problem solving improper fractions

Improper Fractions – Definition, Conversion, Examples, FAQs

What is an improper fraction, fraction conversion, solved examples on improper fractions, practice problems on improper fractions, frequently asked questions on improper fractions.

Fractions are often referred to as numbers between numbers. Fractions are numerical values that represent a part or a portion of a whole. For example, look at the pizza below.

This pizza has been cut into 4 equal parts. So each piece of the pizza represents 1 out of 4 equal parts. So mathematically, we can represent each piece as  $\frac{1}{4}$. This number is called a fraction . 

The number below the bar that represents the total number of equal parts that the whole is divided into is called the denominator .

And the number on the top that represents the number of equal parts we are considering is called the numerator . 

Before discussing improper fraction, we first need to understand proper fractions.

Proper fractions: Fractions for which the numerator is less than the denominator are called proper fractions . For example, if you order a pizza and cut it into 4 equal parts, and then eat one slice out of it, the remaining pizza can be expressed as $\frac{3}{4}$ of the whole pizza. Here, since 3 is less than 4, the fraction is a proper fraction.

Improper fractions: Improper fractions are defined as fractions for which the numerator is greater than the denominator. Let us take an example of improper fraction, imagine you order a pizza that has 4 slices. Your friends eat all 4 slices. And you realize you didn’t get any. You order another pizza. After eating 1 slice from it, you realize you are done eating. So, how much of the pizza did your friends and you have in all?

Your friends first had all 4 slices of 1 pizza, and then you had 1 slice of the same size from the second pizza.

So, the total pizza eaten is $\frac{5}{4}$ slices of pizza. And that’s an improper fraction with a numerator greater than the denominator. 

Mixed numbers: Mixed numbers are just another way of writing fractions greater than a whole. A mixed number has a whole part and a fractional part, which is actually a proper fraction.

Let’s take the same example from above. You and your friends ate one whole pizza and one-fourth of a pizza. As a mixed fraction, we can write this as $1$ and $\frac{1}{4}$ of a pizza or $1\frac{1}{4}$ of a pizza.

Improper Fraction to Mixed Number

An improper fraction can be converted to a mixed fraction and a mixed fraction can be converted to an improper fraction.

Let’s say we want to convert the fraction $\frac{7}{2}$ into a mixed number. 

Step 1: Find the quotient and the remainder by dividing the numerator with the denominator.

Step 2: The quotient will be the number of wholes for the mixed number. The remainder will be the numerator of the fractional part and the denominator will remain the same.

So, the mixed number form of $\frac{7}{2}$will be:

Mixed Number to Improper Fraction

Any mixed number can be converted into an improper fraction.

Let’s say we want to convert the mixed number $1\frac{3}{4}$ into an improper fraction. 

Step 1: Multiply the denominator with the whole. And then add the numerator to the product.

Step 2: Keep the denominator same and solve for the numerator.

Therefore, the improper fraction for the mixed number $1\frac{3}{4}$ will be $\frac{7}{4}$. 

Related Worksheets

Example 1. Identify the improper fractions:

$\frac{1}{5}$, $2\frac{7}{5}$, $\frac{3}{2}$, $\frac{1}{3}$, $\frac{5}{4}$, $6 \frac{1}{6}$

Solution: $\frac{2}{3}$ and $\frac{5}{4}$ are improper fractions as the value of the numerator is greater than the value of the denominator for both of these fractions.

Example 2. Write $4\frac{2}{7}$ as an improper fraction.

Solution: Converting $4\frac{2}{7}$ into an improper fraction,

$4 \frac{2}{7}$ = $\frac{(7\times4) + 2}{7}$ = $\frac{30}{7}$

Example 3. Write $2\frac{9}{6}$ as an improper fraction.

Solution: While dividing 29 by 6, the quotient and remainder are 4 and 5 respectively. So  $\frac{29}{6}$ written as a mixed number is $ 4 \frac{5}{6}$ .

Converting $3\frac{1}{5}$ to an improper fraction:

What are Improper Fractions? Meaning, Definition, Examples

Attend this quiz & Test your knowledge.

