# Fractions

Do you know that the United Kingdom is divided into 4 major regions, this means that each region is a fraction of the UK?

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In this article, we introduce fractions definition, types of fractions as well as arithmetic operations on fractions.

## Fraction definition

A fraction is defined as a part or portion out of a whole. This whole can be anything including even a fraction but not zero.

The idea of fractions is strongly linked with the division operation because effectively, fractions are formed when you try to share something.

Imagine a box of pizza delivered to your doorstep and you want to share it with your friend, you would have to share it (divide) into 2 parts. Then each one of you will be having half of the pizza, which is expressed as, $\frac{1}{2}$

Now if you intend to take the lion's share by splitting it into 4 parts and giving just 1 out of the four pieces to your friend, then you have given 1 part out of 4 parts which are expressed as,

$\frac{1}{4}$.

However, you have actually taken 3 out of for parts, which is expressed as,

$\frac{3}{4}$

Below shows a figure for a better illustration,

An illustration of a fraction by sharing of a whole- StudySmarter Originals

### Components of a fraction

A fraction is composed basically of three components, the numerator, the over sign, and the denominator.

#### Numerator

The numerator is the number or expression on top or above the over sign.

For example, in the fraction $\frac{2}{3}$, 2 is the numerator.

#### The over sign

The over sign ( $\frac{}{}$) separates the numerator from the denominator as seen in the fraction $\frac{2}{3}$.

#### Denominator

The denominator is the number or expression underneath or below the over sign.

For example, in the fraction $\frac{2}{3}$, 3 is the denominator.

An illustration of the components of a fraction - StudySmarter Originals

## Types of fractions

There are three types of fractions which are,

• Proper fraction
• Improper fraction
• Mixed fraction

### Proper fraction

A proper fraction is a fraction where the numerator is less than the denominator.

In the fraction $\frac{3}{5}$, 3 being the numerator is less than the denominator 5.

Further proper fractions are$\frac{1}{5},\frac{4}{7},\frac{6}{11}$ etc.

### Improper fraction

An improper fraction is a fraction where the numerator is greater than the denominator.

In the fraction $\frac{7}{6}$, 7 being the numerator is greater than the denominator 6.

Further improper fractions are $\frac{8}{3},\frac{11}{2},\frac{5}{3}$ etc.

### Mixed fraction

A mixed fraction consists of a whole number and a proper fraction.

The mixed fraction $1\frac{2}{3}$ consists of

1 as the whole number,

2 as the numerator of the proper fraction in the mixed fraction, and

3 as the denominator of the proper fraction in the mixed fraction.

Other examples include, $2\frac{1}{5},4\frac{2}{7},5\frac{3}{8}$ etc.

### Conversion between improper and mixed fraction

Improper fractions can be converted to mixed fractions as well as mixed fractions can be converted to improper fractions.

#### How to convert improper fractions to mixed fractions?

We recall that an improper fraction has its numerator greater than its denominator. This means that the numerator is divisible by the denominator but with a remainder. To convert an improper fraction to a mixed fraction we follow these steps,

• Divide the numerator by the denominator.
• Put the whole number on the left-hand side.
• Put your remainder as your new numerator on the right-hand side of the whole number.

Convert $\frac{7}{3}$ to a mixed fraction.

Solution

$\frac{7}{3}=7÷3$

The quotient from the division of 7 by 3 is 2, and the remainder is 1.

Thus, 2 is your whole number that stays on the left-hand side and 1 is the remainder that becomes your new numerator and 3 is your denominator that remains unchanged.

Thus,

$\frac{7}{3}=2\frac{1}{3}$

#### How to convert mixed fractions to improper fractions?

In order to convert a mixed fraction to an improper fraction follow the following steps,

• Find the product between the denominator and the whole number.
• Find the sum of the numerator and your product.
• Your sum becomes the new numerator.
• Your denominator remains the same.

Convert $2\frac{1}{3}$ to an improper fraction.

Solution

First, we find the product between the denominator which is 3, and the whole number which is 2, thus,

$3×2=6$.

Next, we find the sum between the numerator which is 1 and the product found in the first step which is 6, thus

$1+6=7$

Now your sum is 7, the new numerator is thus 7, and the denominator being 3 remains the same. Therefore,

$2\frac{1}{3}=\frac{7}{3}$.

## Equivalent fractions

Equivalent fractions are fractions that represent the same value, even though they look different.

They differ by a multiplication factor or by the expression of fraction types.

