A Fraction between Two Fractions: A fraction is a small portion of a larger whole or collection. When an object or a whole is divided into equal parts, each part represents a fraction of the given object or whole. When stated in numbers, a fraction has two parts: a numerator and a denominator.

The numerator displays how many selected or shaded portions we have, whereas the denominator tells how many overall pieces of an object or a whole we have. We can find fractions between any two fractions. This article discusses how to find a fraction between two given fractions. Read the full article to get complete details.

Fraction Definition

Fractions are represented as numerical values in mathematics and can be defined as parts of a whole. A fraction is a part or section of a whole that can be any number, a specified value, or an item.

Thus,  

\({\text{Fraction}} = \frac{{{\text{ Number}}\,{\text{of}}\,{\text{selected}}\,{\text{or}}\,{\text{shaded}}\,{\text{parts}}\,{\text{of}}\,{\text{an}}\,{\text{object}}\,{\text{or}}\,{\text{a}}\,{\text{whole }}}}{{{\text{ Total}}\,{\text{number}}\,{\text{of}}\,{\text{equal}}\,{\text{parts}}\,{\text{of}}\,{\text{an}}\,{\text{object}}\,{\text{or}}\,{\text{a}}\,{\text{whole }}}} = \frac{{{\text{ Numerator }}}}{{{\text{ Denominator }}}}\)

Take into account the fraction \(\frac{5}{{12}}\) This fraction is read as “five-twelfth” which means that \(5\) parts out of \(12\) are equal parts divided by the whole. 

In the fraction \(\frac{7}{{12}},7\) is known as a numerator, and \(12\) is known as a denominator.

The following are a few more examples:

Fraction	Meaning of the fraction	Numerator	Denominator
\(\frac{5}{{11}}\) or Five-elevenths	Five equal parts out of \(11\) equal parts in which the whole is divided.	\(5\)	\(11\)
\(\frac{3}{{8}}\) or Three-eighths	Three equal parts out of \(8\) equal parts in which the whole is divided.	\(3\)	\(8\)
\(\frac{1}{{3}}\) or One-third	One part out of \(3\) equal parts in which the whole is divided.	\(1\)	\(3\)

Fraction Examples

A fraction is a number that represents a part of a whole. A single object or a group of objects might make up the whole. Take a rectangle sheet and fold it in half. Fold it horizontally and vertically to divide it into four equal sections. As illustrated in the figure below, one of the four components should be shaded out. The shaded area makes up one-fourth of the overall composition. The number one-fourth is written as \(\frac{1}{{4}},\) which is nothing but a fraction.

If three portions are darkened, as in the figure below, the shaded portion represents three-quarters of the total. Three-fourths is written as \(\frac{3}{{4}}\) and is read as ‘three by four’ or ‘three over four’. Thus, three parts out of \(4\)  equal parts is \(\frac{3}{{4}}.\)

Similarly, \(\frac{3}{{7}}\) is obtained when we divide a whole into \(7\) equal parts and take three parts (see figure below).

For \(\frac{1}{{8}},\) we divide a whole into eight equal parts and take one part of it (see figure below).

Fraction Between Two Fractions

A fraction is made up of two elements. The number on the top of the line or fraction bar is called the numerator. It determines how many equal pieces of the entire collection or whole are taken. The denominator is the number below the line. It displays the total number of equal parts into which the whole is divided or the total number of equal parts in a collection.

Finding a fraction between two fractions calculator is not a hard process. Simply make the sum of the numerators as the new numerator and the sum of the denominators as the new denominator to obtain a fraction between two given fractions.

The following are examples of how to insert a fraction between two fractions:

If \(\frac{p}{q}\) and \(\frac{r}{s}\) are two given fractions and \(\frac{p}{q} < \frac{r}{s}\) then \(\frac{p}{q} < \frac{{p + r}}{{q + s}} < \frac{r}{s}.\)

Where, \(p,q,r\) and \(s\) are the positive integers.

Example: Insert a fraction between the two fractions \(\frac{5}{7}\) and \(\frac{3}{5},\) given \(\frac{5}{7} < \frac{3}{5}.\)

Solution: Required fraction between \(\frac{5}{7}\) and \(\frac{2}{5}\) is \(\frac{{(5 + 2)}}{{(7 + 5)}} = \frac{7}{{12}}\)

Therefore, \(\frac{5}{7} < \frac{7}{{12}} < \frac{3}{5}.\)

Solved Examples – A Fraction Between Two Given Fractions

Q.1. Insert a fraction between the two fractions \(\frac{1}{3}\) and \(\frac{2}{5},\) given \(\frac{1}{3} < \frac{2}{5}.\)
Ans: To insert a fraction between two fractions, make the sum of the numerators the new numerator and the sum of the denominators the new denominator.
Here, the sum of the numerators\(=1+2=3\) and
the sum of the denominators\(=3+5=8\)
So, the new fraction formed between the two fractions \(\frac{1}{3}\) and \(\frac{2}{5}\) is \(\frac{3}{8}.\)

Q.2. Find the fraction between the two fractions \(\frac{2}{7}\) and \(\frac{4}{5},\) given \(\frac{2}{7} < \frac{4}{5}.\)
Ans: To insert a fraction between two fractions, make the sum of the numerators the new numerator and the sum of the denominators the new denominator.
The required fraction between the two fractions \(\frac{2}{7}\) and \(\frac{4}{5}\) is \(\frac{{(2 + 4)}}{{(7 + 5)}} = \frac{6}{{12}} = \frac{1}{2}.\)

