Fraction in Simplest Form: Definition, Examples, Verification

Fraction in Simplest Form: The simplest form of a fraction is one with a reasonably prime numerator and denominator. It signifies that the numerator (upper portion or top) and the denominator (lower part or bottom) of the fraction have no common component other than \(1\). A fraction is a value that represents a portion of a whole. The simplest form of the fraction is also known as the reduced form of a fraction. For example, \(\frac{3}{4}\) is the simplest form of a fraction with a common component of one. However, \(\frac{2}{4}\) is not the simplest form because \(\frac{2}{4}\) may be reduced further and expressed as \(\frac{1}{2}\). In this case, we can also say that \(\frac{1}{2}\) and \(\frac{2}{4}\) are equivalent fractions. Finding the simplest form of any fraction is a simple process. We need to simplify the fraction’s numerator and denominator by dividing them both by the biggest common factor that divides them entirely. Both the numerator and denominator should be entire integers after division. This approach of fractional simplification is also known as reducing fractions.

This post will tell you everything about fractions in their simplest form, their definition, examples, and verification. Read further to find more.

Definition of Fractions 

If a certain quantity of rice is divided into four equal parts, each part obtained is said to be one-fourth \(\left( {\frac{1}{4}} \right)\) of the whole quantity of the rice. Similarly, if an orange is divided into five equal parts, each part is one-fifth \(\left( {\frac{1}{5}} \right)\) of the whole orange. Now, if two parts of these five equal parts are eaten, three parts are left and we say three-fifths \(\left( {\frac{3}{5}} \right)\) of the orange is left.

The numbers \(\left( {\frac{1}{4}} \right),\,\left( {\frac{1}{5}} \right),\,\left( {\frac{3}{5}} \right)\) discussed above, each represents a part of the whole quantity, are called fractions.

A fraction is a quantity that expresses a part of the whole. So, the numbers of the form \(\frac{x}{y}\), where \(x\) and \(y\) are whole numbers, and \(y≠0\), are called fractions.

Here, \(x→\) Numerator and \(y→\) Denominator.

Types of Fractions

There are different types of fractions. Let us understand each type.

1. Proper Fractions

A fraction whose numerator is less than its denominator is called a proper fraction. The value of the proper fraction is always less than \(1\).

For example, \(\frac{3}{4},\,\frac{7}{{10}},\,\frac{1}{4},\,\frac{3}{7}\) are all proper fractions. 

2. Improper Fractions

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.

For example, \(\frac{7}{4},\,\frac{{13}}{5},\,\frac{{11}}{6},\,\frac{{23}}{7}\) are all improper fractions.

3. Mixed Fractions

A combination of a whole number and a proper fraction is called a mixed fraction.

For example, \(2\frac{2}{4},\,6\frac{5}{{10}},\,5\frac{1}{5},\,6\frac{2}{{13}}\) are all mixed fractions.

4. Like Fractions

The group of two or more fractions that have the same denominators are like fractions.

For example, \(\frac{1}{5},\,\frac{2}{5},\,\frac{3}{5},\,\frac{4}{5},\,\frac{7}{5}\) are all like fractions.

5. Unlike Fractions

The group of two or more fractions that have different denominators are unlike fractions.

For example, \(\frac{1}{6},\,\frac{2}{4},\,\frac{2}{5},\,\frac{4}{7},\,\frac{5}{8}\) are all unlike fractions.

6. Unit Fractions

A unit fraction is any fraction with \(1\) as its numerator and a non-zero whole number as the denominator.

For example, \(\frac{1}{9},\,\frac{1}{4},\,\frac{1}{5},\,\frac{1}{3},\,\frac{1}{8}\) are all unit fractions.

7. Decimal Fractions

A decimal fraction is a fraction whose denominator is a power of \(10\) or a multiple of \(10\) like \(100, 1,000, 10,000\), etc.

For example, \(\frac{3}{{10}},\,\frac{4}{{100}},\,\frac{{13}}{{10}},\,\frac{9}{{1000}}\) are all decimal fractions.

Simplest Form of Fraction

A fraction is said to be in its simplest form if \(1\) is the only common factor of its numerator and denominator. So, a fraction can not be said to be in its simplest form if the numerator and the denominator have any common factor other than \(1\).

For example, \(\frac{3}{4}\) is in its simplest form as \(1\) is the only common factor of \(3\) and \(4\) in this fraction. We can simplify fractions as it reduces the complexities in calculations.

Rules to Find the Simplest Form of a Fraction

Let the given fraction be \(\frac{a}{b}\) and the HCF of \(a\) and \(b\) be \(h\). Then, \(\frac{a}{b} = \frac{{a \div h}}{{b \div h}}\) is the simplest form.

For example, the simplest form of \(\frac{8}{{24}}\).

Now, we will find the HCF of \(8, 24\). HCF of two numbers is known as the highest or the largest common factor between two or more numbers.

HCF \((8, 24)=8\)

\(\Rightarrow \frac{8 \div 8}{24 \div 8}=\frac{1}{3}\)

Hence, we can get the simplest form of a fraction by dividing the numerator and the denominator by the same number, and the number should be the HCF of the numerator and the denominator of the fraction.

We can apply this method also, but here we need to do the process several times as after dividing the numerator and the denominator of \(\frac{8}{{24}}\) by \(2\), we are getting \(\frac{4}{{12}}\) which is not the simplest form.

