Equivalent Fractions: Definition, Method and Examples

What are equivalent fractions?

Equivalent fractions are fractions that represent the exact same portion of a whole, even though they use different numbers. They look different but have the same value.

Equivalent fractions have different numerators and denominators but equal the same amount.

For example, if you eat $\dfrac{1}{2}$ of a pizza, you are eating the exact same amount as someone who eats $\dfrac{2}{4}$ of an identical pizza, or $\dfrac{4}{8}$ of that same pizza. The slices are simply cut into different sizes, but the total amount of food is equal.

Every fraction has an infinite number of equivalent fractions. Recognizing them is an essential step before comparing fractions or adding them together.

See equivalence in visual models

Visual models are one of the clearest ways to find equivalent fractions. By comparing identical shapes divided into different numbers of pieces, you can see that the shaded areas remain exactly the same size.

Make equivalent fractions

You can generate an equivalent fraction mathematically by applying a simple rule to the numerator (the top number) and the denominator (the bottom number).

To make an equivalent fraction, multiply or divide both the numerator and the denominator by the exact same nonzero whole number.

When you multiply, you are splitting the existing parts into smaller, more numerous pieces. When you divide, you are grouping smaller pieces together into larger, fewer pieces. Both actions leave the total portion the same.

Example 1: Multiplying to find equivalent fractions

Question: Find two equivalent fractions for $\dfrac{3}{5}$.

Method:

1. Choose any whole number greater than $1$.
2. Multiply the numerator and denominator by that number.
3. Repeat with a different number to find a second equivalent fraction.

Answer: Using $2$, we get $\dfrac{3 \times 2}{5 \times 2} = \dfrac{6}{10}$. Using $3$, we get $\dfrac{3 \times 3}{5 \times 3} = \dfrac{9}{15}$. Both $\dfrac{6}{10}$ and $\dfrac{9}{15}$ are equivalent to $\dfrac{3}{5}$.

Check: If you divide the numerator and denominator of $\dfrac{6}{10}$ by $2$, it simplifies back to $\dfrac{3}{5}$.

When you divide to find an equivalent fraction, the process is known as simplifying fractions. You must choose a number that divides evenly into both the numerator and denominator so that no decimals are left.

Example 2: Dividing to simplify a fraction

Question: Find an equivalent fraction for $\dfrac{12}{18}$ by dividing.

Method:

1. Identify a common factor that divides evenly into both $12$ and $18$.
2. Divide the numerator and denominator by that number.

Answer: The number $6$ divides evenly into both. $\dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3}$. Therefore, $\dfrac{2}{3}$ is an equivalent fraction.

Check: Multiplying $\dfrac{2}{3}$ by $6$ gives back $\dfrac{12}{18}$, confirming the values are equal.

Why multiply or divide both parts

When you multiply the top and bottom by the same number, you are mathematically multiplying the fraction by a form of $1$. Any number multiplied by $1$ remains identical in value.

For example, $\dfrac{2}{2}$ is equal to $1$. If you start with $\dfrac{3}{4}$ and multiply by $\dfrac{2}{2}$, the problem looks like this:

$\dfrac{3}{4} \times \dfrac{2}{2} = \dfrac{6}{8}$

Because you multiplied by $1$, the value of the fraction has not changed. The identical principle applies when dividing: dividing the numerator and denominator by the same number is mathematically the same as dividing the fraction by $1$.

This is also an important rule when you need a common denominator to add or subtract fractions.

Check whether fractions are equivalent

There are multiple ways to verify if two fractions share the same value.

1. Simplify completely: If both fractions simplify to the same fraction, they are equivalent. For example, $\dfrac{4}{12}$ and $\dfrac{5}{15}$ both simplify to $\dfrac{1}{3}$.
2. Look for a multiplier: Check if you can multiply the top and bottom of the smaller fraction by the same whole number to get the larger fraction.
3. Cross-multiply: Multiply the numerator of the first fraction by the denominator of the second fraction. Then, multiply the denominator of the first by the numerator of the second. If the two products are equal, the fractions are equivalent.

Example 3: Checking for equivalence

Question: Are $\dfrac{3}{8}$ and $\dfrac{9}{24}$ equivalent fractions?

Method:

1. Attempt to find a whole-number multiplier linking the numerators.
2. Verify if the identical multiplier connects the denominators.

Answer: Yes. The numerator $3$ multiplies by $3$ to reach $9$. The denominator $8$ multiplies by the same $3$ to reach $24$. The fractions are equivalent.

Check: Using cross-multiplication: $3 \times 24 = 72$, and $8 \times 9 = 72$. The products match, confirming equivalence.

Equivalent fractions on a number line

A number line provides another visual way to prove fractions are equivalent. When you plot equivalent fractions on a number line, they land on the exact same point.

For instance, the fraction $\dfrac{1}{2}$ sits exactly halfway between $0$ and $1$. If you divide the number line into four equal sections instead of two, the point $\dfrac{2}{4}$ is at that same halfway mark.

Common mistakes

The most common error when working with equivalent fractions is adding or subtracting the same number from the numerator and denominator, instead of multiplying or dividing.

Never add or subtract to find an equivalent fraction.

If you take $\dfrac{1}{2}$ and add $1$ to both the numerator and the denominator, you get $\dfrac{2}{3}$. However, $\dfrac{2}{3}$ is noticeably larger than $\dfrac{1}{2}$. The rule of equivalence only works with multiplication and division.

Another common mistake is forgetting to perform the exact same operation to both parts of the fraction. If you multiply the numerator by $2$, you must also multiply the denominator by $2$.

Frequently asked questions

Are simplified fractions always equivalent?

Yes. When you simplify a fraction by dividing both the numerator and the denominator by a common factor, the resulting fraction represents the same value and is entirely equivalent to the original.

How do equivalent fractions help with cooking?

If a recipe requires $\dfrac{1}{2}$ cup of sugar, but you only have a $\dfrac{1}{4}$ measuring cup, knowing equivalent fractions allows you to fill the $\dfrac{1}{4}$ cup exactly two times.

Is it possible to make equivalent fractions with decimals?

While you can mathematically generate equivalent ratios using decimals, standard proper fractions are written with whole numbers. You should choose multipliers or divisors that keep both the top and bottom parts as whole numbers.

Do equivalent fractions apply to mixed numbers?

Yes. For mixed numbers, you can find equivalent fractions by keeping the whole number the same and applying the multiplication or division rule to the fraction part. For example, $1\dfrac{1}{2}$ is equivalent to $1\dfrac{2}{4}$.

Are benchmark fractions equivalent to other fractions?

Yes. Common benchmark fractions, such as $\dfrac{1}{4}$, $\dfrac{1}{2}$, and $\dfrac{3}{4}$, have countless equivalent forms. Recognizing that $\dfrac{50}{100}$ is equivalent to the benchmark fraction $\dfrac{1}{2}$ helps you estimate values quickly.