Section: 复杂分数 | 9.13 复杂分数

Section outline

Gupta knows the area and width of a rectangle. He comes up with this equation for the length of the rectangle $\begin{align*}\frac{\frac{2}{x^2-1}}{\frac{2x}{x+1}}\end{align*}$ . What is the length of the rectangle in simplified form?

Complex Fractions
::复杂分数

A complex fraction is a fraction that has fractions in the numerator and/or denominator. To simplify a complex fraction, you will need to combine all that you have learned about simplifying fractions in general.
::复数分数是指在分子和(或)分母中含有分数的分数。要简化一个复杂的分数,您需要将您所学到的关于一般简化分数的所有分数合并在一起。

Let's simplify the following complex fractions.
::让我们简化以下复杂的部分。

$\begin{align*}\frac{\frac{9x}{x+2}}{\frac{3}{x^2-4}}\end{align*}$
::9xx+23x2-4

Rewrite the complex fraction as a division problem.
::将复杂分数重写为分裂问题。

$\begin{aligned} \frac{\frac{9 x}{x + 2}}{\frac{3}{x^{2} - 4}} = \frac{9 x}{x + 2} \div \frac{3}{x^{2} - 4} \end{aligned}$
$\begin{align*}\frac{\frac{9x}{x+2}}{\frac{3}{x^2-4}} = \frac{9x}{x+2} \div \frac{3}{x^2-4}\end{align*}$
::9xx+23x2 - 4=9xx+2}3x2 - 4

Flip the second fraction, change the problem to multiplication and simplify.
::翻转第二段,将问题改为乘法和简化。

$\begin{aligned} \frac{9 x}{x + 2} \div \frac{3}{x^{2} - 4} = \frac{9 x}{x + 2} \cdot \frac{x^{2} - 4}{3} = \frac{\overset{3}{9 x}}{x + 2} \cdot \frac{(x + 2) (x - 2)}{3} = 3 x (x - 2) \end{aligned}$
$\begin{align*}\frac{9x}{x+2} \div \frac{3}{x^2-4} = \frac{9x}{x+2} \cdot \frac{x^2-4}{3} = \frac{\overset{3}{\bcancel{9}x}}{\cancel{x+2}} \cdot \frac{\cancel{(x+2)}(x-2)}{\bcancel{3}} = 3x(x-2)\end{align*}$
::9xx+23x2-4=9xx+2x2x2-43=9x3xxx2}(x+2)(x-2)(x-2)3=3x(x-2)

$\begin{align*}\frac{\frac{1}{x} + \frac{1}{x+1}}{4- \frac{1}{x}}\end{align*}$
::1x1+1x+14-1x

To simplify this complex fraction, we first need to add the fractions in the numerator and subtract the two in the denominator. The LCD of the numerator is $\begin{align*}x(x+1)\end{align*}$ and the denominator is just $\begin{align*}x\end{align*}$ .
::为了简化这一复杂分数, 我们首先需要在分子中添加分数, 并在分母中减去这两个分数。分子的LCD是 x( x+1) , 分数只是 x 。