Having 3 parts of cake from the total of 6 parts is the same as having half the cake.

### How do equivalent fractions differ by a multiplying factor?

Equivalent fractions can differ by a multiplying factor when one fraction is a multiple of the other fraction. Note that the multiplying factor (multiplier) affects both the numerator and the denominator.

The fractions $\frac{3}{5}$ and $\frac{6}{10}$ are equivalent but differ by a multiplying factor, 2.

In fact,

$\frac{3}{5}=\frac{3×2}{5×2}=\frac{6}{10}$

Similarly, we have

$\frac{6}{10}=\frac{6÷2}{10÷2}=\frac{3}{5}$

Therefore, the best way of knowing if two fractions are equivalent, but differ by a multiplying factor, is by simplifying these fractions to the simplest indivisible forms.

### How do equivalent fractions differ by expression of fraction type?

Equivalent fractions can differ by fraction type, so this happens only between a mixed fraction and an improper fraction.

Remember a mixed fraction can be converted to an improper fraction and vice versa.

Apparently, the two fractions are the same but are only distinct because of their fraction types.

$2\frac{1}{5}$ is the same as $\frac{11}{5}$, the only difference is that $2\frac{1}{5}$ is a mixed fraction while $\frac{11}{5}$ is an improper fraction.

For fractions to be added or subtracted, we follow the following steps,

• If whole numbers are already present add/subtract and keep them aside (left-hand side) because we add these whole numbers to our result afterward.
• We must find the lowest common denominator (LCD) between the denominators of the fractions.
• Use the lowest common denominator as the new general denominator.
• Divide the new denominator by the former denominator and multiply the result by the respective numerator.
• Compute our result. Note that when carrying out subtraction the first (left) value may be less than our second (right) value. In this case, subtract 1 from the whole number which we kept aside. That 1 becomes the value of your denominator once we put it into the expression. Add the denominator value to the former value to get a number larger than our second value before subtracting.
• We may simplify if there is a common factor between the new numerator and the new denominator.
• If any whole numbers were present initially, we add them to our result.

Note that the above steps should also be followed when adding and subtracting as well as operations involving multiplication and division of mixed fractions.

Calculate the following

$\frac{4}{5}+\frac{1}{3}$

Solution

Step 1.

We see that there is no whole number.

Step 2.

The LCD between 5 and 3 is 15, thus 15 is the new general denominator.

Step 3.

We divide the new denominator by the former denominator and multiply the result by the respective numerator. Therefore we have

$\frac{4}{5}+\frac{1}{3}=\frac{15\left(\frac{4}{5}\right)+15\left(\frac{1}{3}\right)}{15}=\frac{{}^{3}\overline{)15}\left(\frac{4}{\overline{)5}}\right)+\overline{){}^{5}15}\left(\frac{1}{\overline{)3}}\right)}{15}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Now, we compute our results to get,

$\frac{3\left(4\right)+5\left(1\right)}{15}=\frac{12+5}{15}=\frac{17}{15}=1\frac{2}{15}$

Calculate

$4\frac{1}{2}-1\frac{3}{4}$

Solution

Step 1.

We solve the whole numbers first,

$4-1=3$

We keep our result on the left-hand side.

Step 2.

The LCD between 2 and 4 is 4. Thus 4 is the new general denominator.

Step 3.

We divide the new denominator by the former denominator and multiply our result by the respective numerator. Therefore we have,

Now, we compute the result to get,

$3\frac{2\left(1\right)-1\left(3\right)}{4}=3\frac{2-3}{4}$

Note that 2 is less than 3, so take 1 from the whole number 3, so that your whole number is now 2. The 1 you have taken when put into the expression becomes equal to your denominator value which is 4. Add it to what is already there. Therefore, we have

$3\frac{2-3}{4}=2\frac{\mathbf{4}\mathbf{+}2-3}{4}=2\frac{6-3}{4}=2\frac{3}{4}$

## Multiplying and dividing fractions

Fractions like other numbers can be multiplied, as well as divided.

### How are fractions multiplied?

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

We can always look out for factors that can be divided through numerators and denominators of any fraction. We just have to ensure that the division is between a numerator and a denominator. We should not divide a numerator and another numerator likewise a denominator and denominator.

Calculate

$\frac{2}{3}×\frac{5}{7}$

Solution

Multiply both numerators and denominators together to get,

$\frac{2}{3}×\frac{5}{7}=\frac{2×5}{3×7}=\frac{10}{21}$

Always confirm that no factor can divide both the numerator and denominator before giving your final answer.

