Contents

- Definition
- Simplest Form of Fractions
- Steps To Find the Simplest Form of a Fraction
- Simplest Form of Fractions With Exponents
- Simplest Form of Fractions With Variables
- Simplest Form of Fractions With Mixed Fractions
- Simplest Form of Fractions With Improper Fractions
- Some Examples of the Simplest Form of Fractions

# Simplest Form (Fractions)|Definition & Meaning

**Definition**

A **fraction’s simplest form** is one with a denominator and numerator that are both fairly prime numbers. It shows that the **numerator and denominator** of the fraction do not have any common factors.

A fraction is a **numerical expression** representing a subset of the whole. The reduced form of a fraction is another name for its most **basic form**. For instance, the simplest representation of a fraction with a common component of 1 is $\frac{3}{4}$. The simplest form, however, is not $\frac{2}{4}$ since $\frac{1}{2}$ is a further **simplification** of $\frac{2}{4}$ that may be written. In this case, we can also claim that the fractions $\frac{1}{2}$ and $\frac{2}{4}$ are equal.

Figure 1 illustrates the example of the simplest form of fraction as $\frac{2}{4}$ can be equivalent or can be written in simplest form as $\frac{1}{2}$.

**Simplest Form of Fractions**

When the top and bottom of a fraction are relatively prime integers, the fraction is said to be in its simplest form. In their most **basic form**, fractions are straightforward to locate. By dividing the numerator and denominator of a fraction by the greatest **common divisor** that divides them exactly, you can easily simplify the **numerator and denominator** of the fraction.

After division, the numerator and denominator must both still be integers. This **fraction-simplifying** procedure is also known as fraction **reduction**. The fraction $\frac{ac}{bc}$ is reduced to $\frac{a}{b}$ by removing the common component “c” from both the **numerator and denominator**.

To simplify a fraction, divide its top and bottom by the largest integer that divides both values equally (they must stay whole numbers).

**Steps To Find the Simplest Form of a Fraction**

- Find the Highest common factor (HCF) of
**the****Numerator**and**Denominator**of**a****Fraction**. - Divide the numerator and denominator by the
**generated**HCF. - Write the
**abbreviated**fraction of the given fraction.

** ****Simplest Form of Fractions With Exponents**

**Fractions** with exponents in the numerator and denominator can be **simplified.** To simplify **fractions** **with** exponents, use the **exponential** **extension** form in the numerator and denominator. **Exponents** **are** **sometimes** **used** to make **numbers** easier to read.

**Simplest Form of Fractions With Variables**

It is also possible to simplify fractions that have variables in the **numerator and denominator**. Use the extended form of each word in the numerator and denominator to **simplify fractions with variables.**

**Simplest Form of Fractions With Mixed Fractions**

A **proper fraction** and a whole are combined to form a mixed fraction. You must only simplify the fractional component of a **mixed fraction** in order to simplify it. To do this, factor the denominator and numerator and eliminate any **shared components**. The new **numerator and denominator** of the mixed fraction will be the outcome.

Steps to form the simplest form of fractions with Mixed Fractions

- Find the fraction’s numerator and denominator’s highest common factor (HCF).
- To obtain the simplified fraction, divide the denominator and numerator by the highest common factor (HCF).
- Together, write the simple fraction and the entire amount.

**Simplest Form of Fractions With Improper Fractions**

If the numerator of **a fraction is higher than or equal to the denominator, the fraction is deemed as an improper fraction.** **Inappropriate** fractions **should** be **converted** **to** mixed fractions **for** **simplification.** **This** **means** dividing the numerator by the denominator. **It** **is** **then** **expressed** **in** **mixed** **number** **form,** **with** the quotient as the **integer,** the **remainder** as the numerator, and the divisor as the **denominator.**

Steps to form the simplest form of fractions with Improper Fractions

- Find the numerator’s and the denominator’s greatest common factor (HCF).
- HCF is divided by the numerator and denominator.

** **In order to reduce the improper fractions completely we convert the improper fractions to mixed fractions. Here are the steps to convert improper fractions to mixed fractions

**Divide**numerator by denominator.- Put the outcome down as a
**whole number.** - Any leftover amount should be used as the
**fraction’s numerator**. - The
**numerator**remains constant.

**Some Examples of the Simplest Form of Fractions**

**Example 1**

Reduce the fraction illustrated in figure 2

### Solution

We can reduce the fraction if we take four common from both the numerator and denominator then $\dfrac{1}{2}$ will be the reduced fraction illustrated in figure 3.

**Example 2**

Reduce the following fractions

a) $\dfrac{15}{35}$

b) $\dfrac{4}{16}$

c) $\dfrac{3}{6}$

### Solution

a) For reducing fractions, we take the highest common factor (HCF) of fifteen and thirty-five. The HCF of fifteen and thirty-five is five.

$\dfrac{3 \times 5}{7 \times 5}$ which is equal to $\dfrac{3}{7}$

b) For reducing fractions, we take the highest common factor (HCF) of four and sixteen. The HCF of four and sixteen is four.

$\dfrac{1 \times 4}{4 \times 4}$ which is equal to $\dfrac{1}{4}$

c) For reducing fractions, we take the highest common factor (HCF) of three and six. The HCF of three and six is three.

$\dfrac{1 \times 3}{2 \times 3}$ which is equal to $\dfrac{1}{2}$

**Example 3**

Check whether $\dfrac{7}{15}$ is in reduced form or not.

**Solution**

We find the factors of seven and fifteen:

Seven: 1,7

Fifteen: 1,3,5,15

One is the only common factor.

So, $\dfrac{7}{15}$ is in its original reduced form.

**Example 4**

Reduce $\dfrac{12}{18}$ to simplest form.

**Solution**

Factors of twelve are 1,2,3,4,6,12

Factors of eighteen are 1,2,3,6,9,18

the highest common factor (HCF) is six, so the fraction will be:

\[\dfrac{6 \times 2}{6 \times 3}\]

Which will be equal to $\dfrac{2}{3}$ hence the reduced form of $\dfrac{12}{18}$ is:

$\dfrac{2}{3}$

**Example 5**

Reduce the following fractions in reduced form.

a) $\dfrac{yz^2}{2z}$

b) $\dfrac{3^2}{3^5}$

**Solution**

a) Express both numerator and denominator in the form of product as the original fraction is a mixed variable.

$\dfrac{y \times z \times z}{2z}$

As we can see z from numerator tor and z from denominator cancel out so the reduced fraction will be equal to:

$\dfrac{yz}{2}$

b) Express both numerator and denominator in the form of product as the original fraction is a mixed variable.

$\dfrac{3 \times 3}{3 \times 3 \times 3 \times 3 \times 3}$

As we can see nine from numerator and nine from the denominator cancel out so the reduced fraction will be equal to $\dfrac{1}{27}$.

*All images/mathematical drawings were created with GeoGebra.*