligned} y = 12 \times 1, 296 = 15, 552 \end{aligned}

::y=12x1,296=15,552

The amount of money taken in is $15,552.
::所收金额是15 552美元。

Step 5: Check the result.
::第五步:检查结果。

If $15,552 is taken at the ticket office and tickets are $12, then we can divide the total amount of money collected by the price per individual ticket.
::如果15 552美元被送到售票处,12美元被罚单,那么,我们可以将每张单张罚单收取的钱款总额除以每张单张罚单的价格。

\begin{aligned} (number of people) = \frac{15552}{12} = 1296 \end{aligned}

人数)=155521212=1296

1296 is indeed the number of people who entered the park. The answer checks out.
::1296确实是进入公园的人数答案是肯定的

Writing/Solving an Equation to Describe a Relationship
::书写/理清描述关系的等同

The following table shows the relationship between two quantities. First, write an equation that describes the relationship. Then, find the value of $\begin{aligned} b \end{aligned}$ when $\begin{aligned} a \end{aligned}$ is 750.
::下表显示两个数量之间的关系。首先, 写入一个描述此关系方程式的方程式。然后, 当 a 为 750 时, 找到 b 的值。

\begin{aligned} a 0 10 20 30 40 50 \\ b 20 40 60 80 100 120 \end{aligned}

::a0 102030 40 50b20406080100120

Step 1: Extract the important information.
::第1步:提取重要信息。

We can see from the table that every time $\begin{aligned} a \end{aligned}$ increases by 10, $\begin{aligned} b \end{aligned}$ increases by 20. However, $\begin{aligned} b \end{aligned}$ is not simply twice the value of $\begin{aligned} a . \end{aligned}$ We can see that when $\begin{aligned} a = 0, b = 20, \end{aligned}$ and this gives a clue as to what rule the pattern follows. The rule linking $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ is:
::从表中可以看出,每次增加10,b 增加20,b 增加10,b 增加20。然而,b 并不是a值的两倍。我们可以看到当a=0,b=20时,我们可以看到,这提供了一种线索,说明该模式遵循什么规则。

“To find $\begin{aligned} b, \end{aligned}$ double the value of $\begin{aligned} a, \end{aligned}$ then add 20.”
::“为了找到b,将a的值翻一番,再加20。”

Step 2: Translate into a mathematical equation:
::第2步:转换成数学公式:

Text	Translates to	Mathematical Expression
“To find $\begin{aligned} b \end{aligned}$ ”	$\begin{aligned} \to \end{aligned}$	$\begin{aligned} b = \end{aligned}$
“double the value of $\begin{aligned} a \end{aligned}$ ”	$\begin{aligned} \to \end{aligned}$	$\begin{aligned} 2 a \end{aligned}$
“add 20”	$\begin{aligned} \to \end{aligned}$	+ 20

Our equation is $\begin{aligned} b = 2 a + 20. \end{aligned}$
::我们的方程式是b=2a+20

Step 3: Solve the equation.
::第三步解决方程式

The original problem asks for the value of $\begin{aligned} b \end{aligned}$ when $\begin{aligned} a \end{aligned}$ is 750. When $\begin{aligned} a \end{aligned}$ is 750, $\begin{aligned} b = 2 a + 20 \end{aligned}$ becomes $\begin{aligned} b = 2 (750) + 20. \end{aligned}$ Following the , we get:
::当 a 是 750 时, 原始问题要求的值为 b。当 a 是 750 时, b= 2a+20 变成 b= 2( 750) +20。之后, 我们得到 :