What type of fraction is $\frac{11}{4}$?

Write the mixed number $3\frac{1}{5}$ as an improper fraction., identify the improper fraction., convert $\frac{5}{2}$ into a mixed fraction..

Which are easier for calculations: improper fractions or mixed fractions?

Improper fractions are easy for arithmetic operations of addition , subtraction , multiplication , and division . Mixed fractions are easier to understand and represent in figures.

Why are fractions important?

Every quantity cannot be expressed in whole numbers . Fractions are employed to indicate a certain part or portion of any item. They are a fundamental concept in computation and applied in algebra , ratios, and several other concepts in mathematics and other fields.

How are decimals related to fractions?

Fractions and decimals are both ways to represent quantities or numbers. Decimals are nothing but a different way to represent fractions whose denominator is 10 or powers of 10. For example, the fraction $\frac{15}{10}$ in decimal form is 1.5.

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Improper Fractions

Improper fractions , with the name, signifying that the fractions are not done in a proper manner for any number, object or element. In Maths, a fraction is a part of a whole. Fractions have two parts, numerator and denominator. If  ⅓ is a fraction, then 1 is the numerator, and 3 is the denominator. An improper fraction has a numerator greater than the denominator. For example, 3/2 is an improper fraction, but 2/3 is a proper fraction , whose  denominator is greater than the numerator.  

In this article, you will find all the topics related to improper fractions. Also, we perform various arithmetic operations on these fractions, such as addition, division, multiplication, etc.

Table of contents:

What are Improper Fractions?

An improper fraction is defined as a fraction whose numerator is greater than the denominator. Suppose, x/y is an improper fraction, such that x > y. It is therefore, the improper fraction is always greater than one.

Examples of Improper Fractions are:

How to Simplify Improper Fractions?

We have understood what an improper fraction is. Now, let us discuss here how to simplify such fractions here.

• Step 1: Determine if the given fraction is improper or not.
• Step 2: Now interpret the denominator and check for how many parts it is dividing the numerator.
• Step 3: Check the common factors of numerator and denominator
• Step 4: Cancel the like terms both from numerator and denominator.

The resulted fraction is the simplified fraction. Let us understand these steps with the help of examples.

Example: Simplify the fraction 24/10.

Solution: Since we know 24>10, thus 24/10 is an improper fraction.

Let us factorise the numerator and denominator.

24 = 2 x 2 x 2 x 3

Since we can see 2 is the common factor for 24 and 10, so after cancelling 2 from the numerator and denominator, we get;

(2 x 2 x 3)/5 = 12/5

Improper Fractions to Mixed Fractions

To convert an improper fraction to a mixed fraction, divide the denominator by the numerator. The quotient becomes the whole number, the remainder will be the numerator, and the divisor becomes the denominator.

Let us consider an example. Convert 17/4 to a mixed fraction.

When we divide 17 by 4, the quotient is 4 and the remainder is 1. So the mixed fraction is 4  1/4.

Although an improper fraction can be converted to a mixed fraction, there are situations where expressing a fraction as an improper fraction minimizes a lot of confusion, especially when the fractions are expressed in calculations.

For example, consider 3 + 6  2/3.

Is it 3 + 6  +  2/3 or 3 + 6  × 2/3 ?

This confusion can be removed by writing it as  3 + 20/3.

Mixed to Improper Fraction

Let us say 3 1 / 4 is a mixed fraction to be converted into an improper fraction. Multiply 4 by 3 and add with the numerator 1, we get; 4 x 3 + 1 = 13 Hence, the required numerator of the improper fraction is 13 while the denominator remains the same, i.e., 4 Therefore, the required fraction is 13/4

Another Method: Write the mixed number as a sum of the whole number and proper fraction, such that; 3 + ¼ Now rationalise the denominators and write the two parts with a common denominator. 12/4 + ¼ Now add the fractions 13/4

Addition of Improper Fractions

The addition of improper fractions can have two scenarios. If the denominators of all the fractions are equal, we can add all the numerators and keep the same denominator.