$\begin{aligned} \frac{\frac{1}{x} + \frac{1}{x + 1}}{4 - \frac{1}{x}} = \frac{\frac{x + 1}{x + 1} \cdot \frac{1}{x} + \frac{1}{x + 1} \cdot \frac{x}{x}}{\frac{x}{x} \cdot 4 - \frac{1}{x}} = \frac{\frac{x + 1}{x (x + 1)} + \frac{x}{x (x + 1)}}{\frac{4 x}{x} - \frac{1}{x}} = \frac{\frac{2 x + 1}{x (x + 1)}}{\frac{4 x - 1}{x}} \end{aligned}$
$\begin{align*}\frac{\frac{1}{x} + \frac{1}{x+1}}{4- \frac{1}{x}} = \frac{{\color{red}\frac{x+1}{x+1}} \cdot \frac{1}{x} + \frac{1}{x+1} \cdot {\color{blue}\frac{x}{x}}}{{\color{blue}\frac{x}{x}} \cdot 4- \frac{1}{x}} = \frac{\frac{x+1}{x(x+1)} + \frac{x}{x(x+1)}}{\frac{4x}{x} - \frac{1}{x}} = \frac{\frac{2x+1}{x(x+1)}}{\frac{4x-1}{x}}\end{align*}$
::1x+1x1x+14-1x=1x1x1x1x1x1x1x1x1x1x1xxxxxxxx1xxxx1xxx1x1x1x1x1xxxxxxxx+1x1x1x1x1x1xxx1x1x1x1x1xx1x1x1xx1xx1x1x1xx1xxx1xxx1xxx1xx1xxx1xxxxxx1xxx1xxx1x1x1x1x1x1xxxx1xxx1x1x1x1x1x1x1x1x1x1xxx1xx1x1xxxx1x1x1x1xx1x1x1x1x1xx1x1x1x1x1x1x1xx1x1x1xx1x1x1x1xxx1x1xx1x1xxx1x1x1x1xx1xx1xx1xxxx1x1x1xx1xxx1xxxx1xxxxx1xxxx1xxxxxxxxxxx1xxxxxxxxxxx1xxxxxxxxxxxxx1xxxxxxx1xxxxxxxx1xxxxxxxxxxxxxxxxxxxxxxx

Divide and simplify if possible.
::在可能的情况下,分开和简化。

$\begin{aligned} \frac{\frac{2 x + 1}{x (x + 1)}}{\frac{4 x - 1}{x}} = \frac{2 x + 1}{x (x + 1)} \div \frac{4 x - 1}{x} = \frac{2 x + 1}{x (x + 1)} \cdot \frac{x}{4 x - 1} = \frac{2 x + 1}{(x + 1) (4 x - 1)} \end{aligned}$
$\begin{align*}\frac{\frac{2x+1}{x(x+1)}}{\frac{4x-1}{x}} = \frac{2x+1}{x(x+1)} \div \frac{4x-1}{x} = \frac{2x+1}{\cancel{x}(x+1)} \cdot \frac{\cancel{x}}{4x-1} = \frac{2x+1}{(x+1)(4x-1)}\end{align*}$
::2x+1x( x+1) 4x -1x=2x1x1x( x+1) 4x -1x=2x1x( x+1) x4x -1x=2x1x( x+1) x4x -1x=2x1x1( x+1) (4x-1)

$\begin{align*}\frac{\frac{5-x}{x^2+6x+8} + \frac{x}{x+4}}{\frac{6}{x+2} - \frac{2x+3}{x^2-3x-10}}\end{align*}$
::5-2x2+6x+8+6x8+xxxxx+46xx+2-2-2x+3x2-3x2-3xx-10

First, add the fractions in the numerator and subtract the ones in the denominator.
::首先,在分子中添加分数,并在分母中减去分数。

