What is an equivalent expression of a fraction?

At the end of this lesson, you will be able to identify equivalent fractions (or equal fractions).

Before we begin, let's first do a quick review of the vocabulary associated with fractions.

Why are Fractions used to represent numbers that are part of a whole?

A fraction indicates a number that is less than a whole. Fractions are easier to understand by looking at a graphic. Take a look....

When you cut an orange in 1/2, you no longer have a whole orange. Each piece is only part of the whole orange.

This represents the fraction 1/2.

When you eat a piece of pie, you are eating a "part" of the pie.

This is also called a "fraction" of the pie.

Fractions that are Equivalent

Now let's take a look at fractions that are equivalent.

Equivalent means "equal". Therefore, equivalent fractions are fractions that are equal in value.

Take a look at the following picture which shows three different fractions, that are all worth exactly the same amount.

Let's pretend these are one of those large Hershey bars.

As you can see from the picture above, if you eat 1/2 of the hershey bar, you will eat the same amount if you eat 2/4 of the hershey bar.

You can also eat 4/8 and you would have eaten the same amount.

These 3 fractions are equal or equivalent.

Writing Equivalent Fractions

Now that you know what equal fractions look like, let's explore how we can write fractions that are equivalent.

We'll first take a look at the fractions we used with our Hershey Bar example above.

This leads us into the Equal Fractions Property.

If the numerator and denominator of a fraction are multiplied (or divided) by the same nonzero number, then the resulting fraction is equivalent to the original fraction.

Let's put this property into action! Take a look at example 1.

Example 1 - Equivalent Fractions

Not too bad, is it? Yes, I know fractions are intimidating, but think about what you are doing and why it makes sense.

In this lesson, in order to identify fractions that are equivalent, you must multiply the numerator and denominator by the same number.

Why the same number? When you multiply the numerator and denominator by the same number, you are actually multiplying by 1.

Think about it: 2/2 = 1    and 5/5 = 1    and 20/20 = 1

When you multiply by 1, you get "the same answer". This is the reason why we are able to identify equivalent or equal fractions.

With this quick refresher, you are now able to move onto the next lesson which is on simplifying fractions.

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2. Fractions
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3. Equivalent 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 $\frac{45+3x}{15}$

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. $\frac{x}{x}=1$)

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.

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 $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$

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

$\color{blue}{3\dfrac{x}{5}}$