Last modified on June 21st, 2024

Improper Fractions

An improper fraction is a fraction in which the numerator is greater than or equal to the denominator. Thus, the value of an improper fraction is always greater than or equal to 1.

${\dfrac{3}{2}}$, ${\dfrac{8}{5}}$, ${\dfrac{28}{11}}$, ${\dfrac{7}{3}}$, ${\dfrac{13}{4}}$, and ${\dfrac{48}{15}}$ are a few improper fractions


Let us suppose that several pizzas are ordered for a house party, and each pizza is cut into 8 slices.

Now, let’s say one of your friends has 11 slices of pizza. Since each pizza has 8 slices, he had more than one whole pizza. This can be represented as ${\dfrac{11}{8}}$, which is an improper fraction.

The above situation is represented in the diagram below:

Improper Fractions to Mixed Numbers

Sometimes, we need to represent an improper fraction as a mixed number. To convert, we first divide the numerator of the given fraction by the denominator and then convert it into a mixed fraction. 

Let us convert the fraction ${\dfrac{11}{8}}$ to its corresponding mixed number using our pizza example.

By dividing 11 by 8, we get the quotient as 1, representing a whole pizza, and the remainder of 3 means there are 3 extra slices (the fractional part).

Representing the remainder as fraction, we get ${\dfrac{3}{8}}$

Now, combining the whole and the fractional part, we get the improper fraction as ${\dfrac{11}{8}}$. 

Now, dividing the numerator (11) by the denominator (8), we get the mixed fraction ${1\dfrac{3}{8}}$

Improper Fraction to Decimals

To convert an improper fraction to a decimal number, we simply divide the numerator by the denominator of the given fraction.

Dividing the improper fraction ${\dfrac{11}{8}}$, we get

11 ÷ 8 = 1.375

On a Number Line

Let us now represent the same improper fraction ​${\dfrac{11}{8}}$ on a number line. 

1. Drawing the Number Line

We start by drawing a number line and marking each interval by the corresponding whole numbers (0, 1, 2, 3…) 

2. Converting the Improper Fraction into a Mixed Number

${\dfrac{11}{8}}$ = ${1\dfrac{3}{8}}$

3. Marking the Whole Numbers on the Number Line

Since the whole number is 1, we mark 1 and 2 on the number line as ${1\dfrac{3}{8}}$ lies between 1 and 2.

4. Dividing the Number Line into Intervals

The region between 1 and 2 is divided into 8 equal parts (as the denominator is 8), and each part is marked as ${\dfrac{1}{8}}$, ${\dfrac{2}{8}}$, and so on. 

5. Locating the Mixed Number

Now, the third interval from 1 is marked as ${1\dfrac{3}{8}}$


To simplify, an improper fraction means to express the fraction in its lowest term. Thus, we divide the numerator and denominator by their greatest common factor (GCF). 

Let us simplify the improper fraction ${\dfrac{28}{20}}$

Factorizing the numerator and the denominator, we get

28 = 2 × 2 × 7

20 = 2 × 2 × 5

Thus, the GCF of 28 and 20 is 4

Now, by dividing the numerator and the denominator by 4, we get


= ${\dfrac{7}{5}}$


When adding improper fractions, we can have two cases:

With the Same Denominators

To add improper fractions with the same denominator, we follow the following steps:

Let us add ${\dfrac{7}{4}}$ and ${\dfrac{11}{4}}$

Step 1: Adding the numerators together


= ${\dfrac{7+11}{4}}$

Step 2: Simplifying

= ${\dfrac{18}{4}}$

= ${\dfrac{9}{2}}$

With Different Denominators

Let us add ${\dfrac{7}{2}+\dfrac{9}{4}}$

Step 1: Finding the LCM of the denominators

Here, the denominators are 2 and 4

The LCM of 2 and 4 is 4

Step 2: Rationalizing the denominators

Now, to rationalize, we will multiply the first fraction by 2 and the second fraction by 1

${\dfrac{7\times 2}{2\times 2}+\dfrac{9\times 1}{4\times 1}}$

= ${\dfrac{14}{4}+\dfrac{9}{4}}$.

Step 3: Adding the numerators together

= ${\dfrac{14+9}{4}}$

Step 4: Simplifying

= ${\dfrac{23}{4}}$


Subtraction of improper fractions is similar to addition.

With the Same Denominators

If we subtract ${\dfrac{4}{3}}$ from ${\dfrac{14}{3}}$ 


= ${\dfrac{14-4}{3}}$

= ${\dfrac{10}{3}}$

With Different Denominators

If we subtract ${\dfrac{4}{3}}$ from ${\dfrac{9}{5}}$ 


= ${\dfrac{9\times 3}{5\times 3}-\dfrac{4\times 5}{3\times 5}}$

= ${\dfrac{27}{15}-\dfrac{20}{15}}$

= ${\dfrac{27-20}{15}}$

= ${\dfrac{7}{15}}$


While multiplying improper fractions, we multiply the numerators and the denominators separately.

Let us multiply ${\dfrac{7}{3}}$ and ${\dfrac{11}{5}}$

${\dfrac{7}{3}\times \dfrac{11}{5}}$

= ${\dfrac{7\times 11}{3\times 5}}$

= ${\dfrac{77}{15}}$


While dividing, we multiply the first fraction by the reciprocal of the second fraction.

Let us divide ${\dfrac{7}{2}}$ by ${\dfrac{11}{5}}$

${\dfrac{7}{2}\div \dfrac{11}{5}}$

= ${\dfrac{7}{2}\times \dfrac{5}{11}}$

= ${\dfrac{7\times 5}{2\times 11}}$

= ${\dfrac{35}{22}}$

Solved Examples

Write ${2\dfrac{1}{3}}$ as an improper fraction.


The denominator = 3
The numerator = 1
The whole number = 2
Now, by multiplying 2 by 3 and then adding 1 to it, we get
(2 × 3) + 1 = 7
Thus, ${2\dfrac{1}{3}}$ = ${\dfrac{7}{3}}$

Identify the improper fractions:
${\dfrac{5}{3}}$, ${\dfrac{7}{9}}$, ${\dfrac{37}{13}}$, ${\dfrac{25}{12}}$, ${\dfrac{9}{43}}$, and ${\dfrac{2}{3}}$


Here, the improper fractions are: ${\dfrac{5}{3}}$, ${\dfrac{37}{13}}$, and ${\dfrac{25}{12}}$

Last modified on June 21st, 2024