Calculate

$\frac{3}{4}×\frac{8}{9}$

Solution

This can be solved using two methods.

Method 1.

Just multiply directly the numerators and denominators.

$\frac{3}{4}×\frac{8}{9}=\frac{3×8}{4×9}=\frac{24}{36}$

A common mistake of students is to leave the fraction in this form. If you do this, you would be wrong because the fraction is not in the simplified form.

The next step to take is to divide both the numerator and denominator by their greatest common divisor (GCD). The GCD between 24 and 36 is 12. Hence,

$\frac{3}{4}×\frac{8}{9}=\frac{24}{36}=\frac{24÷12}{36÷12}=\frac{2}{3}$

Moreover, if your not sure of how to find the GCD, start by dividing with the smallest number from 2, 3,..

Method 2.

You can directly simplify the fractions while multiplying them as seen below.

$\frac{{}^{1}\overline{)3}}{{}^{1}\overline{)4}}×\frac{{}^{2}\overline{)8}}{{}^{3}\overline{)9}}=\frac{1×2}{1×3}=\frac{2}{3}$

We notice that this method is direct and faster.

### How are fractions divided?

In order to divide fractions, we apply the multiplication principles to the multiplication of the first fraction with the inverse of the second fraction.

The inverse of a fraction can be found while interchanging the positions of the denominator and the numerator.

Once this fraction is inversed, the division sign is replaced with the multiplication sign.

Calculate

$\frac{2}{5}÷\frac{3}{7}$

Solution

Step 1.

Inverse the fraction at the right of the division sign and change the division sign to multiplication sign to get,

$\frac{2}{5}÷\frac{3}{7}=\frac{2}{5}×\frac{7}{3}$

Step 2.

Multiply the numerators and denominators.

$\frac{2}{5}×\frac{7}{3}=\frac{14}{15}$

Calculate,

$\frac{5}{14}÷\frac{100}{28}$

Solution

Step 1.

Inverse the fraction at the right of the division sign and change the division sign to multiplication sign, to get,

$\frac{5}{14}÷\frac{100}{28}=\frac{5}{14}×\frac{28}{100}\phantom{\rule{0ex}{0ex}}$

Step 2.

We now simplify to get,

$\frac{5}{14}×\frac{28}{100}=\frac{{}^{1}\overline{)5}}{{}^{1}\overline{)14}}×\frac{{}^{2}\overline{)28}}{{}^{20}\overline{)100}}=\frac{1×2}{1×20}=\frac{1×\overline{)2}}{1×\overline{){}^{10}20}}=\frac{1}{10}$

## Further examples of fractions

We shall be looking at word problems that are fraction problems.

What is the sum of one-third of a pie and half of a pie?

SolutionOne-third of a pie is $\frac{1}{3}$ and half of a pie is $\frac{1}{2}$.

The sum means the addition of both fractions. Thus their sum is given by

$\frac{1}{3}+\frac{1}{2}$

Their LCD is 6, thus, we have

If one-fifth of a man's income is used on accommodation and half of the rest is used on groceries, what fraction of his income is used on groceries?

Solution

Step 1.

First, the fraction of the rest of the man's income after deducing that of the accommodation is

Step 2.

The rest of the man's income is four-fifth. We are told that he used half of the rest for groceries. Therefore, the fraction of the groceries is

$\frac{4}{5}×\frac{1}{2}=\frac{{}^{2}\overline{)4}}{5}×\frac{1}{\overline{)2}}=\frac{2}{5}$.

## Fractions - Key takeaways

• A fraction is defined as a part or portion out of a whole. This whole can be anything including even a fraction but not zero.
• A fraction is composed basically of three components: the numerator, the over sign and the denominator.
• There are three types of fractions which are proper, improper and mixed fractions.

• Improper fractions can be changed to mixed fractions as well as mixed fractions can be converted to improper fractions.

• Equivalent fractions are two or more fractions that are the same.

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What is a fraction and examples?

A fraction is defined as a part or portion out of a whole. An example is 2/3.

What are the 3 types of fractions?

The  three types of fractions are proper, improper and mixed fraction.

How to divide fractions?

In order to divide fractions, we apply the multiplication principles to the multiplication of the first fraction with the inverse of the second fraction.

Fractions are added by finding the lowest common denominator between fractions it being used as the new denominator before they can then be calculated.

How to simplify fractions?

You simplify a fraction by using the common factor between the denominator and numerator to divide both until they are no longer divisible.

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