\begin{aligned} b & = 2 (750) + 20 \\ = 1500 + 20 \\ = 1520 \end{aligned}

::b=2(750)+20=1500+20=1520

Step 4: Check the result.
::第四步:检查结果。

In some cases you can check the result by plugging it back into the original equation. Other times you must simply double-check your math. In either case, checking your answer is always a good idea. In this case, we can plug our answer for $\begin{aligned} b \end{aligned}$ into the equation, along with the value for $\begin{aligned} a, \end{aligned}$ and see what comes out. $\begin{aligned} 1520 = 2 (750) + 20 \end{aligned}$ is TRUE because both sides of the equation are equal. A true statement means that the answer checks out.
::在某些情况下,您可以通过将其插回原始方程式来检查结果。有时您必须简单重复检查您的数学。在任一情况下, 检查您的答案总是一个好主意。在这种情况下, 我们可以将b的答案插入方程式, 加上一个值, 并查看结果。 1520=2( 750)+20是 TRUE, 因为方程式的两边是相等的。真实的语句意味着答案检查出来。

Solve Problems Using Equations
::使用等同的解决问题

Let’s solve the following real-world problem by using the given information to write a mathematical equation that can be solved for a solution.
::让我们用给定的信息来写一个数学方程式, 解决以下现实世界的问题。

A group of students are in a room. After 25 students leave, it is found that $\begin{aligned} \frac{2}{3} \end{aligned}$ of the original group is left in the room. How many students were in the room at the start?
::25名学生离开后,发现原组中的23名学生留在房间里。开始时有多少学生在房间里?

Step 1: Extract the important information
::第1步:摘录重要信息

We know that 25 students leave the room.
::我们知道有25名学生离开房间。

We know that $\begin{aligned} \frac{2}{3} \end{aligned}$ of the original number of students are left in the room.
::我们知道,原来学生人数的23人留在会议室里。

We need to find how many students were in the room at the start.
::开始的时候我们需要找到有多少学生在教室里

Step 2: Translate into a mathematical equation. Initially we have an unknown number of students in the room. We can refer to this as the original number.
::步骤2: 转换成数学方程。最初我们房间里的学生人数不详, 我们可以称之为原数。

Let’s define the variable $\begin{aligned} x \end{aligned}$ as the original number of students in the room. After 25 students leave the room, the number of students in the room is $\begin{aligned} x - 25 \end{aligned}$ . We also know that the number of students left is $\begin{aligned} \frac{2}{3} \end{aligned}$ of $\begin{aligned} x . \end{aligned}$ So we have two expressions for the number of students left, and those two expressions are equal because they represent the same number. That means our equation is:
::让我们将变数x定义为室内学生的最初人数。 25名学生离开房间后,室内学生人数为x- 25。我们还知道剩下的学生人数是x的23。因此,我们有两种剩余学生人数的表达方式,而这两个表达方式是相同的,因为它们代表相同的人数。这意味着我们的等式是:

\begin{aligned} \frac{2}{3} x = x - 25 \end{aligned}

::23x=x-25 23x=x-25

Step 3: Solve the equation.
::第三步解决方程式

Add 25 to both sides.
::双方增加25个。

\begin{aligned} x - 25 & = \frac{2}{3} x \\ x - 25 + 25 & = \frac{2}{3} x + 25 \\ x & = \frac{2}{3} x + 25 \end{aligned}

::x- 25= 23xx- 25+25= 23x+25x= 23xx= 23x+25

Subtract $\begin{aligned} \frac{2}{3} x \end{aligned}$ from both sides.
::从两边减23x

\begin{aligned} x - \frac{2}{3} x & = \frac{2}{3} x - \frac{2}{3} x + 25 \\ \frac{1}{3} x & = 25 \end{aligned}

::x-23x=23x-23x+2513x=25

Multiply both sides by 3.
::将两边乘以 3 。

\begin{aligned} 3 \cdot \frac{1}{3} x & = 3 \cdot 25 \\ x & = 75 \end{aligned}

::313x=325x=75

Remember that $\begin{aligned} x \end{aligned}$ represents the original number of students in the room. So there were 75 students in the room to start with.
::记住 x 代表了房间里学生的最初人数。因此这里有75名学生需要开始学习。

Step 4: Check the answer:
::第4步:检查答案:

