Section: 分数积表达式求值 | 7.5分数积表达式求值

Section outline

Jae is creating a model building for a design competition. Each floor will be $\begin{aligned} 3 \frac{1}{2} \end{aligned}$ inches tall. How can Jae find the total height of his building model. How tall will it be if Jae decides to have 10 floors?
::杰正在为设计竞赛建立一个模范建筑。每层楼将达到312英寸高。杰如何能找到他的建筑模型的总高度。如果杰决定拥有10层,那么它会有多高?

In this concept, you will learn how to evaluate expressions and products of fractions.
::在这个概念中,你们将学会如何评价分数的表达方式和产品。

Evaluating Expressions with Products of Fractions
::带有分数产品的评价表达式

An expression is a numerical phrase that combines numbers and operations but no equal sign.
::表达式是一个数字短语,它结合数字和操作,但无等号。

There are two kinds of expressions. Numerical expressions include numbers and operations only. Algebraic expressions include numbers, operations, and variables.
::有两种表达式。数字表达式只包括数字和操作。代数表达式包括数字、操作和变量。

Types of Expressions
::表达式类型

	Includes	Examples
numerical expressions	numbers, operations	$\begin{aligned} 3 + 4 \end{aligned}$ $\begin{aligned} \frac{3}{4} \times \frac{2}{3} \end{aligned}$ $\begin{aligned} 15.6 - 8 \end{aligned}$ $\begin{aligned} 4 (\frac{3}{4}) \end{aligned}$
algebraic expressions	numbers, operations, variables	$\begin{aligned} 3 + x \end{aligned}$ ::3+xx 3+x $\begin{aligned} \frac{3}{4} \cdot \frac{b}{3} \end{aligned}$ ::3 3 4 4 b 3 $\begin{aligned} 15.6 - q \end{aligned}$ ::15.6 - q $\begin{aligned} c (\frac{3}{4}) \end{aligned}$ :c) 3 4 )

Includes

Examples

numerical expressions

numbers, operations

$\begin{aligned} 3 + 4 \end{aligned}$

$\begin{aligned} \frac{3}{4} \times \frac{2}{3} \end{aligned}$

$\begin{aligned} 15.6 - 8 \end{aligned}$

$\begin{aligned} 4 (\frac{3}{4}) \end{aligned}$

algebraic expressions

numbers, operations, variables

$\begin{aligned} 3 + x \end{aligned}$
::3+xx 3+x

$\begin{aligned} \frac{3}{4} \cdot \frac{b}{3} \end{aligned}$
::3 3 4 4 b 3

$\begin{aligned} 15.6 - q \end{aligned}$
::15.6 - q

$\begin{aligned} c (\frac{3}{4}) \end{aligned}$
:c) 3 4 )

Since a numerical expression includes numbers and operations, perform the operation required to evaluate. To evaluate means to find the value of the expression.
::由于数字表达式包括数字和操作,所以执行评估所需的操作。要评价查找表达式价值的手段。

Evaluate the numerical expression.
::评估数字表达式。

$\begin{aligned} (\frac{1}{4}) (\frac{3}{4}) \end{aligned}$

Notice that there are two sets of " data-term="Parentheses" role="term" tabindex="0"> parentheses . Remember that the parentheses can identify groups and also indicate multiplication .
::注意有两组括号。记住括号可以识别组, 也可以表示乘法。

Evaluate by multiplying and then simplifying or by simplifying first then multiplying.
::通过乘数和简化或简化评价,或者首先简化评价,然后增加评价。

\begin{aligned} \frac{1}{4} \times \frac{3}{4} = \frac{3}{16} \end{aligned}

The answer is in simplest form.
::答案是最简单的。

The value of the expression is $\begin{aligned} \frac{3}{16} \end{aligned}$ .
::表达式的值为 3 16 。

As you learn about algebra and higher levels of math, you will be working with algebraic expressions.
::当你学习代数和更高数学水平时, 你会用代数表达式工作。

An algebraic expression is similar to a numeric expressions, except that it uses variables. Variables are letters that represent an unknown number. Sometimes the values of the variables are given.
::代数表达式与数字表达式相似,但使用变量。变量是代表未知数字的字母。有时给出变量的值。

Evaluate the algebraic expression.
::评估代数表达式。

$\begin{aligned} x y \end{aligned}$ when $\begin{aligned} x = \frac{3}{4} \end{aligned}$ , and $\begin{aligned} y = \frac{1}{3} \end{aligned}$
::x = 3 4 y = 1 3xxxx = 3 4 y = 1 3

