## What are fractions?

Fractions refer to the number of parts that equal the whole.

Suppose we have a cake divided into equal portions, the fraction comes to represent each of the portions into which we have cut the cake. Thus, if we have four equal portions, each of them represents a quarter of the pie. This is expressed numerically as follows: $1 \over 4$.

The number $1$ refers to the specific slice of the total pie set. We can look at it in the following way: we are talking about one slice and, therefore, we express it with a $1$. If we were talking about two slices, instead of $1$ we would write $2$.

The number $4$ refers to all equal portions of the pie. Since we have divided the pie into four equal portions, the number that should represent this division is $4$.

## Test yourself on simple fractions!

Without calculating, determine whether the quotient in the division exercise is smaller than 1 or not:

$$2:1$$

Fractions can seem like a scary subject because sometimes we have to deal with decimal numbers and fraction lines that appear out of nowhere, and suddenly we have to multiply or simplify the numbers on either side of the fraction line. This may seem like a mess to us, but to make it clear to you we have prepared an article that covers everything about fractions and how to solve them in the simplest way.

## Visualizing Fractions

Let's try to understand what we are seeing in the pictures:

For younger children, the easiest way to learn fractions is through drawings that represent division visually.

For example, below we have our pie divided visually:

The fraction $1 \over 1$ Represents the whole pie. We have a pie that we have kept whole, without dividing it into portions. Therefore, we have a pie divided into a single slice, i.e., whole.

The fraction $1 \over 2$ Represents half of the pie. We have two portions and therefore, the number $2$ comes to represent those two portions. From both, we have taken one, something that we represent with the number $1$.

The fraction $3 \over 4$ is a bit more complicated, but there is an easy way to understand it: this fraction represents a pie that we have divided into four slices (quarters), from which we have taken $3$. Therefore, the number $3$ comes to represent the number of portions we have taken and $4$, the number of total portions there were.

The fraction $1 \over 4$, like that of , $3 \over 4$shows us that the pie has been divided into $4$ portions and therefore we represent it with the number $4$. In the case of $ΒΌ$, this fraction tells us that of the four portions, we have taken only one.

Sometimes we can see that fractions are represented in other ways, such as:

$(\frac{1}{1})$ or $(1:1)$ and sometimes $(1/1)$

but don't worry, they all mean the same thing.

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## Fraction structure: numerator and denominator

The parts and the total are represented by numbers that appear both above and below the fraction line.

## Numerator

The numerator is written on the fraction line and represents the parts of the whole.

Do you know what the answer is?

## Denominator

The denominator is written below the fraction line and represents the whole (in our example, the pie).

## Addition and subtraction of fractions

When we want to add or subtract fractions, there are a series of points that we must check before getting down to work.

## Addition and subtraction of fractions with equal denominator

When fractions share a denominator, all we have to do is add or subtract the numerators of the fractions.

For example:

${3 \over 10}+{6 \over 10}-{5 \over 10}+{1 \over 10}$

${3+6-5+1 \over 10} = {5 \over 10}$

## Subtraction of fractions with different denominator

When the denominator of the fractions is different, the first thing to do is to find thelowest common denominator.

Do you think you will be able to solve it?

## Sums of fractions with different denominators

When the denominator of the fractions is different, the first thing to do is to find the lowest common denominator.

## How to find the lowest common denominator

To find the lowest common denominator, we must find multiples of the larger denominator until we arrive at a number that can be divided by the smallest denominator.
Let's see an example: ${3 \over 2}+{6 \over 4}$In this case, the common denominator will be $4$

Why?

For the reason that: $2Γ2=4$ and $4Γ1=4$.

In the case of fraction ${3 \over 2}$, we multiply it by $2$ (both numerator and denominator) and in the case of fraction ${6 \over 4}$, we multiply it by $1$. In this way, we obtain the following result:

${3 \over 2}+{6 \over 4} = {6 \over 4} + {6 \over 4} = {12 \over 4}$

## Mixed fractions

So far we have only talked about simple fractions. In this section we will focus on another more complicated type of fractions, mixed fractions, which are also called mixed numbers. Before we dive into the addition and subtraction of mixed numbers, let's understand what they are. Mixed numbers are numbers that combine whole numbers and fractions, such as: $2{1 \over 2}$

## Addition and subtraction of mixed numbers

You can add and subtract mixed numbers in two different ways.