If we start with 75 students in the room and 25 of them leave, then there are $\begin{aligned} 75 - 25 = 50 \end{aligned}$ students left in the room.
::如果我们从房间的75名学生开始,其中25名学生离开,那么房间就只剩下75-25=50名学生。

$\begin{aligned} \frac{2}{3} \end{aligned}$ of the original number is $\begin{aligned} \frac{2}{3} \cdot 75 = 50 \end{aligned}$ .
::原号为23-75=50的23。

This means that the number of students who are left over equals $\begin{aligned} \frac{2}{3} \end{aligned}$ of the original number. The answer checks out .
::这意味着留学生人数等于原人数的23人。

The method of defining variables and writing a mathematical equation is the method you will use the most in an algebra course. This method is often used together with other techniques such as making a table of values, creating a graph, drawing a diagram and looking for a pattern.
::定义变量和写入数学方程式的方法,是代数课程中使用最多的方法。这个方法通常与其他技术一起使用,例如制作数值表、创建图表、绘制图表和寻找模式。

Write a Verbal Equation
::a 写音量方程式。

In the examples above, we had a rule, written in words, and from that developed an algebraic equation. In the following example, we will develop a verbal equation based on a table, and use that to write an algebraic equation.
::在以上的例子中,我们有一个用文字写成的规则,以及从发展代数方程式中得出的规则。在下面的例子中,我们将在表格的基础上发展一个口头方程式,用它来写代数方程式。

The following table shows the values of two related quantities. Write an equation that describes the relationship mathematically.
::下表显示两个相关数量的值。写入一个数学描述关系的方程式。

$\begin{aligned} x - \end{aligned}$ value	$\begin{aligned} y - \end{aligned}$ value
-2	10
0	0
2	-10
4	-20
6	-30

Step 1: Extract the important information.
::第1步:提取重要信息。

We can see from the table that $\begin{aligned} y \end{aligned}$ is five times bigger than $\begin{aligned} x \end{aligned}$ . The value for $\begin{aligned} y \end{aligned}$ is negative when $\begin{aligned} x \end{aligned}$ is positive, and it is positive when $\begin{aligned} x \end{aligned}$ is negative. Here is the rule that links $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ :
::从表格中可以看出, Y 是 x 的五倍。 x 是正的, y 的值是负的, x 是正的。这是连接 x 和 y 的规则 :

“ $\begin{aligned} y \end{aligned}$ is the negative of five times the value of $\begin{aligned} x \end{aligned}$ ”
::“y是x值的五倍负值”

Step 2: Translate this statement into a mathematical equation.
::第2步:将该语句转换成数学公式。

Text	Translates to	Mathematical Expression
“ $\begin{aligned} y \end{aligned}$ is”	$\begin{aligned} \to \end{aligned}$	$\begin{aligned} y = \end{aligned}$
“negative 5 times the value of $\begin{aligned} x \end{aligned}$ ”	$\begin{aligned} \to \end{aligned}$	$\begin{aligned} - 5 x \end{aligned}$

Our equation is $\begin{aligned} y = - 5 x \end{aligned}$ .
::我们的方程式是5x

Step 3: There is nothing in this problem to solve for. We can move to Step 4.
::步骤3:在这个问题上没有任何问题需要解决,我们可以转向步骤4。

Step 4: Check the result.
::第四步:检查结果。

In this case, the way we would check our answer is to use the equation to generate our own $\begin{aligned} x, y \end{aligned}$ pairs. If they match the values in the table, then we know our equation is correct. Substitute -2, 0, 2, 4, and 6 for $\begin{aligned} x \end{aligned}$ and solve for $\begin{aligned} y \end{aligned}$ :
::在此情况下, 我们检查答案的方式是使用方程式生成自己的 x, y 配对。如果它们匹配表格中的值, 那么我们就会知道我们的方程是正确的。 x 的替代 - 2, 0, 2, 4 和 6 和 y 的解答 :

