Equivalent Forms of Algebraic Fractions

What are algebraic fractions? How do we recognize equivalent fractions? And how do we multiply algebraic fractions? Watch the video below to find out.

 
This next video uses the distributive property to create equivalent fractions, and then gives two examples of how this can be helpful when simplifying expressions.
 

Remember:

In algebra, division is represented by a fraction. Algebraic fractions are fractions with a variable in the numerator, denominator or both. Algebraic fractions are multiplied same as regular fractions, and can be simplified before you multiply.

Try on your own:

1. Which expression does not equal  $\color{blue}{\frac{5n}{8}}$
     a.   $\frac{5}{8}n$       b.   $5n*\frac{1}{8}$       c.  $\frac{5}{n}*8$
 
2. Multiply. Express your answer in simplest form.
      $a.\quad \frac{4x}{y}*\frac{3y}{x}$
 
     $b.\quad \frac{-2n}{3}*\frac{5n}{6}*a $
 
3. One rectangle is half as wide and one-fourth as long as another rectangle. How do their areas compare?
 
4.   a. Show that $\frac{30x}{40x}$ and $\frac{3}{4}$ are equivalent by letting $x=5$.
 
      b. Show that $\frac{30 + x}{40 + x}$ and $\frac{30}{40}$ are not equivalent by letting $x=5$.
 
      c. Show that $\frac{30 + x}{40 + x}$ and $\frac{3 + x}{4 + x}$ are not equivalent by letting $x=5$.
 
5.   Simplify $\frac{45+3x}{15}$

flower fractions

Solutions:

1. Which expression does not equal  $\color{blue}{\frac{5n}{8}}$
     a.   $\frac{5}{8}n$       b.   $5n*\frac{1}{8}$        c.  $\frac{5}{n}*8$
 
To find out which answer is different we will multiple out each problem so they are all fractions. By doing this we can see that c does not equal  $\frac{5n}{8}$.

$$
\begin{array}{c|c|c}
a. \quad \color{blue}{\frac{5}{8}n} \quad & b.\quad \color{purple}{5n*\frac{1}{8}}\quad & \color{red}{(c)}\quad \color{#d3376f}{\frac{5}{n}*8} \\
\quad \color{blue}{\frac{5}{8}*\frac{n}{1}}\quad & \quad \color{purple}{\frac{5n}{1}*\frac{1}{8}}\quad & \quad \color{#d3376f}{\frac{5}{n}*\frac{8}{1}} \\
\quad \color{blue}{\frac{5n}{8}}\quad & \quad \color{purple}{\frac{5n}{8}}\quad & \quad \color{#d3376f}{\frac{40}{n}}
\end{array}
$$
 
2. Multiply. Express your answer in simplest form.
Remember: When multiplying fractions you make all the terms into fractions and then multiply across. Also if the same variable is in the numerator and the denominator of the fraction (with the same exponent) they can be simplified to 1. (ex. $\frac{x}{x}=1$)

a. $\quad \dfrac{4x}{y}*\dfrac{3y}{x} \quad$            b. $\quad \color{#008080}{\dfrac{-2n}{3}*\dfrac{5n}{6}*a}$
             
  $\quad \dfrac{4\color{red}{x}*3\color{purple}{y}}{\color{red}{x}*\color{purple}{y}}\quad $              $\quad \color{#008080}{\dfrac{-2n}{3}*\dfrac{5n}{6}*\dfrac{a}{1}} $
             
  $\quad \dfrac{4*3}{1}\quad $              $\quad \color{#008080}{\dfrac{\color{purple}{-10}n^2a}{\color{purple}{18}}} $
             
  $\quad 12 \quad $              $\quad \color{#008080}{\dfrac{-5n^2a}{9}} $

 

3. One rectangle is half as wide and one-fourth as long as another rectangle. How do their areas compare?
The first step is to draw a picture of the situation.
Drawing of problem 3 in algebraic fractions
 
The area for figure A is $\color{purple}{\frac{1}{2}w*\frac{1}{8}L=\frac{1}{8}wL}$
The area for figure B is $\color{green}{w*L=wL}$
So figure B is 8 times larger than figure A.

 

4.   a. Show that $\frac{30x}{40x}$ and $\frac{3}{4}$ are equivalent by letting $x=5$.

$\dfrac{30x}{40x}=\dfrac{30*\color{#d3376f}5}{40*\color{#d3376f}5}=\dfrac{150}{200}=\dfrac{3}{4}$

 
      b. Show that $\color{blue}{\frac{30 + x}{40 + x}}$ and $\color{purple}{\frac{30}{40}}$ are not equivalent by letting $x=5$.

$\color{blue}{\dfrac{30+x}{40+x}=\dfrac{30+\color{#d3376f}5}{40+\color{#d3376f}5}=\dfrac{35}{45}=\dfrac{7}{9}}$

$\color{purple}{\dfrac{30}{40}=\dfrac{3}{4}}$

$\color{blue}{\dfrac{7}{9}} \neq \color{purple}{\dfrac{3}{4}}$

 
      c. Show that $\color{blue}{\frac{30 + x}{40 + x}}$ and $\color{purple}{\frac{3 + x}{4 + x}}$ are not equivalent by letting $x=5$.

$\color{blue}{\dfrac{30+x}{40+x}=\dfrac{30+\color{#d3376f}5}{40+\color{#d3376f}5}=\dfrac{35}{45}=\dfrac{7}{9}}$
 
$ \color{purple}{\dfrac{3+x}{4+x}=\dfrac{3+\color{#d3376f}5 }{4+\color{#d3376f}5}=\dfrac{8}{9}}$
 
$\color{blue}{\dfrac{7}{9}} \neq \color{purple}{\dfrac{8}{9}}$

 

5.   To simplify $\color{#2f6683}{\frac{45+3x}{15}}$ we need to remember that $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$
 

$\color{#2f6683}{\dfrac{45+3x}{15}=\dfrac{45}{15}+\dfrac{3x}{15}=3+\dfrac{x}{5}}$
 
$\color{blue}{3\dfrac{x}{5}}$