﻿ What are fractions and what they are used for. Equivalent fractions.

# What are fractions and what they are used for. Equivalent fractions.

Don’t you understand what fractions are? When you have to trade fractions you don’t know what to do?

Then what follows interests you. I’m going to explain to you what fractions are, but not only from a real point of view, but from the point of view of numbers, which will help you understand operations with fractions.

Do you know what equivalent fractions are? In this section we will explain what equivalent fractions are and how to obtain them and what they are used for.

## What are fractions? What are fractions used for?

A definition of fractions that is a way of representing the parts of a whole. For example, we have this rectangular bar: We could divide it into 2 parts: The part that is coloured green is represented as a fraction of this form: Where:

• The bottom number, the denominator, represents the total parts into which the whole

has been divided.

• The number above, the numerator, represents the parts we want to refer to

We can divide the whole into the parts we want, yes, the parts have to be equal.

Now we divide the bar into 3 parts: How do we represent in fraction form the part painted green? Thus: Now in the denominator (the number below) we have a 3, which are the parts into which we have divided the bar and in the numerator (the part above) we have a 2, which are the parts painted green and to which we want to refer.

If we divide it into 4 parts and paint only 1 part green: This part would be represented: Now, this definition of what fractions are is only valid if the number we are representing is less than 1, as we are referring to one or more parts of the whole.

However, we can also represent the complete whole in the form of a fraction, that is, if we divide the bar into 3 parts and want to refer to the 3: It would be represented: The numerator and denominator are equal, because the parts into which the bar is divided and to which we refer are the same.

In fact, if we do the division, it results in the unit: And this leads me to explain to you what fractions are in another way.

## The fractions: Another way to represent a division

I’m going to give you another definition of what the fractions are that will help you solve operations with them.

Fractions are another way of representing numbers, both decimals and integers. It’s like leaving the indicated division without reaching the final result.

In the last example, we have represented the 3 parts that make up the complete bar, that is, the whole. If we make the division, it gives us the unit: The previous fractions, we can also express them as decimal numbers if we finish making the division:   Up to now, the numerator has always been smaller than or equal to the denominator, but can the numerator be larger than the denominator?

Of course.

In that case, the fractions no longer represent parts of a whole, but a division, or rather, a division.

For example, imagine you have to distribute 5 kg of flour among 2 other smaller equal sacks, how much would each sac weigh?

Then you have to divide 5 by 2, and it would give you the result, wouldn’t it? Each bag would weigh 2.5 kg

O you can also express the result as a fraction, without having to give the result of the division: Do you see it? The fraction is the previous step to solve the division

## What is the value of a fraction?

As we have just seen, the fraction is a division and its value is the result of solving the division. This value can be any integer or decimal.

Therefore, any number can be represented as a fraction:   Fractions can have different values depending on their numerator:

• If the numerator is less than the denominator, the fraction value will be less than 1 Is the case when fractions represent parts of a whole

• If the numerator is the same as the denominator, the fraction value will be 1 When the fraction represents the whole, i.e. the unit

• If the numerator is greater than the denominator, the value of the fraction will be greater than 1  In this case, the fractions represent distributions.

In any case, fractions are a way of representing the numbers, both decimal and integer, of any value, leaving the division indicated without getting its value.

Now that you have a little more clarity about fractions from a mathematical point of view, let’s see another concept: equivalent fractions.

## What are equivalent fractions

Equivalent fractions are those which, when divided, have the same value or, in other words, represent the same number (decimal or integer).

Let’s take a slower look at it with some examples:

We have a bar, divided in 2 equal parts and we want to represent in fraction form the colored part of green: The 2 parts into which we have divided the bar will be the denominator and the green part will be the numerator: If we do the division, the fraction has a value of: Now, we want to represent the same part painted green, but in this case the bar will be divided into 4 parts. So that the green part is the same as the green part of the previous case, 2 parts of the bar will correspond: The 4 parts into which we have divided the bar will be the denominator and the 2 parts in green will be the numerator: Y this fraction has a value of: As the green part of both cases are equal, the value of each fraction is the same, as it could not be otherwise.

Therefore, the fractions: These are equivalent fractions and are two different ways of representing the same number.

In the same way if we want to represent unity as a fraction, we have infinite ways to do it, as long as the numerator and denominator are equal: The value of all these fractions is 1: Therefore, they are also equivalent fractions and are different fractions to represent the same value.

Finally, it is the same to distribute 10 units among 4, 5 units among 2 or 100 units among 40. The result of all distributions is the same: We are therefore talking about equivalent fractions.

As you can see, there are several ways of representing the same number in the form of a fraction, and each of those fractions representing the same value are the equivalent fractions.

## How to obtain the equivalent fractions.

Equivalent fractions are obtained by multiplying or dividing the numerator and denominator by the same number, i.e., up and down. Let’s see examples of equivalent fractions: We see that the value of the fraction does not change, when multiplying or dividing numerator and denominator by the same number.

### How to obtain the equivalent fractions that interest us

In certain cases, which we will see later, we need to transform the fraction to operate with it at our convenience.

Let’s see how to do it. Let’s imagine we have the following fraction Y we are interested in transforming it into an equivalent fraction with 24 as denominator.

Therefore, in other words, we want to transform this fraction into its equivalent with denominator 24: I need to find a numerator, so that dividing it by 24 results in the same value as the original fraction. How do I get the numerator of the equivalent fraction?

We have said before that for the fraction to be equivalent, the numerator and the denominator must be multiplied or divided by the same number and thus their result does not vary.

Then if before we had a 2 and now we have a 24, indirectly we have multiplied the denominator by a number, therefore, I have to know what that number is, to multiply the numerator by the same number.

This number is obtained by dividing the denominator of the equivalent fraction by the denominator of the original fraction: And in our case: Therefore, I have indirectly multiplied the denominator by 12, and for the fraction to be equivalent, the numerator must also be multiplied by 12. It is always a matter of multiplying or dividing by the same number so that the value of the original fraction does not vary.

To do this more directly, place the new denominator in the fraction with the original numerator and multiply it by the corresponding number, that is: This way of calculating the equivalent fraction will be used when we obtain a common denominator in an addition or subtraction of fractions with a different denominator.

## What are the equivalent fractions for?

We are interested in working with equivalent fractions, when we need to transform the fraction, to obtain a common denominator or also when simplifying fractions.