First option:

On the one hand, we can calculate integers and fractions separately according to the rules we have already seen.

For example:

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

$(1+2)+ββ({1 \over 3}+{1 \over 3})=3+{1+1 \over 3}=3{2 \over 3}$

Second option:

On the other hand, the second way we can add and subtract mixed numbers is to convert the whole number into a fraction whose denominator is equal to that of the accompanying fraction. To do this, we just multiply the whole number by the denominator of the fraction and then add the result in the numerator. It sounds a bit messy, but it's pretty straightforward.

For example:

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

We convert whole numbers into fractions and then add according to the rules we have already seen:

$1{1 \over 3}={3 \over 3}+ {1 \over 3}={4 \over 3}$

$2{1 \over 3}={6 \over 3}+ {1 \over 3}={7 \over 3}$

The result will be as follows:

${4 \over 3}+{7 \over 3}={11 \over 3}$

So far we have covered many topics and concepts such as mixed numbers or improper fractions without getting too deep into them, so let's do a brief review.

Do you know what the answer is?

## Improper fractions

Improper fractions are those whose numerator is greater than the denominator. That is, it is a fraction whose result is greater than $1$ or, in other words, the improper fraction contains more than the whole.

For example:

${4 \over 2}$

In this case we see that the denominator is $2$. That is, the whole is equal to $2$ parts and, in addition, we have another $2$ parts of another whole.

${4 \over 2}= {2 \over 2}+{2 \over 2} = 2$

## Mixed number

A mixed number is a number that combines a whole number with a fraction, such as $2{ββ1 \over 2}$.

When we encounter an improper fraction, we can simplify it to a mixed number.

We can do this if we decompose the fraction into several smaller fractions based on the same denominator.

For example:

The fraction ${5 \over 2}$ can be divided into: ${ββ2 \over 2} + {ββ2 \over 2} + {1 \over 2} = 2{ββ1 \over 2}$

## Simplification of fractions - Fraction reduction

Sometimes we can simplify fractions to make them easier to work with.

For example, the following fraction can be simplified:

${3 \over 15}$

Simplifying fractions can be done by dividing both the numerator and denominator by the same number, preferably by the one that transforms the numerator into the smallest possible number.

For example:

In the fraction ${3 \over 15}$, the numerator is $3$ and the denominator, $15$. Both can be divided by $3$, thus obtaining a much simpler fraction:

$3\div3=1$

$15\div3=5$

By simplifying $3\div15$ we get $1Γ·5$ or . ${1 \over 5}$

You can read more about simplifying fractions in the article "Simplifying fractions" on our website to find out how to do it.

Once the principles and concepts are understood, we can move on to other topics related to fractions.

## Multiplication of fractions

To multiply fractions, what we must do is multiply the numerators with each other and the denominators with each other.

Do you think you will be able to solve it?

## Multiplication of proper fractions: multiplying a fraction by another fraction

Multiplying proper fractions is quite easy. All we have to do is multiply the denominator of the first fraction by the denominator of the second fraction and do the same with the numerators.

For example:

${{1\over2}\times{3\over4}}={3 \over 8}$

## Multiplication of proper fractions: multiplying a fraction by a whole number

When we have an exercise where we have to multiply a fraction by a whole number, all we have to do is convert the whole number into a fraction as we have already learned.

For example:

${2\times{3\over4}}={2 \over 1}\times{3\over4}={6\over4}={3\over2}$

## Multiplication of mixed fractions: multiplying a fraction by a mixed number

As with multiplying a fraction by a whole number, the simplest way is to convert mixed numbers into improper fractions.

For example:

${2{1\over2}\times{3\over4}}$

First, we convert $2{1\over2}$ into a simple fraction:

$2{1\over2}={4\over2}+{1\over2}={5\over2}$

Then we continue with the exercise:

${{5\over2}\times{3\over4}}={15\over8}=1{7\over8}$

## Multiplication of mixed fractions: multiply whole number by a mixed number

There are two ways to multiply a whole number by a mixed number.

