# Fractions - with calculator, rules and examples

Fractions ﹣ Calculator

With the fraction calculator, you can add, subtract, multiply and divide any fraction. The calculation of fractions is explained in detail. Among other things, expanding and shortening fractions or equating two fractions for addition are covered. The inverse fraction for division is considered, as well as the final conversion of an improper fraction into a mixed fraction.

## Input 'Help' for the Fraction Calculator

With the help of the fraction calculator, two fractions can be linked together using all four basic arithmetic operations. Both proper and mixed fractions can be added, subtracted, multiplied or divided. All transformations of the fractions suitable for calculating the result are displayed and derived step by step in the result window.

### Proper or mixed fractions

Please select whether you want to enter ’proper fractions’ or ‘mixed fractions’ for the calculation. Below, you will find more information on the difference between ordinary and mixed fractions.

## What are Fractions?

Fractions form a special notation for division, where the numerator above the fraction bar is divided by the denominator or divisor below the fraction bar. This notation can be used, for example, to perform the addition of two divisions and thus the addition of two fractions using certain rules for fractions, which we will discuss below. First, the definitions of the different types of fractions are explained here.

### What is a proper fraction?

A proper fraction represents the fractional part of a whole. The denominator at the bottom indicates how many parts the whole has been divided into. The numerator at the top indicates how many parts of the whole are meant. For example, you can think of ¾, i.e., three quarters, as three pieces of pizza, where the pizza has been divided into a total of four pieces, i.e., four quarters.

#### Example

34 is a proper fraction, because 3 ÷ 4 = 0.75 is less than 1, so it is a proper fraction of a whole.

### What is a improper fraction?

An improper fraction exists if the amount of the numerator is greater than or equal to the amount of the denominator. Then the result is no longer a fraction of a whole, but greater than or equal to one.

#### Example

54 is an improper fraction, because 5 ÷ 4 = 1.75 is greater than 1, so it is not a fraction of a whole.

### What is a common (ordinary) fraction?

A common fraction, also called an ordinary fraction, has the representation of a whole number as its numerator and denominator.

#### Example

34 or 54 are common or ordinary fractions.

### What is a mixed fraction?

A mixed fraction, also called a mixed number, is made up of a whole number and a common fraction. The whole number and the fraction are added together. For example, the mixed fraction is 2¼ = 2 + ¼. While both the real and the improper fraction are ordinary or common fractions, the mixed fraction, as described earlier, is the composition of an integer and a common fraction added together. An improper fraction can be split in this way into its integer part and the remaining proper fraction. For example, the improper fraction 3/2 can be split into 1 and ½, i.e., it can be transformed into the mixed fraction 1½.

#### Example

114 is a mixed fraction.

### What is a decimal fraction?

A fraction whose denominator is 10, 100, 1,000, etc., i.e., a fraction whose denominator forms a power of ten, is called a decimal fraction (fraction of ten). In many cases, you can transform a fraction into a decimal fraction by expanding or reducing it, provided the conversion results in a denominator is in powers of ten. Any decimal fraction can also be converted into a decimal number and vice versa. For example: 43/100 = 0.43.

#### Example

3100 or 541000 are decimal fractions.

## How are Fractions Transformed?

Transformations of fractions, i.e., changes of fractions without changing their value (fraction number), are usually the prerequisite for calculating with fractions. For the addition and subtraction of fractions, for example, it is necessary that the two fractions initially have the same denominator, which in turn makes it necessary to extend or shorten the fractions. These and other transformations are explained below. The transformations presented here are also explained in detail in the result window of the fraction calculator behind the corresponding info buttons and assigned to the respective fraction calculation.

### How are fractions expanded?

Fractions are expanded by multiplying both the numerator and the denominator by the same number. This serves to transform a fraction in which the value of the fraction, i.e., the fraction number, is not changed. This is because the fraction represented by the fraction is only divided into smaller sections. The division is therefore refined. For example, in the case of an addition of two fractions, an expansion serves to multiply the smaller denominator of one fraction by its numerator so that it is equal to the larger denominator of the other fraction.

#### Example: Extending fractions

• To extend the fraction 34 by 5, multiply the numerator and denominator by 5 and get 3 × 54 × 5 = 1520.
• The fraction 34 was thus extended by 5 to 1520, where the two fractions retain the same value.

### How are fractions shortened?

Just as fractions can be expanded, they can also be shortened. Fractions are shortened by dividing both the numerator and the denominator by the same number. This does not change the value of the fraction or the fractional number, because the part represented by the fraction is only divided into larger sections, i.e., the division is coarsened. Shortening is also used, for example, to bring the addition and subtraction of fractions to the same denominator, as described below. Also, possibly large numerators and denominators in the result after multiplying two fractions can be converted into smaller values by shortening.

