Simplifying Fractions: Methods, Examples and Practice

What does simplifying fractions mean?

Simplifying, or reducing to lowest terms, means rewriting a fraction using the smallest possible whole numbers without changing its mathematical value.

For example, $\dfrac{4}{8}$ represents exactly the same proportion as $\dfrac{1}{2}$. The fraction $\dfrac{1}{2}$ is the simplest way to write it because the numbers cannot be divided down any further.

Identify common factors

Before a fraction can be simplified, you must identify numbers that divide evenly into both the top and bottom parts. These numbers are called common factors.

A common factor is a whole number that divides two or more numbers without leaving a remainder.

• If the numerator and denominator are both even, they share the factor $2$.
• If both numbers end in $0$ or $5$, they share the factor $5$.
• If the sum of the digits is a multiple of $3$, they share the factor $3$.

Simplify step by step

The first method for reducing a fraction is to divide the numerator and denominator by common prime factors repeatedly.

Follow these steps:

1. Find any common factor that divides both the numerator and denominator.
2. Divide both parts by that number.
3. Check the new fraction. If the numbers still share a common factor, divide again.
4. Stop when the only common factor left is $1$.

Use the greatest common factor

Instead of dividing in multiple small steps, you can simplify a fraction in a single step using the `greatest common factor`.

The greatest common factor is the largest number that divides evenly into both the numerator and the denominator. Find all the factors for both numbers, identify the largest factor they share, and divide the top and bottom by that single number.

For the fraction $\dfrac{8}{12}$, dividing both the numerator $8$ and denominator $12$ by the greatest common factor $4$ produces the simplest form, $\dfrac{2}{3}$.

How to know when a fraction is simplest

A fraction is in simplest form, or lowest terms, when its numerator and denominator have no common factors other than $1$.

Always check your final fraction. If the top and bottom numbers can still be divided evenly by a number larger than 1, the fraction is not yet in simplest form.

For example, $\dfrac{2}{9}$ is in simplest form. The factors of $2$ are $1$ and $2$. The factors of $9$ are $1$, $3$, and $9$. Because they share no factor other than $1$, the fraction cannot be simplified further.

Worked examples

Simplifying fractions is a helpful skill in many areas of mathematics, including `comparing fractions`.

Example 1: Multi-step reduction

Question: Simplify the fraction $\dfrac{18}{24}$ using a step-by-step method.

Method:

1. Identify a common factor for $18$ and $24$. Both are even, so divide by $2$.
2. Dividing gives $\dfrac{9}{12}$.
3. Check for further factors. Both $9$ and $12$ are divisible by $3$.
4. Dividing by $3$ gives $\dfrac{3}{4}$.

Answer: The simplified fraction is $\dfrac{3}{4}$.

Check: The factors of $3$ are $1$ and $3$. The factors of $4$ are $1$, $2$, and $4$. The only common factor is $1$, so it is in simplest form.

Example 2: Greatest common factor reduction

Question: Simplify the fraction $\dfrac{20}{35}$ using its greatest common factor.

Method:

1. List the factors of $20$: $1$, $2$, $4$, $5$, $10$, $20$.
2. List the factors of $35$: $1$, $5$, $7$, $35$.
3. Identify the largest common factor, which is $5$.
4. Divide the numerator and denominator by $5$.

Answer: The simplest form is $\dfrac{4}{7}$.

Check: The only common factor of $4$ and $7$ is $1$.

Example 3: A fraction that cannot be simplified

Question: Write the fraction $\dfrac{15}{22}$ in simplest form.

Method:

1. List the factors of $15$: $1$, $3$, $5$, $15$.
2. List the factors of $22$: $1$, $2$, $11$, $22$.
3. Identify the common factors.
4. Because the only shared factor is $1$, no division will reduce the fraction further.

Answer: The fraction $\dfrac{15}{22}$ is already in simplest form.

Check: $15 \div 1 = 15$ and $22 \div 1 = 22$. The values remain unchanged.

Common mistakes

One frequent mistake is stopping the simplification process too early. For example, reducing $\dfrac{12}{16}$ to $\dfrac{6}{8}$ is a correct first step, but the fraction is not yet in simplest form because $6$ and $8$ can still be divided by $2$.

Another major error is attempting to cancel any matching digits across the numerator and denominator. Crossing out the digit $4$ in $\dfrac{14}{24}$ to get $\dfrac{1}{2}$ is mathematically invalid. You must divide the entire number by a common factor, never just cross out individual digits.

Frequently asked questions

Does simplifying a fraction change its value?

No. A simplified fraction has the exact same mathematical value as the original fraction; it is simply written using smaller numbers.

Why is it important to simplify fractions?

Simplified fractions are easier to read, understand, and use in calculations. Working with smaller numbers also makes it easier to find a `common denominator` when adding or subtracting fractions.

Can you simplify improper fractions?

Yes. An improper fraction, where the numerator is larger than the denominator, can be simplified using the exact same methods. If needed, you can simplify the numbers first and then `convert improper fractions to mixed numbers`.