$\begin{aligned} \frac{\frac{5 - x}{x^{2} + 6 x + 8} + \frac{x}{x + 4}}{\frac{6}{x + 2} - \frac{2 x + 3}{x^{2} - 3 x - 10}} = \frac{\frac{5 - x}{(x + 4) (x + 2)} + \frac{x}{x + 4} \cdot \frac{x + 2}{x + 2}}{\frac{x - 5}{x - 5} \cdot \frac{6}{x + 2} - \frac{2 x + 3}{(x + 2) (x - 5)}} = \frac{\frac{5 - x + x (x + 2)}{(x + 4) (x + 2)}}{\frac{6 (x - 5) - (2 x + 3)}{(x + 2) (x - 5)}} = \frac{\frac{x^{2} + x + 5}{(x + 4) (x + 2)}}{\frac{4 x - 36}{(x + 2) (x - 5)}} \end{aligned}$
$\begin{align*}\frac{\frac{5-x}{x^2+6x+8} + \frac{x}{x+4}}{\frac{6}{x+2} - \frac{2x+3}{x^2-3x-10}} = \frac{\frac{5-x}{(x+4){\color{red}(x+2)}} + \frac{x}{x+4} \cdot {\color{red}\frac{x+2}{x+2}}}{{\color{blue}\frac{x-5}{x-5}} \cdot \frac{6}{x+2} - \frac{2x+3}{(x+2){\color{blue}(x-5)}}} = \frac{\frac{5-x+x(x+2)}{(x+4)(x+2)}}{\frac{6(x-5)-(2x+3)}{(x+2)(x-5)}} = \frac{\frac{x^2+x+5}{(x+4)(x+2)}}{\frac{4x-36}{(x+2)(x-5)}}\end{align*}$
::5 -xx2+6x8+6x8+6x8+xx2+46x+2x2+2x2+2x2x2+2x2x2x3x3x3x3xxx10=5x(x+4)(x+2)+x2x+2xx-2x5x5x5x2x5x2x5x6x2+2x6x+6x6x+6x6x+6x6x+2x2x2x2x2xx2xx2xx2x2xx2x6x+5x2xx5xx2x4x+4x2x5x5x2x5xxxx2x2x6xxx5x2x5x+2x2x(x+2x2x)(x+2x)(x-5)6x5xxxxxx3x3x3x3x(x2+2)(x5)

Now, rewrite as a division problem, flip, multiply, and simplify.
::现在,重写为分裂问题,翻转,乘法和简化。

$\begin{aligned} \frac{\frac{x^{2} + x + 5}{(x + 4) (x + 2)}}{\frac{4 x - 36}{(x + 2) (x - 5)}} = \frac{x^{2} + x + 5}{(x + 4) (x + 2)} \div \frac{4 x - 36}{(x + 2) (x - 5)} = \frac{x^{2} + x + 5}{(x + 4) (x + 2)} \cdot \frac{(x + 2) (x - 5)}{4 (x - 9)} \end{aligned}$
$\begin{align*}\frac{\frac{x^2+x+5}{(x+4)(x+2)}}{\frac{4x-36}{(x+2)(x-5)}} = \frac{x^2+x+5}{(x+4)(x+2)} \div \frac{4x-36}{(x+2)(x-5)} = \frac{x^2+x+5}{(x+4)\cancel{(x+2)}} \cdot \frac{\cancel{(x+2)}(x-5)}{4(x-9)}\end{align*}$
::x2+x+2(x+4)(x+2)(x+2)(x-5)=x2+5(x+4)(x+2)(x+2)**4x-36(x+2)(x-5)=x2+5(x+4)(x+4)(x+2)(x+2)(x+2)__(x+2)(x+2)(x+2)(x-2)(x-5)4(x-9)

$\begin{aligned} = \frac{(x^{2} + x + 5) (x - 5)}{4 (x + 4) (x - 9)} \end{aligned}$
$\begin{align*}= \frac{(x^2+x+5)(x-5)}{4(x+4)(x-9)}\end{align*}$

::=(x2+x+5)(x-5)(x-5)4(x+4)(x-9)

Examples
::实例

Example 1
::例1

Earlier, you were asked to determine the length of a given rectangle in simplified form.
::早些时候,有人要求你以简化的形式确定给定矩形的长度。

Rewrite the complex fraction as a division problem.
::将复杂分数重写为分裂问题。

$\begin{aligned} \frac{\frac{2}{x^{2} - 1}}{\frac{2 x}{x + 1}} = \frac{2}{x^{2} - 1} \div \frac{2 x}{x + 1} \end{aligned}$
$\begin{align*}\frac{\frac{2}{x^2-1}}{\frac{2x}{x+1}} = \frac{2}{x^2-1} \div \frac{2x}{x+1}\end{align*}$
::2x2 - 12xxx+1=2x2 - 1=2x2 - *2xxx+1