For example, (17/4) + (9/4) + (5/4)

Since the denominator of all three fractions is equal, we add all the numerators by keeping the common denominator as 4.

Hence, we get the following:

(17/4) + (9/4) + (5/4) = (17 + 9 + 5) / 4

(17/4) + (9/4) + (5/4) = 31/4.

The other case is that denominators of fractions are not equal. Here, the process is slightly different and involves the calculation of the least common multiple of the denominators.

You can learn the calculation of the least common multiples here.

Consider the example, (15/3) + (3/4) + (5/2)

Here, the denominators of the fractions are different. (i.e., 3, 4, 2), and take the LCM of 2, 3, and 4.

Hence, to add the given fractions, follow the below steps:

• The least common multiple of 3, 4, and 2 is 12.
• Now the fraction we get by adding all these fractions will have 12 as the denominator.
• Divide the LCM by each of the denominators and multiply the quotient by the numerators.
• The numerator of the new fraction will be the sum of all the numbers obtained in the previous step.
• Now, in the first fraction, the denominator is 3. LCM 12 divided by 3 is 4. The numerator is 15. 15 x 4 is 60. Similarly, for the other two fractions, the numbers are 9, and 30.

Therefore, (15/3) + (3/4) + (5/2) = (60/12) + (9/12) + (30/12)

(15/3) + (3/4) + (5/2) = (60 + 9 + 30) / 12

(15/3) + (3/4) + (5/2) = 99/12.

Now, the sum of fractions can be simplified as:

(15/3) + (3/4) + (5/2) = 33/4

convert into mixed fractions, if required.

Hence, the mixed fraction is:

(15/3) + (3/4) + (5/2) = 8  1/4.

Subtraction of Improper Fractions

In a similar manner, as we added the improper fractions, we can also subtract them. 

• The first step is to check if denominators are the same or not
• The second is to rationalise the denominators 
• The last step is to subtract the given fractions and simply, if required

Let us see an example:

Subtract 5/2 – 7/3.

LCM (2,3) = 6

(5/2 x 3/3) – (7/3 x 2/2) 

= 15/6 – 14/6

= (15-14)/6

Related Articles

Solved examples of improper fractions.

Multiply 3 by 9/7.

Given: 3 x 9/7 

3 x 9/7  = 27/7

Convert the improper fraction 5/4 into a mixed fraction.

To convert an improper fraction into a mixed fraction, first, divide the number 5 by 4.

If we divide 5 by 4, we get the quotient as 1 and the remainder as 1.

Hence, keep the quotient as the whole number part and the remainder as the fraction’s numerator part and the denominator as 4.

Hence, 5/4 written in the mixed fraction is 1  1/4.

i.e., 5/4 = 1  1/4.

Convert the mixed number 4  2/3 into an improper fraction.

Given: Mixed fraction = 4  2/3

To convert the mixed fraction into an improper fraction, follow the below steps:

Step 1: Multiply 3 by 4, and we get 12.

Step 2: Now, add 12 and 2, we get 14

I,e 3(4) + 2 =14

Now, put the number 14 in the fraction’s numerator and keep the number 3 as the denominator.

So, 14/3 is the required fraction.

Write 3 1/4 as an improper fraction.

Given that 3 1/4 is a mixed number.

Now, multiply denominator 4 by the whole number 3.

Now, add 12 to the numerator.

12 + 1 = 13

Hence, 13 is the new numerator of the improper fraction, whereas the denominator remains the same.

3 1 / 4  = 13/4

Video Lesson

Practice Questions

Convert the given improper fractions into mixed numbers.

Convert into improper fractions.

• 4 1 / 2 = ?
• 6 1 / 9 = ?

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Improper Fractions

Fractions were first used in Ancient Egypt around 1600 BC, which makes the concept quite old. Fractions are of different types. When the numerator is smaller than the denominator i.e. 1/2, it is a proper fraction. However, when the denominator is smaller than or equal to the numerator, then it is an improper fraction. An example of this is 16/15.