Q.3. Find the fraction between the two fractions \(\frac{4}{7}\) and \(\frac{1}{3},\) given \(\frac{4}{7} < \frac{1}{3}.\)
Ans: To insert a fraction between two fractions, make the sum of the numerators the new numerator and the sum of the denominators the new denominator.
The required fraction between the two fractions \(\frac{4}{7}\) and \(\frac{1}{3}\) is \(\frac{{(4 + 1)}}{{(7 + 3)}} = \frac{5}{{10}} = \frac{1}{2}.\)

Q.4. Insert a fraction between the two fractions \(\frac{5}{6}\) and \(\frac{7}{11},\) given \(\frac{5}{6} < \frac{7}{{11}}.\)
Ans: To insert a fraction between two fractions, make the sum of the numerators the new numerator and the sum of the denominators the new denominator.
The required fraction between the two fractions \(\frac{5}{6}\) and \(\frac{7}{11}\) is \(\frac{{(5 + 7)}}{{(6 + 11)}} = \frac{{12}}{{17}}.\)

Q.5. Insert a fraction between the two fractions \(\frac{5}{7}\) and \(\frac{8}{11},\) given \(\frac{5}{7} < \frac{8}{{11}}.\)
Ans: To insert a fraction between two fractions, make the sum of the numerators the new numerator and the sum of the denominators the new denominator.
Here, the sum of the numerators\(=5+8=13\) and
the sum of the denominators\(=7+11=18\)
So, the new fraction formed between the two fractions \(\frac{5}{7}\) and \(\frac{8}{11}\) is \(\frac{13}{18}.\)

Summary 

In this article, we learnt about the definition of the fractions, examples of fractions, finding the fraction between two fractions calculator, solved examples on a fraction between two given fractions, and FAQs on a fraction between two given fractions. You can also find “How to Find a fraction between two fractions worksheet” on the Embibe app and website.

The learning outcome of this article is, we understood that how to insert a fraction between the two given fractions. To insert a fraction between two fractions, make the sum of the numerators the new numerator and the sum of the denominators the new denominator.

Learn About Different Types of Fractions

Frequently Asked Questions (FAQs)

Q.1. How to find a fraction between two fractions?
Ans. Finding a fraction between two fractions is not a hard process. Simply make the sum of the numerators the new numerator and the denominators the new denominator to obtain a fraction between two fractions.
The following are examples of how to insert a fraction between two provided fractions:
If (\frac{p}{q}) and (\frac{r}{s}) are two given fractions and (\frac{p}{q} < \frac{r}{s}) then (\frac{p}{q} < \frac{{p + r}}{{q + s}} < \frac{r}{s}.)

Q.2. Is there always a fraction between any two fractions?
Ans: There is a fraction between any two whole numbers. There is \(\frac{1}{2}\) between \(0\) and \(1,\frac{3}{2}\) between \(1\) and \(2,\) and so on. Any two whole integers can have an endless number of fractions between them.
There are also \(\frac{1}{3},\frac{1}{4},\frac{1}{5},\) and any other number that may be expressed as \(\frac{1}{n},\) where \(n\) is a whole number, between \(0\) and infinity, and the value of the fraction lies between \(0\) and \(1.\) There are also fractions such as \(\frac{2}{3},\frac{3}{4},\frac{4}{5},\) and so on. \(\frac{m}{n}\) is a fraction between \(0\) and \(1\) if m and n are both positive whole numbers and \(m\) is smaller than \(n.\) In the same way, there exists an infinite number of fractions between any two whole numbers.

Q.3. What fraction is between 1/3 and 2/3?
Ans: To find a fraction between two fractions, make the sum of the numerators the new numerator and the sum of the denominators the new denominator.
The given two fractions are \(\frac{1}{3}\) and \(\frac{2}{3},\frac{1}{3} < \frac{2}{3}\)
Here, the sum of the numerators\(=1+2=3\) and
the sum of the denominators\(=3+3=6\)
So, the new fraction formed between the two fractions \(\frac{1}{3}\) and \(\frac{2}{3}\) is \(\frac{3}{6} = \frac{1}{2}.\)

Q.4. What fraction is between 1 and 2?
Ans: To find a fraction between two fractions, make the sum of the numerators the new numerator and the sum of the denominators the new denominator.
The given two fractions are \(\frac{1}{1}\) and \(\frac{2}{1},1 < 2\)
Here, the sum of the numerators\(=1+2=3\) and
the sum of the denominators\(=1+1=2\)
So, the new fraction formed between the two fractions \(1\) and \(2\) is \(\frac{3}{2}.\)

Q.5. What is a fraction?
Ans: Fractions are represented as numerical values in mathematics and can be defined as parts of a whole. A fraction is a portion or section of a whole that can be any number, a specified value, or an item.
Consider the fraction \(\frac{3}{5}.\) This fraction is read as “three-fifth”, which means that \(3\) parts out of \(5\) equal parts in which the whole is divided. In the fraction \(\frac{3}{5},3\) is called the numerator, and \(5\) is called the denominator.

Q.6. What is equivalent in fractions?
Ans: Equivalent fractions are fractions with the same value but differing numerators and denominators. \(\frac{6}{9}\) and \(\frac{10}{15},\) for example, are equivalent fractions since they are both equal to \(\frac{2}{3}.\)

Q.7. How to determine if fractions are equivalent?
Ans: When given different fractions are simplified and reduced to a single fraction, they are equivalent fractions. Apart from that, there are several alternative approaches for determining whether the supplied fractions are comparable. Here are a few examples:
1. Making the numerators and denominators the same.
2. Finding the decimal version of both fractions is the first step.
3. Method of cross multiplication.
4. Using a visual way.

Now you are provided with all the necessary information on a fraction between two fractions and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.

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