Flip the second fraction, change the problem to multiplication and simplify.
::翻转第二段,将问题改为乘法和简化。

$\begin{aligned} \frac{2}{x^{2} - 1} \div \frac{2 x}{x + 1} = \frac{2}{x^{2} - 1} \cdot \frac{x + 1}{2 x} = \frac{2}{(x + 1) (x - 1)} \cdot \frac{(x + 1)}{2 x} = \frac{1}{x^{2} - x} \end{aligned}$
$\begin{align*}\frac{2}{x^2-1} \div \frac{2x}{x+1} = \frac{2}{x^2-1} \cdot \frac{x+1}{2x} = \frac{\bcancel{2}}{\bcancel{(x+1)}(x-1)} \cdot \frac{\bcancel{(x+1)}}{\bcancel{2}x} = \frac {1}{x^2-x}\end{align*}$
::2x2 - 1 - 1x2 - 2xx2+1=2x2 - 1xxxx+12x=2(x+1)(x-1)(x-1)(x+1)(x+1)__(x+1)_(x+1)2x=1x2-x

Therefore , the length of the rectangle in simplified form is $\begin{align*}\frac {1}{x^2-x}\end{align*}$ .
::因此,简化形式的矩形长度为1x2-x。

S implify the complex fractions.
::简化复杂的分数。

Example 2
::例2

$\begin{align*}\frac{\frac{5x-20}{x^2}}{\frac{x-4}{x}}\end{align*}$
::5x-20x2x-4x

Rewrite the fraction as a division problem and simplify.
::将分数重写为分裂问题并简化。

$\begin{align*}\frac{\frac{5x-20}{x^2}}{\frac{x-4}{x}} = \frac{5x-20}{x^2} \div \frac{x-4}{x} = \frac{5 \cancel{(x-4)}}{x^{\cancel{2}}} \cdot \frac{\cancel{x}}{\cancel{x-4}} = \frac{5}{x}\end{align*}$
::5x - 20x2x - 4x=5x - 20x2x - 4x=5(x-4)x2x - 4x=5x

Example 3
::例3

$\begin{align*}\frac{\frac{1-x}{x} - \frac{2}{x-1}}{1 + \frac{1}{x}}\end{align*}$
::1-xx-2-2x-11+1x

Add the fractions in the numerator and denominator together.
::在分子和分母中同时添加分数。

$\begin{align*}\frac{\frac{1-x}{x} - \frac{2}{x-1}}{1+\frac{1}{x}} = \frac{\frac{x-1}{x-1} \cdot \frac{1-x}{x} - \frac{2}{x-1} \cdot \frac{x}{x}}{\frac{x}{x} \cdot 1+ \frac{1}{x}} = \frac{\frac{(x-1)(1-x)-2x}{x(x-1)}}{\frac{x+1}{x}} = \frac{\frac{-x^2+1}{x(x-1)}}{\frac{x+1}{x}} \end{align*}$
::1-xxxxxx-1-1xx+1x=x-1x-1x-1x-11x-1=x-2x-1x-1xxxxxxxxxx-1xxx-1(1-xx)-2xxx(x-1-1)(1-x)-2xx(x-1)xxxx+1xx(x-1)xxxxxxxxxxxxxxxx2+1xx(x-1)xxx+1xxxxxxx

Now, rewrite the fraction as a division problem and simplify.
::现在,将分数重写为分数问题并简化。