An improper fraction is one type of fraction in which the numerator is equal to or greater than the denominator. Its value is always one or greater than one. Improper fractions are usually written in mixed number form in a simplified manner, as mixed fractions are easier to comprehend.

What is an Improper Fraction?

An improper fraction is a type of fraction where the numerator is greater than or equal to the denominator. For example, 5/2 and 8/5, are improper fractions. Every fraction has two parts, numerator, and denominator. In mathematics, there are two main types of fractions based on the values of numerator and denominator, and those are proper fractions and improper fractions.

Improper Fraction Definition

A fraction whose numerator is greater than or larger than the denominator is defined as an improper fraction such as 7/3 and 12/5. Improper fractions are easier to solve using addition and subtraction compared to the type of fractions such as mixed fractions.

Improper Fraction and Mixed Fraction

An improper fraction is a fraction whose numerator is greater than or equal to its denominator. For example, 9/4, 4/3 are improper fractions. Numerically, they are always equal to or greater than 1. On the other hand, a mixed fraction is a fraction that is written as a combination of a natural number and a proper fraction. It is a simplified form of an improper fraction. For example, \( 21\dfrac{4}{5}, 16\dfrac{2}{3}\) are mixed fractions. Numerically, a mixed fraction is always greater than 1. Also, any mixed fraction can be written as an improper fraction.

Generally, in real life, mixed fractions are easier to interpret and compare, as compared to improper fractions. We can easily convert any improper fraction to a mixed number or mixed fraction to an improper fraction by following some basic steps that you will study in later sections on this page.

Converting Improper Fractions to Decimals

Fractions and decimals are two ways to represent numbers. Improper fractions can be converted to decimals easily by dividing the numerator with the denominator . Let’s look at a quick and easy example of how we can convert improper fractions to decimals . Let’s consider the following example.

Example: Convert the given improper fraction to a decimal: 10/4.

The first step is to divide 10 by 4. When we do so, we will get 10 ÷ 4 = 2.5.

Here, 10/4 is an improper fraction and 2.5 is a decimal. Similarly, the value of 3/2 is 1.5 in decimals and 5/2 is 2.5 in decimals. When we convert an improper fraction to a decimal, the decimal value is always greater than 1.

Converting Improper Fractions to Mixed Numbers

The denominator of the mixed fraction form of an improper fraction is always the same as that of the original fraction. Mixed numbers are considered as the simplified form of improper fractions, that is why it is important to learn this conversion. For converting an improper fraction to a mixed number, we have to follow the below-listed steps:

• Step 1- Divide the numerator with the denominator.
• Step 2- You will get values of quotient and remainder.
• Step 3- Arrange those values of the quotient, remainder, and divisor in the following manner to express a fraction as a mixed number: \(Quotient\dfrac{Remainder}{Divisor}\).

Let’s look at a quick and easy example of how we can convert improper fractions to mixed numbers. Let’s say you have an improper fraction, 13/4. The first step is to divide 13 by 4. We get 3 as the quotient with a remainder of 1. Next, we will place 1 as the numerator, 4 as the denominator, and 3 as the whole number . Thus, we get the mixed fraction: \( 3\dfrac{1}{4}\).

Similarly, let’s solve another example. Here, we have an improper fraction: 9/2. On dividing 9 by 2, we get 4 as the quotient with a remainder of 1. Again, we will repeat the same process. We will place 1 as the numerator, 2 as the denominator, and 4 as the whole number. Thus, we get the mixed fraction: \(4\dfrac{1}{2}\).

How to Solve Improper Fractions?

Solving improper fractions mean performing arithmetic operations on them and simplifying the value of the answer so obtained. There are mainly four arithmetic operators in mathematics and those are addition, subtraction, multiplication, and division. Solving an improper fraction is the same as solving any other proper fraction, the only difference is that, here, we have to simplify the answer and write it in mixed numbers.

Let's solve the improper fraction: 4/3 + 7/3.

Step 1: We have the same denominator for both the fractions. Therefore, we will directly add the numerators 4 and 7. We get 11. Thus, on adding improper fractions, we get 11/3.