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y \end{aligned}$
-2	$\begin{aligned} - 5 (- 2) = 10 \end{aligned}$
0	$\begin{aligned} - 5 (0) = 0 \end{aligned}$
2	$\begin{aligned} - 5 (2) = - 10 \end{aligned}$
4	$\begin{aligned} - 5 (4) = - 20 \end{aligned}$
6	$\begin{aligned} - 5 (6) = - 30 \end{aligned}$

The $\begin{aligned} y - \end{aligned}$ values in this table match the ones in the earlier table. The answer checks out.
::此表格中的 y- 值与先前表格中的 y- 值匹配。答案会检查出来。

Example
::示例示例示例示例

Example 1
::例1

Zarina has a $100 gift card, and she has been spending money on the card in small regular amounts. She checks the balance on the card weekly and records it in the following table.
::Zarina有一张100美元的礼品卡,她经常在卡上花费少量资金,每周检查卡片的余额,在下表记录。

Week Number	Balance ($)
1	100
2	78
3	56
4	34

Write an equation for the money remaining on the card in any given week.
::写一个卡片上所剩钱的方程任何特定星期。

Step 1: Extract the important information.
::第1步:提取重要信息。

The balance remaining on the card is not just a constant multiple of the week number; 100 is 100 times 1, but 78 is not 100 times 2. There is still a pattern though: the balance decreases by 22 whenever the week number increases by 1. This suggests that the balance is somehow related to the amount “-22 times the week number.”
::2. 不过,仍然存在着一种模式:每当周数增加1时,余额就会减少22倍。这意味着余额在某种程度上与“周数22倍”的数额有关。

In fact, the balance equals “-22 times the week number, plus something .” To determine what that something is, we can look at the values in one row on the table. For example, the first row, where we have a balance of $100 for week number 1.
::事实上,平衡等于“周数的22乘以周数,加某东西 ” 。为了确定什么是某事,我们可以在表格上用一行查看数值。例如,第一行,我们第1周的余额为100美元。

Step 2: Translate into a mathematical equation.
::步骤2: 转换成数学等式。

First, define the variables. Let $\begin{aligned} n \end{aligned}$ stand for the week number and $\begin{aligned} b \end{aligned}$ for the balance.
::首先,定义变量。让 n 表示周数和 b 表示余数。

Then we can translate our verbal expression as follows:
::然后,我们可以将我们的口头表述翻译如下:

Text	Translates to	Mathematical Expression
Balance equals -22 times the week number, plus something .	$\begin{aligned} \to \end{aligned}$	$\begin{aligned} b = - 22 n + ? \end{aligned}$

To find out what that ? represents, we can plug in the values from that first row of the table, where $\begin{aligned} b = 100 \end{aligned}$ and $\begin{aligned} n = 1 \end{aligned}$ . This gives us $\begin{aligned} 100 = - 22 (1) + ? \end{aligned}$ .
::要找到什么 ? 表示, 我们可以插入表格第一行的值, 即 b=100 和 n= 1 的值。这给我们 100 \\ 22 (1) +? 。

So what number gives 100 when you add -22 to it? The answer is 122, so that is the number the ? stands for. Now our final equation is:
::添加 - 22 时, 数字给 100 多少 ? 答案是 122 , 这就是 ? 代表的数字。现在, 我们最后的方程式是 :

\begin{aligned} b = - 22 n + 122 \end{aligned}

::b22n+122

Step 3: All we were asked to find was the expression . We weren't asked to solve it, so we can move to Step 4.
::步骤3:我们被要求找到的只是表达方式。我们没有被要求解决它,所以我们可以搬到步骤4。

Step 4: Check the result.
::第四步:检查结果。

To check that this equation is correct, we see if it really reproduces the data in the table. To do that we plug in values for $\begin{aligned} n \end{aligned}$ :
::要检查此方程式是否正确, 我们看看它是否真的复制了表格中的数据。要做到这一点, 我们用 n 的值插入 :

\begin{aligned} n = 1 \to b & = - 22 (1) + 122 = 122 - 22 = 100 \\ n = 2 \to b & = - 22 (2) + 122 = 122 - 44 = 78 \\ n = 3 \to b & = - 22 (3) + 122 = 122 - 66 = 56 \\ n = 4 \to b & = - 22 (4) + 122 = 122 - 88 = 34 \end{aligned}