To evaluate this expression, substitute the given values for $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ into the expression. Notice that the expression has $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ next to each other. When two variables are next to each other the operation is multiplication.
::要评价此表达式, 请在表达式中替换 x 和 y 的给定值。注意表达式彼此相邻有 x 和 y 。当两个变量相邻时, 操作是乘法。

\begin{aligned} \frac{3}{4} \times \frac{1}{3} \end{aligned}

Then, simplify the expression. Cross simplify the 3s with the greatest common factor (GCF) of 3. Each three becomes a one.
::然后,简化表达式。交叉简化三点,最大共同系数(GCF)为三点,每三点就变成一个。

\begin{aligned} \frac{3}{4} \times \frac{1}{3} = \frac{1}{4} \times \frac{1}{1} \end{aligned}

Next, multiply to evaluate the expression.
::下一步,乘以来评价表达式。

\begin{aligned} \frac{1}{4} \times \frac{1}{1} = \frac{1}{4} \end{aligned}

The value of the expression is $\begin{aligned} \frac{1}{4} \end{aligned}$ .
::表达式的值为 14。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Jae and his design competition.
::早些时候,你得到一个问题关于Jae和他的设计比赛。

Jae is making a model building where each floor is $\begin{aligned} 3 \frac{1}{2} \end{aligned}$ inches tall. Create an expression that will help determine the total height of the building depending on the number of floors he decides to make.
::Jae正在建造一座建模大楼,每层楼高312英寸。创建一个表达式, 帮助确定大楼的总高度, 取决于他决定建造的楼层数量。

For Jae's model building, he can create an algebraic expression to determine the height of his model. The total height will be the the height of each floor times the number of floors.
::对于Jae的模型建筑来说,他可以创建代数表达式来确定模型的高度。总高度将是每层地板的高度乘以地板数。

$\begin{aligned} 3 \frac{1}{2} x, where x is equal to the total number of floors \end{aligned}$
::1 2xxxx,x等于楼层总数

Find the height of the model when x = 10.
::x = 10 时查找模型的高度。

First, substitute the variable with the given value.
::首先,用给定值替换变量。

$\begin{aligned} 3 \frac{1}{2} \times 10 \end{aligned}$

Then, convert the mixed number and whole number into an improper fraction .
::然后,将混合数和整数转换成不适当的部分。

$\begin{aligned} 3 \frac{1}{2} \times 10 = \frac{7}{2} \times \frac{10}{1} \end{aligned}$

Next, simplify the fractions.
::接下来,简化分数。

$\begin{aligned} \frac{7}{2} \times \frac{10}{1} = \frac{7}{1} \times \frac{5}{1} \end{aligned}$

Finally, multiply the fractions.
::最后,将分数乘以。

$\begin{aligned} \frac{7}{1} \times \frac{5}{1} = 35 \end{aligned}$

Jae's 10 floor model will be 35 inches tall.
::Jae的10层模型将高35英寸

Example 2
::例2

Evaluate $\begin{aligned} x y \end{aligned}$ when $\begin{aligned} x \end{aligned}$ is $\begin{aligned} \frac{2}{3} \end{aligned}$ and $\begin{aligned} y \end{aligned}$ is $\begin{aligned} \frac{8}{12} \end{aligned}$ .
::当 x 是 2 3 , y 是 8 12 时, 评估 x y 。

First, substitute the values into the expression.
::首先,将值替换为表达式。

\begin{aligned} \frac{2}{3} \times \frac{8}{12} \end{aligned}

Then, simplify the expression.
::然后,简化表达方式。

\begin{aligned} \frac{2}{3} \times \frac{8}{12} = \frac{1}{3} \times \frac{4}{3} \end{aligned}

Next, multiply the fractions.
::接下来,乘以分数。

\begin{aligned} \frac{1}{3} \times \frac{4}{3} = \frac{4}{9} \end{aligned}

The answer is $\begin{aligned} \frac{4}{9} \end{aligned}$ .
::答案是4 9。

Example 3
::例3

Evaluate $\begin{aligned} (\frac{4}{7}) (\frac{21}{28}) \end{aligned}$ . Answer in simplest form.
::评价(4 7) (21 28). 答复最简单。

First, simplify the fractions.
::首先,简化分数。

\begin{aligned} \frac{4}{7} \times \frac{21}{28} = \frac{1}{1} \times \frac{3}{7} \end{aligned}

Then, multiply.
::然后,乘数。

\begin{aligned} 1 \times \frac{3}{7} = \frac{3}{7} \end{aligned}

The value of the expression is $\begin{aligned} \frac{3}{7} \end{aligned}$ .
::表达式的值为 3 7 。

Example 4
::例4

Evaluate $\begin{aligned} (x y) \end{aligned}$ when $\begin{aligned} x \end{aligned}$ is $\begin{aligned} \frac{3}{5} \end{aligned}$ and $\begin{aligned} y \end{aligned}$ is $\begin{aligned} \frac{10}{11} \end{aligned}$ . Answer in simplest form.
::当 x 3 5 和 y 10 11 时评价( x y ) 。回答最简单。