Step 2: Simplifying the improper fraction (dividing 11 by 3), we will get 3 as a whole, 2 as a numerator, and 3 as the denominator.

The answer is \(3\dfrac{2}{3}\).

Related Articles on Improper Fractions

Check the following topics related to the concept of improper fractions.

• Improper Fraction to Mixed Number
• Mixed Number to Improper Fraction
• Improper Fraction to Mixed Number Calculator
• Types of Fractions

Improper Fraction Examples

Example 1: Identify improper fractions out of the following: 12/5, 3, 1/9, 4/2, 4/5.

Improper fractions are the ones in which the numerator is either equal to or greater than the denominator. Out of the given fractions, 12/5, 3, and 4/2 are improper fractions. As, in 12/5, 12 > 5, 3 can be written as 3/1 where 3 > 1, and 4/2 can be simplified as 2/1 where 2 > 1.

Example 2: There are five-fourth liters of plain milk in the refrigerator. Convert the given quantity into a mixed number.

To convert the given improper fraction 5/4 to a mixed number, we have to divide 5 by 4. By dividing, we will get 1 as the quotient, and 1 as the remainder. Therefore, the answer is \(1\dfrac{1}{4}\).

Example 3: What is 4 1/3 as an improper fraction?

To convert the given mixed number to an improper fraction, we have to first multiply the denominator and the whole number and then we need to add the numerator to the sum. ⇒ \(4\dfrac{1}{3}\) = (4 × 3 + 1)/3 ⇒ (12 + 1)/3 ⇒ 13/3 Therefore, \(4\dfrac{1}{3}\) is equivalent to 13/3.

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Practice Questions on Improper Fractions

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FAQs on Improper Fractions

Fractions in which the numerator is equal to or less than the denominator are known as improper fractions in math. 3/2, 6/5, 18/11 are a few examples of improper fractions.

What is Proper Fraction and Improper Fraction?

Proper fractions are those in which the numerator is less than the denominator, while improper fractions are opposite of proper fractions where numerator ≥ denominator.

Are Whole Numbers Examples of Improper Fraction?

Yes, whole numbers are examples of improper fractions, as we can write any whole number in the form of a fraction in which the numerator is greater than the denominator. For example, 3= 3/1, 5= 5/1, etc.

How can we Add Improper Fractions?

The addition of two like improper fractions is done by adding the values of their numerator and writing the common denominator as the denominator of the sum obtained. While, in the case of two unlike improper fractions, we first take the LCM of the denominators and convert them into like fractions. Then, we add those two fractions. Let's take an example of addition of improper fractions 5/4 and 7/4.

• Step 1: We have the same denominator in these fractions 5/4 + 7/4. Therefore, we will simply add the numerators 5 and 7. We get 12 as the sum. Thus, on adding, we have 12/4.
• Step 2: By simplifying the improper fraction (dividing 12 by 4), we get 3 as the answer.

How do we Simplify an Improper Fraction?

Simplification of improper fractions means finding the lowest value of the fraction by dividing the numerator with the denominator. Let us take an example to understand how to simplify an improper fraction. Consider the fraction 100/5. By dividing 100 by 5, we get 20 as the answer. Thus, 100/5 = 20 in the simplified form.

How to Subtract Improper Fractions?

Subtraction of two like improper fractions is done by subtracting the values of their numerator and writing the common denominator as the denominator of the difference obtained. While, in the case of two unlike improper fractions, we first take the LCM of the denominators and convert them into like fractions. Then, we subtract those two fractions.

How to Convert Improper Fraction to Mixed Numbers?

Improper fractions are converted to mixed numbers by dividing the numerator with the denominator and write the answer in this form: \(Quotient\dfrac{Remainder}{Divisor}\).

How to Multiply Improper Fractions?

The numerator and denominator of an improper fraction are multiplied with the numerator and denominator of the other fraction to find the product. After multiplying, we simplify the fraction so obtained. In this way, we multiply two improper fractions.