**Writing and Identifying Equivalent Fractions**

Fractions that are the same in value but “look” different are known as **equivalent fractions**. How can that be? As long as you multiply or divide the top and bottom of a fraction by the same nonzero whole number, the resulting fraction will be equivalent to the original.

**Examples of Showing or Converting Equivalent Fractions**

**Example 1:** Are the fractions , , and equivalent to each other?

If we model all these fractions as the shaded parts of circles with the same size but subdivided into different equal parts, it becomes obvious that all colored regions occupy the same amount of area.

This is our visual/geometric proof why they are equivalent fractions.

**Example 2:** Are the fractions and equivalent?

There are two ways we can show why these fractions are equivalent using some arithmetic.

- One way is to start with and
**multiply**its top and bottom by**3**to get the target fraction .

- Another way is to reverse the order. I start with and
**divide**its top and bottom by**3**to arrive at .

Therefore, fractions and are equivalent fractions!

In addition, here are the two fractions represented in circles having the same size but with different equal subdivisions. The two seemingly different fractions occupy the same area.

This is how it looks when they overlap. (Animated)

**Example 3:** Show that , , , and are equivalent fractions.

The strategy is to pick any of the four fractions and using some arithmetic, transform it into the other three fractions. For this example, I would pick the smallest fraction which is .

**Step 1**: Convert to to show that they are equivalent fractions.

Multiply by to get . Thus, .

**Step 2**: Convert to to show that they are equivalent fractions.

Multiply by to get . Thus, .

**Step 3**: Convert to to show that they are equivalent fractions.

Multiply by to get . Thus, .

Since can be converted into , and , all of them are equivalent fractions.

**Example 4:** Show that , , , and are equivalent fractions.

This time, I will pick the largest fraction which is and work myself backward by transforming it to the other three fractions with lesser values. Since the values would be going down, it makes sense to use division instead of multiplication.

**Step 1**: Transform to .

Divide both the top and bottom of by **2** to obtain . Thus, .

**Step 2**: Transform to .

Divide both the top and bottom of by **10** to obtain . Thus, .

**Step 3**: Transform to .

Divide both the top and bottom of by** 20** to obtain . Thus, .

This can be a short arithmetic “proof” demonstrating that they are indeed equivalent fractions.

There is also an easier way to show that two fractions are equivalent. We may call it the “Cross-Multiplication” Rule.

**Cross-Multiplication Rule**

Here are the steps:

**Step 1**: To check if and are equivalent fractions, set them equal to each other.

**Step 2**: Perform the cross-multiplication procedure. The diagram below should help.

- Multiply the left numerator to the right denominator (red line segment). Write it as
.*ad* - Then write the equal symbol (=).
- Finally, multiply the left denominator to the right numerator (blue line segment). Write it as
**bc**.

**Step 3**: If *ad = bc* is a true statement then and are equivalent fractions. Otherwise, if *ad *≠ *bc* then the two fractions are not equivalent.

**Examples of How to Apply the Cross-Multiplication Rule to Verify if the Two Given Fractions are Equivalent**

**Example 5:** Are the fractions and equivalent?

The cross products are equal. This means that they are equivalent fractions.

**Example 6:** Are the fractions and equivalent?

These two fractions may seem to be totally different in value. But the cross multiplication rule should reveal their equivalency.

Yes, they are indeed equivalent fractions!