• The first way is like the previous multiplication cases: we convert the mixed number into an improper fraction and the whole number into an improper fraction. We must make sure that both have a common denominator.

For example:

$2{1\over2}\times{2}$

$2{1\over2}={4\over2}+{1\over2}={5\over2}$

$2={4\over2}$

Then, we proceed with the exercise:

${5\over2}\times{4\over2}={20\over4}={5\over1}=5$

The second way to do this is by applying the distributive property, but we will talk about that in detail later.

Do you know what the answer is?

## Division of fractions

Dividing proper fractions is simple. All we have to do is divide both numerators into each other and both denominators. But what happens when we are dealing with whole and mixed numbers?

## Division of proper fractions: dividing a fraction by another fraction

As we have said before, dividing proper fractions is quite simple. All we have to do is divide the denominator of the first fraction by the denominator of the second fraction and do the same with the numerators.

For example:

${{3\over4}\div{1\over2}}={3 \over 2}=1{1\over2}$

## Cross Multiplication

When the numbers are large and in order, everything is fine, but what happens if we have a reverse exercise like the following one?

${{1\over2}\div{3\over4}}$

In this case, we should resort to cross multiplication. What we must do is leave the first fraction as it is, transform the division into a multiplication and invert the numerator and denominator of the second fraction.

For example:

${{1\over2}\div{3\over4}}={{1\over2}\times{4\over3}}$

Now it is much easier for us to solve the exercise:

${{1\over2}\times{4\over3}}={4\over6}={2\over3}$

## Division of proper fractions: dividing a fraction by a whole number

When we have an exercise in which we have to divide a fraction by a whole number, what we must do is to convert the whole number into a fraction as we already learned in the section Multiplication of proper fractions: multiplying a fraction by a whole number.

For example:

${{3\over4}\div2}={{3\over4}\times{1\over2}}={3\over8}$

Do you think you will be able to solve it?

## Division of proper fractions: dividing a whole number by a fraction

When we have an exercise where we have to divide a whole number by a fraction, all we have to do is convert the whole number into a fraction as we have already learned and then multiply the fraction we have created by the other cross fraction. Actually, this is the same procedure that we apply to divide a fraction by a whole number. The only thing that changes is that we must invert the fraction and not the whole number.

For example:

$2\div{3\over4}={2\over1}\times{4\over3}={8\over3}=2{2\over3}$

## Division of improper fractions: dividing a mixed number by a whole number or fraction

We have already understood the procedure. In exercises of this type, what we have to do is to convert the mixed number into an improper fraction and the whole number as well. Then we divide according to the rules we have learned. The principles of division hold.

There are other types of fractions, such as decimals, but we will study them in another article.

## Exercises

### Exercises on Addition of fractions

• $\frac{1}{3}+\frac{2}{6}=$
• $\frac{7}{2}+\frac{1}{4}=$
• $2\frac{2}{5}+4\frac{5}{6}=$
• $\frac{4}{8}+\frac{1}{4}=$
• $7\frac{4}{8}+5\frac{1}{8}+\frac{3}{7}=$

Do you know what the answer is?

### Fraction subtraction exercises

• $\frac{1}{3}-\frac{2}{4}=$
• $4\frac{6}{8}-\frac{7}{9}=$
• $\frac{15}{20}-\frac{1}{5}=$
• $\frac{18}{6}-\frac{7}{5}=$
• $3\frac{3}{9}-\frac{1}{8}-2\frac{11}{6}=$

### Fraction exercises with addition and subtraction

• $\frac{1}{2}-\frac{1}{2}+\frac{1}{2}=$
• $2\frac{1}{2}-2\frac{1}{4}+1\frac{1}{3}=$
• $\frac{1}{3}+\frac{1}{3}-\frac{1}{3}=$
• $7\frac{2}{3}+\frac{1}{4}-2\frac{2}{3}+1\frac{1}{4}=$
• $1\frac{3}{9}+\frac{3}{3}-1\frac{3}{18}+7\frac{3}{6}+\frac{1}{3}=$