::n=1(b)-22(1)+122=122-22=100n=2(b)-22(2)+122=122-44=78n=3(b)-22(3)+122=122-66=56n=4(b)-22(4)+122=122-88=34

The equation perfectly reproduces the data in the table. The answer checks out.
::方程式完全复制了表格中的数据答案已经查出来了

Review
::回顾

Day	Profit
1	20
2	40
3	60
4	80
5	100

For 1-3, use the above table that depicts the profit in dollars taken in by a store each day.
::对于1-3,使用上表来说明每天一家商店以美元购买的利润。

Write a mathematical equation that describes the relationship between the variables in the table.
::写入一个数学方程式, 描述表格变量之间的关系。

What is the profit on day 10?
::第十天有什么好处?

If the profit on a certain day is $200, what is the profit on the next day?
::如果某一天的利润是200美元,那么第二天的利润是多少?

For 4-6, Write a mathematical equation that describes each situation, assuming the cookie jar starts with 24 cookies.
::4 -6,写一个数学方程式描述每个情况,假设饼干罐头是24个饼干开始。

How many cookies are left in the jar after you have eaten some?
::你吃过之后罐子里还剩多少饼干?

How many cookies are in the jar after you have eaten 9 cookies?
::你吃过9个饼干后罐子里有多少饼干?

How many cookies are in the jar after you have eaten 9 cookies and then eaten 3 more?
::罐子里有多少饼干? 你吃了9个,然后又吃了3个?

For 7-12, write a mathematical equation for the situation and solve.
::7 -12,写一个数学方程式来说明情况并解决问题。

Seven times a number is 35. What is the number?
::数字7乘以35 数字是多少?

Three times a number, plus 15, is 24. What is the number?
::3乘以1乘以1乘以15是24 数字是多少?

Twice a number is three less than five times another number. Three times the second number is 15. What are the numbers?
::数字的两倍是3比5乘以另外的数字。第二个数字的三倍是15。数字是多少?

One number is 25 more than 2 times another number. If each number were multiplied by five, their sum would be 350. What are the numbers?
::一个数字是25乘以2乘以2,另一个数字。如果每个数字乘以5,它们的总和将是350。数字是多少?

The sum of two consecutive integers is 35. What are the numbers?
::两个连续整数的总和是35,数字是多少?

Peter is three times as old as he was six years ago. How old is Peter?
::彼得比六年前大三倍彼得多大了?

For 13-16, Jae just took a math test with 20 questions, each worth an equal number of points. The test is worth 100 points total.
::13 - 16, Jae刚刚做了数学测试, 共问了20个问题, 每个问题各值相等的点数。测试总值为 100 点。

Write an equation relating the number of questions Jae got right to the total score he will get on the test.
::写一个方程式说明Jae得到的问题数量与他在测试中的总分一致

If a score of 70 points earns a grade of $\begin{aligned} C - \end{aligned}$ , how many questions would Jae need to get right to get a $\begin{aligned} C - \end{aligned}$ on the test?
::如果得分70分得C级,Jae需要回答多少问题才能得到C级的测试?

If a score of 83 points earns a grade of $\begin{aligned} B \end{aligned}$ , how many questions would Jae need to get right to get a $\begin{aligned} B \end{aligned}$ on the test?
::如果得分83分得B分, Jae需要回答多少问题才能得到B分?

Suppose Jae got a score of 60% and then was allowed to retake the test. On the retake, he got all the questions right that he got right the first time, and also got half the questions right that he got wrong the first time. What is his new score?
::假设Jae得分为60%,然后又被允许重考。在重考时,他得到了所有他第一次答对的问题,还得到了一半问题,他第一次错了。他的新得分是多少?

