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Equivalent Fractions

When two (or more) fractions are equivalent, they have the same value.

Sometimes, equivalent fractions don't look alike at first. In fact, it can sometimes be hard to tell right away that two fractions actually have the same value.

The good news is that it's pretty easy to find out if two fractions are equivalent once you simplify and reduce them.

It might be a little gross and slimy, but think of equivalent fractions like a slug and a snail. They might not look the same at first—after all, the snail has a shell. But, when you simplify the snail by getting rid of its shell, it looks just like a slug. The snail becomes equivalent to the slug.

The same is true with fractions. To tell if they are equivalent, just reduce them down into their simplest forms (i.e., their slug form).

(NOTE: Please do not pull the shells off your neighborhood snails. Be nice to them).

Let's take a look at some examples of equivalent fractions.

These $3$ fractions don't look the same right now. But (spoiler alert), they are equivalent.

So, once all of the fractions are reduced, we get:

Because the $3$ original fractions all have the same value once they're simplified, we can say that they are all equivalent.

Let's try another approach, just to get a little more practice.

Can't Spell Equivalent Without Equal

We say that two fractions are equivalent if they equal the same value. In other words, fractions are equivalent if they're equal.

While this might seem like a no brainer, it can sometimes be difficult to tell if two fractions are equal. For example, $\frac9{51}$ and $\frac{21}{119}$ are equivalent fractions. I clearly can't tell.

Let's look at some methods at figuring out if two fractions are equivalent.

Draw a Picture

Drawing a picture is a good way to visually prove two fractions are equivalent but it can be time consuming for bigger fractions. Let's look at a simpler example with the fractions $\frac12$ and $\frac24$.

The top Nick Jonas face is divided into $2$ equal parts with $1$ part shaded in. This represents the fraction $\frac12$. The bottom Nick Jonas face is divided into $4$ equal parts with $2$ parts shaded in. This represents the fraction $\frac24$.

Notice that in both Nick Jonas faces, the filled in part is the exact same. This is because the fractions are equivalent. See now how this could get cumbersome to represent the fraction $\frac{21}{119}$?

Reduce

A more general way to see if fractions are equivalent is to reduce them to simplest form. Simplest form is when the numerator and denominator share no common factors other than $1$.

Let's see if the fractions $\frac3{18}$ and $\frac{7}{42}$ are equivalent. Start with the first fraction and look to see what common factors they share by writing out the prime factorization of the top and bottom. $$\frac3{18}=\frac{\color{#8925AE}{3}}{2\times\color{#8925AE}{3}\times3}$$ Since they share a $3$, we can divide both the top and bottom by that. $$\frac3{18}{{\div\,3}\atop{\div\,3}}=\frac16$$ This is the simplest form of the first fraction.

Let's check the second fraction now. $$\frac{7}{42}=\frac{\color{#8925AE}{7}}{2\times3\times\color{#8925AE}{7}}$$ They both share a factor of $7$, so we can remove that from both. $$\frac7{42}{{\div\,7}\atop{\div\,7}}=\frac16$$ This is the simplest form of the second fraction. Notice that it is the same as the first fraction. Therefore, we know these two fractions are equivalent. $$\frac{3}{18}=\frac{7}{42}$$

Divide

The final method is to just divide and find the decimal representations of each number.

Let's try this with $\frac38$ and $\frac{12}{32}$.

Notice they equal the same decimal value. Therefore, they're equivalent.

Creating Equivalent Fractions

In this area, a common problem you might encounter is as follows:

Find a fraction with a denominator of $40$ that is equivalent to $\frac15$.

These problems will either ask you to find the numerator or the denominator of the equivalent fraction. To do this, we can rewrite the problem as follows: $$\frac15=\frac{?}{40}$$

This resembles a problem about proportions. First, we look at the denominators, since that was what was given. What can we multiply $5$ by to give us $40$? The answer is $8$.

Second, multiply the other part of the known fraction by this number. We are multiplying the top and bottom of the fraction by the same number so we know we aren't changing its value. $$\frac15{{\times\,8}\atop{\times\,8}}=\frac{8}{40}$$

This gives us our equivalent fraction! Use any of the methods above to double check your work.

(Video) Equivalent Fractions

Christine Munafo

Here's a video by Christine Munafo that explains equivalent fractions.

$\large{\text{Summary}}$

The first key piece of information is the word equivalent. Equivalent means equal amounts or an equal value of something.

An example of equivalent fractions would be $\frac12$ and $\frac24$. They may look different, but they each refer to the same amount of something.

Since $\frac12$ and $\frac24$ are equivalent, we say that: $$\frac12=\frac24$$

Creating Equivalent Fractions

Creating an equivalent fraction is simple. Suppose we have the fraction $\frac38$ and want to find the equivalent version of it with 16 in the denominator. $$\frac38=\frac{{?}}{16}$$

To figure this out, we look at what number we can multiply $8$ by to get $16$. In other words, $16\div8$. The answer is $2$.

To get the new numerator, we just multiply the old one by the number we found, $2$. $$\frac{3}{8}\cdot{2\atop2}=\frac{6}{16}$$

This works with multiplication and division.

In general, if a fraction can be reduced, then its simplest form is equivalent to the original fraction. The example $\frac{1}{2} = \frac{2}{4}$ works well for this, because $\frac{2}{4}$ reduces to $\frac{1}{2}$. By the same logic, the following fractions are all equivalent to $\frac{1}{2}:$

$$\frac{3}{6}, \frac{4}{8}, \frac{5}{10}, \frac{6}{12}, ...$$