#### Example: Shortening fractions

• To reduce the fraction 1040 by 5, divide the numerator and denominator by 5 and get 10 ÷ 540 ÷ 5 = 28
• 1040 was therefore shortened by 5 to 28, where both fractions retain the same value.
• 28 could even be shortened again by 2, so that you get 14, which then cannot be shortened any further.

### Why do we need the greatest common divisor when shortening?

In order to continue calculating with numbers that are as small as possible, i.e., manageable, fractions should also be shortened as much as possible. This is achieved by dividing the numerator and the denominator of the fraction by their greatest common divisor (GCD)..

#### Example: Shortening with greatest common divisor

• Based on the above example with the fraction 1040, the greatest common divisor of 10 and 40 is 10.
• So you can shorten the fraction by 10 to get the fraction that cannot be shortened further. Then the numerator and denominator no longer have any common divisors except 1.

### How do you make fractions have the same name?

Common fractions that have the same denominator are called homonymous. If fractions are expanded so that they have the same denominators, they are called homonymous. For example, two fractions can be made homonymous by expanding one fraction with the denominator of the other fraction. This means that both the numerator and the denominator of one fraction are multiplied by the denominator of the other fraction. Since the two denominators are always multiplied, the values of the expanded fractions can often become very large, which could make further calculations more complicated. In practical calculations, the smallest common denominator (main denominator) of the fractions should therefore be determined in order to make them equal. The main denominator is the lowest common multiple (LCM) of the denominators, which is often smaller than the multiplication of the two denominators. We will explain this in more detail in the next section.

Making them have the same denominator is used, for example, in the addition and subtraction of fractions: If the two fractions have the same denominator, the numerators of the two fractions can be added or subtracted, while the denominator, which is the same for both fractions, remains unchanged.

#### Example: Making fractions equal

• The fractions 16 and 38 are to be given the same name.
• Widening the left fraction 16 with 8, i.e., with the denominator of the right-hand fraction.
• Extend the right fraction 38 with the denominator 6 of the left fraction.
• This is how you get the fractions of the same denominator 848 and 1848.

### Why do you need the lowest common denominator when determining the same value?

In order to be able to continue calculating with the least possible, manageable numbers in the course of a calculation, the lowest possible common denominator should be determined. This denominator, also called the main denominator, is the lowest common multiple (LCM) of the two denominators.

#### Example: Same-named with lowest common denominator

• Based on the above example with the fractions 16 and 38, the lowest common multiple of the two denominators 6 and 8 is 24.
• So you can only expand the left-hand fraction by 4 instead of, say, 8, and expand the right-hand fraction by 3 instead of, say, 6.
• This way you get the fractions with the same denominator 424 and 924 with 24 as the lowest common denominator.

### How do you form the reciprocal of a fraction?

The reciprocal of a fraction is obtained by exchanging the numerator and denominator of the fraction. If you want to divide a fraction by another fraction, you can also form the reciprocal fraction from a fraction and then multiply the two fractions together.

#### Example: Fractions with reciprocal fractions

34  ÷  13  =  34  ×  31

### How do you convert a fraction into a decimal number?

To calculate the decimal number for a fraction, simply divide the numerator by the denominator.

#### Example: Conversion of fraction to decimal number

34  = 3 ÷ 4 = 0.75

### How do you reshape a fictitious fraction into a mixed fraction?

An improper fraction can be divided into its integer part and the remaining proper fraction. The integer part is the division of the numerator by the denominator. The remaining proper fraction is obtained by dividing the numerator by the denominator using the remainder (modulo calculation).

#### Example: Transforming improper fraction to mixed fraction

• 54  = 5 ÷ 4 = 1.25 ⇒ The integer part of the mixed fraction is 1
• 5 modulo 4 = 0.25 ("The remainder of 5 ÷ 4 is 0.25")
• 0.25 = 25100 = 14 ⇒ The proper fraction of the mixed fraction is ¼.
• Therefore, the mixed fraction is 114.

Fractions are added by first making them equal. The numerators are added while the common denominator remains unchanged.

### Example for the addition of fractions

34 + 13 = 912 + 412 = 9+412 = 1312 = 1112

For a more detailed description of the rules for adding fractions and a comprehensive example, see our article on the topic Add Fractions.

## How to Subtract Fractions

Fractions are subtracted by first making them equal. The numerators are subtracted while the common denominator remains unchanged.

### Example of subtraction of fractions

3413 = 912412 = 9−412 = 512

For a more detailed description of the rules for subtracting fractions and a comprehensive example, please see our article on the topic Subtract fractions.