$\begin{aligned} \frac{- x^{2} + 1}{x (x - 1)} \div \frac{x + 1}{x} & = \frac{- (x^{2} - 1)}{x (x - 1)} \cdot \frac{x}{x + 1} \\ = \frac{- (x - 1) (x + 1)}{x (x - 1)} \cdot \frac{x}{x + 1} \\ = - 1 \end{aligned}$
$\begin{align*}\frac{-x^2+1}{x(x-1)} \div \frac{x+1}{x} &= \frac{-(x^2-1)}{x(x-1)} \cdot \frac{x}{x+1} \\ &= \frac{-\cancel{(x-1)} \cancel{(x+1)}}{\cancel{x} \cancel{(x-1)}} \cdot \frac{\cancel{x}} {\cancel{x+1}} \\ &= -1\end{align*}$
::-x2+1x(x- 1) x+1x(x2- 1)xxxxxxxx(x1)xxxxxx+1}(x-1)xxx(x-1)xx(x-1)xxxxxxxxxx+1*1

Example 4
::例4

$\begin{align*}\frac{\frac{\text{-}3}{\text{-}4x^2-5x+6} + \frac{\text{-}4}{\text{-}4x+3}}{\text{-}\frac{1}{\text{-}4x^2-5x+6} + \frac{2}{\text{-}4x+3}} \end{align*}$
::-3-4x2-5x+6+4-4x+3-1-4x2-5x+6+2-4x+3

Add the numerator and the denominator of this complex fraction.
::添加此复杂分数的分子和分母。

$\begin{aligned} \frac{\frac{- 3}{- 4 x^{2} - 5 x + 6} + \frac{- 4}{- 4 x + 3}}{- \frac{1}{- 4 x^{2} - 5 x + 6} + \frac{2}{- 4 x + 3}} & = \frac{\frac{- 3}{(- 4 x + 3) (x + 2)} + \frac{(- 4)}{(- 4 x + 3)} \cdot \frac{(x + 2)}{(x + 2)}}{\frac{- 1}{(- 4 x + 3) (x + 2)} + \frac{2}{- 4 x + 3} \cdot \frac{(x + 2)}{(x + 2)}} \\ = \frac{\frac{- 4 x - 11}{- 4 x^{2} - 5 x + 6}}{\frac{2 x + 3}{- 4 x^{2} - 5 x + 6}} \\ = \frac{- 4 x - 11}{- 4 x^{2} - 5 x + 6} \cdot \frac{- 4 x^{2} - 5 x + 6}{2 x + 3} \\ = \frac{- 4 x - 11}{2 x + 3} \end{aligned}$
$\begin{align*}\frac{\frac{\text{-}3}{\text{-}4x^2-5x+6} + \frac{\text{-}4}{\text{-}4x+3}}{\text{-}\frac{1}{\text{-}4x^2-5x+6} + \frac{2}{\text{-}4x+3}} &= \frac{\frac{\text{-}3}{( \text{-}4x+3)(x+2)} + \frac{( \text{-}4)}{(\text{-}4x+3)} \cdot \frac{(x+2)}{(x+2)}}{\frac{\text{-}1}{( \text{-}4x+3)(x+2)} + \frac{2}{\text{-}4x+3} \cdot \frac{(x+2)}{(x+2)}} \\ \\ &= \frac{\frac{\text{-}4x-11}{\text{-}4x^2-5x+6}}{\frac{2x+3}{\text{-}4x^2-5x+6}} \\ \\ &= \frac{\text{-}4x-11}{\text{-}4x^2-5x+6} \cdot \frac{\text{-}4x^2-5x+6}{2x+3}\\ \\ &= \frac{\text{-}4x-11}{2x+3}\end{align*}$
::- 3-4x2-5x6+6x6(x+2)-5x+4+4+3-4-4x2-5x6+6+2-4-4x3+3=-3(-4x+3)(x+2+3)+2(4x+3)+(4x+3)+(x+2)×-1(x+2)(x+2)(x+2)+2)+2-4x3x3+3(x2)(x+2)(x+2)(4x+3)(x+2)+2)+2+2+2+2-44x+3(x+2)(x+2)(x+2)(x+3)(x+3)(x+2)+2+2+2+2+2+2+2-44x+3(x+2)(x+2)(x+2)(x+2)(x+2)(x)(x+2)=-4-4x+3x+3x+3(x)+(x+2)=4x-11-44x2-4x2-4x2x2x2x2x-2-5x2=5x2=+5x=+5x+5x+5x+6x+6x+6x+6x+6x+6x+6x+6x6x6x6x+6x6x6x6x6x6x+6x6x6x6x6x6x6x+4x6x6x=4x5x=4x=4x5x5x5x5x5x=4x=4x=4x=4x=4x=4x=4x=4x=4x=4x=4x=4xxxxx4xxxxxxxxxxxxxxx4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x=4x=4x=4x=4xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Review
::回顾