For 17-22, solve the problem by writing an equation.
::17 -22,通过写一个方程来解决问题。

How much water should be added to one liter of pure alcohol to make a mixture of 25% alcohol?
::在一升纯酒精中加多少水才能混合25%的酒精?

A mixture of 50% alcohol and 50% water has 4 liters of water added to it. It is now 25% alcohol. What was the total volume of the original mixture?
::50%酒精和50%水的混合物加了4升水,现在为25%酒精。原混合物的总量是多少?

In Crystal’s silverware drawer there are twice as many spoons as forks. If Crystal adds nine forks to the drawer, there will be twice as many forks as spoons. How many forks and how many spoons are in the drawer right now?
::在Crystal的银器抽屉里,勺子是叉子的两倍。如果Crystal在抽屉里加9个叉子,叉子将比勺子多一倍。现在抽屉里有多少叉子和多少勺子?

Mia is exploring different routes to drive to Javier's house.
1. Mia drove to Javier’s house at 40 miles per hour. Javier’s house is 20 miles away. Mia arrived at Javier’s house at 2:00 pm. What time did she leave?
  ::米亚以每小时40英里的速度开车到哈维尔家,哈维尔家在20英里之外,米亚下午2点到达哈维尔家。她什么时候离开?
2. Mia left Javier’s house at 6:00 pm to drive home. This time she drove 25% faster. What time did she arrive home?
  ::米娅下午6点离开哈维尔家开车回家。这次她开的车更快25 % 。她几时到家?
3. The next day, Mia took the expressway to Javier’s house. This route was 24 miles long, but she was able to drive at 60 miles per hour. How long did the trip take?
  ::第二天,米娅搭上高速公路前往哈维尔家。这条路长24英里,但她能以每小时60英里的速度开车。行程需要多长时间?
4. When Mia took the same route back, traffic on the expressway caused her to drive 20% slower. How long did the return trip take?
  ::当米娅从同一条路线返回时,高速公路上的交通让她开慢了20%。返回旅行需要多长时间?
::米娅正在探索通往哈维尔家的不同路线。米娅开往哈维尔家的路程是不同的。米娅每小时40英里。米娅开往哈维尔家的路程是40英里。哈维尔家离家20英里。哈维尔家离家20英里。米娅是几点离开哈维尔家的?米娅是什么时候离开的?米娅在下午6点离开的家开车回家。这次米娅开车更快了25 % 。她什么时候回家? 第二天,米娅开高速公路到哈维尔家。这条路长达24英里,但她可以每小时60英里开车。行程持续了多久?当米娅走回同一条路时,高速公路上的交通使她开车慢了20 % 。返回行程又花了多长时间?

The price of an mp3 player decreased by 20% from last year to this year. This year the price of the player is $120. What was the price last year?
::MP3播放器的价格从去年到今年下降了20%。今年的播放器价格是120美元。去年的价格是多少?

SmartCo sells deluxe widgets for $60 each, which includes the cost of manufacture plus a 20% markup. What does it cost SmartCo to manufacture each widget?
::SmartCo销售豪华部件,每件60美元,包括制造成本加上20%的加价。SmartCo制造每件部件的成本是多少?

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。

Section outline

General

模式和表达式

Patterns and Equations ::模式和等价

Wr ite an Equation ::写一个方程式

Writing and Solving an Equation ::书写和解决方程式

Writing/Solving an Equation to Describe a Relationship ::书写/理清描述关系的等同

Solve Problems Using Equations ::使用等同的解决问题

Write a Verbal Equation ::a 写音量方程式。

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Patterns and Equations
::模式和等价

Wr ite an Equation
::写一个方程式

Writing and Solving an Equation
::书写和解决方程式

Writing/Solving an Equation to Describe a Relationship
::书写/理清描述关系的等同

Solve Problems Using Equations
::使用等同的解决问题

Write a Verbal Equation
::a 写音量方程式。

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)