So, we will continue the process till we get \(1\) as the common factor of the numerator and the denominator. The first method is easier to find the simplest form than the second method as the second one is a lengthy process.

Solved Examples – Fraction in Simplest Form

Q.1. Is the fraction \(\frac{3}{7}\) in its simplest form?
Ans: Given fractions is  \(\frac{3}{7}\).
The numerator is \(3\), and the denominator is \(7\). To check whether the fraction is in its simplest form, we will find the HCF of \(3, 7\).
Now, HCF \((3, 7)=1\) as \(3, 7\) are coprime numbers.
Therefore, \(\frac{3}{7}\) is in its simplest form as the highest common factor of the numerator, and the denominator is \(1\).

Q.2. Find the fractions in the simplest form from the following. \(\frac{3}{6},\,\frac{1}{3},\,\frac{2}{5},\,\frac{4}{6},\,\frac{5}{{15}}\).
Ans: Given fractions are, \(\frac{3}{6},\,\frac{1}{3},\,\frac{2}{5},\,\frac{4}{6},\,\frac{5}{{15}}\)
To find the simplest form, first, we will find the highest common factor of the numerator and the denominator of each fraction. The fractions whose HCF of the numerator and the denominator is only \(1\), can be said the fractions are in the simplest form.
In \(\frac{3}{6}\), the HCF of \(3,\,6\) is \(3.\)
In \(\frac{1}{3}\), the HCF of \(1, 3\) is \(1.\)
In \(\frac{2}{5}\), the HCF of \(2, 5\) is \(1.\)
In \(\frac{4}{6}\), the HCF of \(4,6\) is \(2.\)
In \(\frac{5}{15}\), the HCF of \(5, 15\) is \(5.\)
Hence, \(\frac{1}{3}\) and \(\frac{2}{5}\) are in their simplest form.

Q.3. What is the simplified form of the fraction \(\frac{144}{36}\)?
Ans: The given fraction is \(\frac{144}{36}\)
We need to find the lowest form or the simplest form of the given fraction.
Now, the highest common factor of \(144, 36\) is \(36.\)
So, \(\frac{{144 \div 36}}{{36 \div 36}} = \frac{4}{1} = 4.\)
Hence, the simplest form is \(4.\)

Q.4. Is \(\frac{5}{15}\) the lowest form of \(\frac{25}{75}\)?
Ans: No, \(\frac{5}{15}\) is the not lowest form of \(\frac{25}{75}.\)
\(\frac{5}{15}\) can be simplified again if we divide the numerator and the denominator by \(5\).
So, the lowest form of \(\frac{25}{75}\) is \(\frac{1}{3}\),  as \(\frac{{25 \div 25}}{{75 \div 25}} = \frac{1}{3}.\)
Hence, \(\frac{5}{15}\) is not the lowest form of \(\frac{25}{75}.\)

Q.5. What is the simplest form of the fraction \(\frac{2}{5}\) ?
Ans: Given fractions is \(\frac{2}{5}.\)
The numerator is \(2\), and the denominator is \(5.\) To check whether the fraction is in its simplest form or not, we will find the HCF of \(2, 5.\)
Now, HCF \((2, 5)=1\) as \(2, 5\) are coprime numbers.
Therefore, \(\frac{2}{5}\) is in its simplest form, and we can not reduce or simplify more.

Summary

In this article, we covered the definition of the fraction, types of fractions, the simplest form of a fraction, ways to identify if a fraction is in its simplest form or not and how to find the simplest form of a fraction.

Frequently Asked Questions (FAQs) – Fraction in Simplest Form

Q.1. Explain the fraction in the simplest form with an example.
Ans: A fraction is said to be in its simplest form if \(1\) is the only common factor of its numerator and denominator. So, a fraction can not be said in its simplest form if the numerator and the denominator have a common factor other than \(1\). For example, \(\frac{3}{4}\) is in its simplest form as \(1\) is the only common factor of \(3\) and \(4\) in this fraction.

Q.2. What is the simplest fraction form of \(1.33\)?
Ans: The fraction form of the decimal number \(1.33\) is \(\frac{{133}}{{100}}\).
\(100\) and \(133\) are the coprime numbers and their HCF is \(1\).
Hence, the simplest fraction form of the given decimal number is \(\frac{{133}}{{100}}\).

Q.3. How do you express the fractions in the simplest form?
Ans: First, we will find the highest common factor (HCF) of the numerator and denominator of the given fraction. If the HCF is \(1\), then the fraction is in its simplest form. If the HCF is other than \(1\), then divide both the numerator and denominator by HCF, and get the simplest form of the fraction.

Q.4. Can we find the simplest form of a fraction by multiplying the numerator and the denominator by the same non-zero number?
Ans: No, we can not. We can get the lowest or the simplest form of a fraction by dividing the numerator and the denominator by the HCF of the numerator and the denominator.

Q.5. What is the difference between the simplest form of a fraction and the equivalent fractions?
Ans: The simplest form of a fraction is also an equivalent fraction, but the ways to find the equivalent fraction and the simplest form of a fraction are slightly different. If we multiply or divide the numerator and the denominator of a fraction by the same non-zero number, we will get the equivalent fractions. To find the simplest form of a fraction, we need to divide the numerator and the denominator by the HCF of the numerator and the denominator.

Now that you are provided with a detailed article on the simplest form of a fraction, we hope all your doubts are cleared on this topic. If you have any queries or questions, you can ask them in the comment box below. We will be more than happy to help you. Best of luck with your studies!