Simplify the complex fractions.
::简化复杂的分数。

$\begin{align*}\frac{\frac{2x}{5}}{\frac{8}{7}}\end{align*}$
::2x587

$\begin{align*}\frac{\frac{4}{x^2-9}}{\frac{6x}{x+3}}\end{align*}$
::4x2-96xxx+3

$\begin{align*}\frac{\frac{7x^3}{x^2+5x+6}}{\frac{35x^2}{x+2}}\end{align*}$
::7x3x2+5x+635x2x+2

$\begin{align*}\frac{\frac{24x+3}{3x+1}}{\frac{16x+2}{6x^2-13x-5}}\end{align*}$
::24x+33x+116x+26x2-13x-5

$\begin{align*}\frac{\frac{4}{x-1} + \frac{1}{x}}{\frac{1}{x} -5}\end{align*}$
::4-1+1x1x-5

$\begin{align*}\frac{\frac{3x}{x+4} - \frac{1}{x}}{\frac{3x-4}{x^2+6x+8}}\end{align*}$
::3x+4-1x3x-4x2+6x+8

$\begin{align*}\frac{8- \frac{3x}{x+5}}{\frac{10}{x+5} + \frac{5}{x+1}}\end{align*}$
::8-3xx+510x+5+5x+5x+1

$\begin{align*}\frac{\frac{x}{x+3} - \frac{4}{2x+1}}{\frac{3}{2x+1} + \frac{6}{x^2-9}}\end{align*}$
::xx+3-42x+132x+1+6x2-9

$\begin{align*}\frac{\frac{x+3}{x} + \frac{2x}{5-x}}{\frac{3}{2x} - \frac{4x}{x-5}}\end{align*}$
::x+3x+2x5-x32x-4xxx-5

$\begin{align*}\frac{\frac{2x}{5x^2-13x-6} + \frac{1}{x-3}}{\frac{4}{5x+2} - \frac{5x}{5x^2-3x-2}}\end{align*}$
::2x5x2-13x-6+1x-345x+2-5x5x2-3x-2

$\begin{align*}\frac{\frac{3x}{x^2-4} + \frac{x+4}{x^2+3x+2}}{\frac{x+1}{x^2-x-2} - \frac{2x}{x^2+2x+1}}\end{align*}$
::3xx2-4+x4x2+3x2x2+2x1x2-x-2-2x2+2x2+2x1+1

Use the following pattern to answer the next four questions.
::使用以下模式回答下四个问题。

$\begin{align*}2+\frac{1}{1+\frac{1}{2}}, \ 2+\frac{1}{1+\frac{1}{2+\frac{2}{3}}}, \ 2 + \frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{4}}}}\end{align*}$

Find the next two terms in the pattern.
::查找图案中的下两个术语。

Using your graphing calculator, simplify each term in the pattern to a decimal.
::使用您的图形计算计算器,将模式中的每个词简化为小数点后的一个词。

Make a conjecture about this pattern and the number the terms appear to be approaching.
::对这种模式和术语似乎即将到来的数量进行猜测。

Find the sixth term in the pattern. Does it support your conjecture?
::查找模式中的第六学期。它是否支持您的猜测 ?

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。

复杂分数

Section outline

Complex Fractions
::复杂分数

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)

Section outline

Complex Fractions ::复杂分数

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)

Complex Fractions
::复杂分数

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)