## How to Multiply Fractions

Fractions are multiplied by multiplying the two numerators and the two denominators.

### Example of multiplication of fractions

34 × 13 = 3×14×3 = 312

For a more detailed description of the rules for multiplying fractions and a comprehensive example, see our article on the topic Multiply Fractions.

## How to Divide Fractions

Fractions are divided by multiplying one fraction by the reciprocal of the other fraction.

### Example of division of fractions

34 ÷ 13 = 34 × 31 = 3×34×1 = 94

For a more detailed description of the rules for dividing fractions and a comprehensive example, see our article on the topic Divide Fractions.

## How Does the Fraction Calculator Calculate?

The fraction calculator masters all the basic arithmetic operations presented here for calculating fractions. In a first step, the fraction calculator first arranges any negative signs of the entered fractions. If mixed fractions have been entered, the fraction calculator then converts them into unmixed fractions. In the next step, the calculator truncates the fractions as far as possible.

If the fractions are to be added or subtracted, the fraction calculator makes the two fractions equal and then adds or subtracts the numerators. If the two fractions are to be multiplied or divided, the calculator does this for both the numerator and the denominator, whereby the inverse fraction of one of the two fractions is first generated for the division.

The result calculated in this way is still an unreal fraction in some calculations. This fraction is finally converted by the fraction calculator into a mixed fraction.

### Calculation Examples by the Fraction Calculator

1−5−8 + 224

#### 1. Sort negative signs

In this step, the fraction calculator removes the negative signs from fractions with both negative numerators and negative denominators. And, if only the denominator is negative, the calculator makes the corresponding numerator negative instead.

• If both the numerator and the denominator of a fraction are negative, the two negative signs can be removed because dividing two negative values in the same way as dividing two positive values gives a positive result ("minus divided by minus results in plus").
• If, in the case of fractions, only the denominator is negative, the negative sign can be placed before the numerator instead. This is because dividing a positive value by a negative value leads to a negative result in the same way as, conversely, dividing a negative value by a positive value.

These transformations help to improve the order and thus clarity of the following calculations.

158 + 224

#### 2. Convert mixed fractions into unmixed ones

The fraction calculator converts the previously mixed fractions into unmixed fractions here, i.e., the whole number before the fraction is added to the associated fraction:

• The integer associated with the fraction on the left, i.e., 1, is first converted to 8/8 and then added to the associated fraction.
• The integer associated with the fraction on the right, i.e., 2, was first converted to 8/4 and then added to the associated fraction.

138 + 104

#### 3. Shorten fractions

Here, the right-hand fraction is shortened by the fraction calculator. In order to be able to calculate with the smallest possible numbers in the further course, the fractions should be shortened as much as possible by dividing the numerator and denominator of each fraction by their greatest common divisor.

The fraction on the left cannot be shortened because its numerator and denominator have no common divisor except for the one.

The greatest common divisor of the right-hand fraction, i.e., the greatest common divisor of the numerator 10 and the denominator 4, is 2. Therefore, both the numerator and the denominator can be divided by 2 to shorten the fraction: 104 = 52

138 + 52

#### 4. Make all fractions have the same denominator

To add the two fractions, the fraction calculator makes them equal. To do this, the lowest common multiple of the denominators is calculated here. The lowest common multiple (LCM) of the two denominators 8 and 2 is 8.

• The left-hand fraction is therefore expanded by 1, i.e., the numerator 13 is multiplied by 1 and the denominator 8 is multiplied by 1 to give the denominator the value 8.
• The right-hand fraction is extended by 4, i.e., the numerator 5 is multiplied by 4 and the denominator 2 is multiplied by 4, so that the denominator has the value 8 here too.

138 + 208

#### 5. Add fractions with the same denominator

This leads to the intermediate result of the fraction task entered. To do this, the fraction calculator adds the numerators of the two fractions with the same denominator. The denominator remains unchanged.

13 + 208 = 338

#### 6. Result (Finally convert improper fractions to mixed ones)

This is finally the result of the fraction calculation task entered. Here, the fraction calculator finally converts the improper fraction of the intermediate result into the corresponding mixed fraction. This mixed fraction is calculated by dividing, with remainder (modulo calculation), the numerator by the denominator of the improper fraction:

33 ÷ 8 = 4 Remainder 1

So the mixed fraction consists of the integer part 4 and the remainder part 18.

= 418

## Source information

As source for the information in the 'Fractions' category, we have used in particular:

## Last update

This page of the 'Fractions' category was last edited or reviewed by Michael Mühl on February 20, 2023. It corresponds to the current status.

### Changes in this category "Fractions"

• Publication of the topic Fraction together with the corresponding texts.