# Improper Fractions

Improper fractions, with the name, signifies that the fractions are not done in a proper manner for any number, object or any 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.

## 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:

• 17/5
• 9/4
• 13/4
• 16/3
• 5/2

## 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

10 = 2 x 5

Since, we can see 2 is the common factor for 24 and 10, so after cancelling 2 from 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 $$\frac{17}{4}$$ to a mixed fraction.

When we divide 17 by 4, the quotient is 4 and the remainder is 1. So the mixed fraction is $$4\frac{1}{4}$$.

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 $$3 + 6 \frac{2}{3}$$.

Is it $$3 + 6 + \frac{2}{3}$$ or $$3 + 6 \times \frac{2}{3}$$?

This confusion can be removed by writing it as $$3 + \frac{20}{3}$$

## Mixed to Improper Fraction

Let us say 31/4 is a mixed fraction to be converted into improper fraction.
Multiply 4 by 3 and add with the numerator 1, we get;
4 x 3 + 1 = 13
Hence, the required numerator of 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 whole number and proper fraction, such that;
3 + ¼
Now rationalise the denominators and write the two parts with common denominator.
12/4 + ¼
13/4

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, $$\frac{17}{4}$$ + $$\frac{9}{4}$$ + $$\frac{5}{4}$$

Since the denominator of all three fractions are equal, we just add all the numerators:

17+9+5 = 31

Therefore,

$$\frac{17}{4}$$ + $$\frac{9}{4}$$ + $$\frac{5}{4}$$ = $$\frac{31}{4}$$

If the denominators of fractions are not equal, the process is slightly different and involves calculation of least common multiple of the denominators.

You can learn the calculation of least common multiples here.

Consider the example,

$$\frac{15}{3}+\frac{3}{4}+\frac{5}{2}$$

In the above fractions, the denominators are different, thus follow the below given steps to add them.

• The least common multiple of 3, 4, and 2 is 24.
• Now the fraction we get by adding all these fractions will have 24 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 24 divided by 3 is 8. The numerator is 15. 15 x 8 is 120. Similarly, for the other two fractions, the numbers are 18, and 60.

Therefore, $$\frac{15}{3}+\frac{3}{4}+\frac{5}{2}$$

= $$\frac{120+18+60}{24}$$

= $$\frac{198}{24}$$

= $$8\frac{1}{4}$$

## Subtraction of Improper Fractions

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

• The first step is to check if denominators are same or not
• Second is to rationalise the denominators
• 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

Therefore,

(5/2 x 3/3) – (7/3 x 2/2)

= 15/6 – 14/6

= (15-14)/6

= ⅙

## Solved Examples on Improper Fractions

Q.1: Multiply 3 by 9/7.

Solution: 3 x 9/7

= 27/7

Q.2: Convert the improper fraction 5/4 into mixed fraction.

Solution: Given, 5/4

5/4 = 4/4 + ¼

= 1 + ¼

= 11/4

Q.3: Convert the mixed number 42/3 into improper fraction.

Solution: Given, 42/3

Multiply 3 by 4 and 2

3 x 4 + 2 = 14

So, 14/3 is the required fraction.

Q.4: Write 3 1/4 as an improper fraction.

Solution: Given, 3 1/4 is a mixed number.

Multiply denominator 4 by the whole number 3.

4 x 3 = 12

Now, add 12 to the numerator.

12 + 1 = 13

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

So,

3 1/= 13/4

## Practice Questions

Convert the given improper fractions into mixed numbers.

• 9/2 = ?
• 12/5 = ?
• 22/9 = ?
• 19/7 = ?
• 20/9 = ?

Convert into improper fractions.

• 4 1/2 = ?
• 7 ⅓ = ?
• 4 ⅖ = ?
• 6 1/9 = ?
• 4 ⅞ = ?

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## Frequently Asked Questions on Improper Fraction

### What is an Improper fraction? Give examples.

A fraction is called improper, if the numerator is greater than the denominator. Examples are 9/2, 8/5, 10/3, etc.

### How to convert improper fraction into mixed fraction?

Divide the numerator by denominator. Write the whole number separately and put the remainder in the numerator. If 11/4 is a fraction, then;
11 divided by 4 = 2 + 3 remainder
Therefore, 11/4 = 2 3/4

### Convert 1 ⅔ as an improper fraction?

Multiply 3 by 1 and add 2 to get the denominator of the required fraction and keep the denominator common.
1 ⅔ = (3 x 1 + 2)/3 = 5/3

### What is the difference between proper and improper fraction?

A proper fraction has numerator smaller than the denominator but an improper fraction has denominator smaller than the numerator.

### Solve 31/5.

3 ⅕ = (5 x 3 + 1)/5 = 16/5