### Exercise on multiplication of fractions

• $\frac{1}{4}\times\frac{1}{4}\times1\frac{1}{4}=$
• $\frac{2}{3}\times\frac{2}{3}\times\frac{2}{3}=$
• $2\frac{2}{3}\times\frac{2}{6}\times5\frac{2}{9}=$
• $\frac{6}{5}\times\frac{2}{2}\times\frac{1}{9}=$
• $2\frac{3}{7}\times3\frac{7}{2}\times1\frac{2}{16}=$

### Exercises on division of fractions

• $\frac{2}{5}:\frac{2}{4}=$
• $\frac{1}{2}:3\frac{2}{10}=$
• $\frac{1}{5}:\frac{2}{5}:\frac{4}{5}=$
• $1\frac{2}{5}:2\frac{2}{4}:\frac{2}{6}=$
• $\frac{3}{7}:\frac{4}{8}:\frac{1}{6}=$

Do you think you will be able to solve it?

### Exercises on fractions with multiplication and division

• $\frac{1}{3}\times\frac{1}{3}:1\frac{1}{3}=$
• $\frac{6}{20}:\frac{2}{4}\times\frac{3}{2}=$
• $\frac{3}{7}\times2\frac{4}{8}:\frac{1}{6}\times2\frac{7}{2}=$
• $1\frac{2}{17}\times1\frac{1}{8}\times\frac{1}{4}:1\frac{8}{3}=$
• $\frac{1}{35}:2\frac{2}{7}\times5\frac{1}{4}:\frac{1}{1}=$

### Combined fraction exercises with (addition, subtraction, multiplication and division).

• $\frac{2}{3}\times\frac{2}{5}:\frac{3}{2}-3\frac{3}{4}=$
• $\frac{27}{3}-\frac{2}{5}\times4\frac{9}{5}:\frac{3}{1}+3\frac{2}{44}=$
• $\frac{1}{11}+\frac{7}{3}\times1\frac{2}{8}:\frac{1}{9}+7\frac{2}{3}=$
• $8\frac{1}{7}-\frac{7}{3}:7\frac{12}{11}+\frac{5}{15}\times8\frac{1}{3}=$
• $100\frac{1}{100}-200\frac{7}{100}:300\frac{12}{100}+400\frac{5}{100}\times500\frac{1}{100}=$

## Examples and Exercises with Solutions

### Exercise #1

Without calculating, determine whether the quotient in the division exercise is less than 1 or not:

$5:6=$

### Step-by-Step Solution

Note that the numerator is smaller than the denominator:

5 < 6

As a result, we can claim that:

\frac{5}{6} < 1

Therefore, the quotient in the division problem is indeed less than 1

Yes

### Exercise #2

Without calculating, determine whether the quotient in the division exercise is less than 1 or not:

$7:11$

### Step-by-Step Solution

Note that the numerator is smaller than the denominator:

7 < 11

As a result, we can claim that:

\frac{7}{11}<1

Therefore, the quotient in the division problem is indeed less than 1

Yes

### Exercise #3

Without calculating, determine whether the quotient in the division exercise is less than 1 or not:

$1:2=$

### Step-by-Step Solution

Note that the numerator is smaller than the denominator:

1 < 2

As a result, we can claim that:

\frac{1}{2}<1

Therefore, the fraction in the division problem is indeed less than 1

Yes

### Exercise #4

Without calculating, determine whether the quotient in the following division is less than 1 or not:

$11:8$

### Step-by-Step Solution

Note that the numerator is smaller than the denominator:

11 > 8

As a result, we can claim that:

\frac{11}{8} > 1

Therefore, the quotient in the division problem is not less than 1

No.

### Exercise #5

Solve the following:

$\frac{29}{29}=$

### Step-by-Step Solution

We will divide the numerator and denominator by the highest number that both are divisible by.

In this case, the number is 29

We will divide the fraction as follows:

$\frac{29:29}{29:29}=\frac{1}{1}=1$

$1$