First, substitute the values into the expression.
::首先,将值替换为表达式。

\begin{aligned} \frac{3}{5} \times \frac{10}{11} \end{aligned}

Then, simplify the fractions.
::然后,简化分数。

\begin{aligned} \frac{3}{5} \times \frac{10}{11} = \frac{3}{1} \times \frac{2}{11} \end{aligned}

Next, multiply.
::下一个,乘数。

\begin{aligned} \frac{3}{1} \times \frac{2}{11} = \frac{6}{11} \end{aligned}

The value of the expression is $\begin{aligned} \frac{6}{11} \end{aligned}$ .
::表达式的值为 6 11 。

Example 5
::例5

Evaluate $\begin{aligned} (\frac{5}{9}) (\frac{45}{60}) \end{aligned}$ . Answer in simplest form.
::评价(5 9) (45 60). 答复最简单。

First, simplify the fractions.
::首先,简化分数。

\begin{aligned} \frac{5}{9} \times \frac{45}{60} = \frac{5}{3} \times \frac{1}{4} \end{aligned}

Next, multiply.
::下一个,乘数。

\begin{aligned} \frac{5}{3} \times \frac{1}{4} = \frac{5}{12} \end{aligned}

The value of the expression is $\begin{aligned} \frac{5}{12} \end{aligned}$ .

::表达式的值为 5 12 。

Review
::回顾

Evaluate each expression. Answer in simplest form.
::评估每个表达式,回答最简单。

Evaluate $\begin{aligned} (x y) \end{aligned}$ when $\begin{aligned} x = \frac{2}{3} \end{aligned}$ and $\begin{aligned} y = \frac{6}{10} \end{aligned}$
::当x=2 3和y=6 10时评价(xy)

Evaluate $\begin{aligned} (x y) \end{aligned}$ when $\begin{aligned} x = \frac{1}{3} \end{aligned}$ and $\begin{aligned} y = \frac{4}{10} \end{aligned}$
::评估(xy)x=1 3和y=4 10

Evaluate $\begin{aligned} (x y) \end{aligned}$ when $\begin{aligned} x = \frac{12}{13} \end{aligned}$ and $\begin{aligned} y = \frac{2}{6} \end{aligned}$
::当x = 12 13 y = 2 6时评价 (x y)

Evaluate $\begin{aligned} (x y) \end{aligned}$ when $\begin{aligned} x = \frac{1}{3} \end{aligned}$ and $\begin{aligned} y = \frac{4}{5} \end{aligned}$
::评估(xy)x=1 3和y=4 5

Evaluate $\begin{aligned} (x y) \end{aligned}$ when $\begin{aligned} x = \frac{7}{9} \end{aligned}$ and $\begin{aligned} y = \frac{3}{21} \end{aligned}$
::当x=7 9和y=3 21时评价(xy)

Evaluate $\begin{aligned} (x y) \end{aligned}$ when $\begin{aligned} x = \frac{4}{5} \end{aligned}$ and $\begin{aligned} y = \frac{16}{20} \end{aligned}$
::评估(xy)x=4 5和y=16 20

Evaluate $\begin{aligned} (\frac{4}{6}) (\frac{1}{2}) \end{aligned}$
::评价(4 6) (1 2)

Evaluate $\begin{aligned} (\frac{1}{9}) (\frac{6}{18}) \end{aligned}$
::评价(1 9) (6 18)

Evaluate $\begin{aligned} (\frac{4}{9}) (\frac{1}{4}) \end{aligned}$
::评价(4 9) (1 4)

Evaluate $\begin{aligned} (\frac{4}{11}) (\frac{11}{12}) \end{aligned}$
::评价(4 11) (11 12)

Evaluate $\begin{aligned} (\frac{9}{10}) (\frac{5}{6}) \end{aligned}$
::评价(9 10) (5 6)

Evaluate $\begin{aligned} (\frac{8}{9}) (\frac{3}{6}) \end{aligned}$
::评价(8 9) (3 6)

Evaluate $\begin{aligned} (\frac{18}{19}) (\frac{3}{6}) \end{aligned}$
::评价(18 19) (3 6)

Evaluate $\begin{aligned} (\frac{4}{9}) (\frac{36}{40}) \end{aligned}$
::评价(4 9) (36 40)

Evaluate $\begin{aligned} (\frac{12}{14}) (\frac{7}{8}) \end{aligned}$
::评价(12 14) (7 8)

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

Evaluating Expressions with Products of Fractions ::带有分数产品的评价表达式

Types of Expressions ::表达式类型

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)

Resources ::资源