Fractions are one of the most recurring operations that the little ones usually encounter during their school years in Primary Education. They start with the natural numbers, learning to add, subtract, multiply and divide before later doing the same applications with the fractions themselves. They are the intermediate step before facing some somewhat more complicated operations such as equations or logarithms. For this reason, it is of vital importance that they are correctly understood, internalized and know how to apply them.
Fractions are still a mathematical representation of a real situation. In this sense, when we make a fraction we are expressing the portion of a part or of an object that can be divided into different portions.
What are fractions?
Fractions are a really well-known mathematical expression that expresses the division between two numbers, which are known as the numerator and denominator. In this sense, the number that is established at the top of the fraction itself is called the numerator, while the one at the bottom is called the denominator. A dividing line is the one that is responsible for dividing both the numerator and the denominator to complete the different parts that make up the fraction itself. In order for it to be considered a fraction, the norm of A must be fulfilled as the numerator divided by B, which is the denominator, expressed as A/B.
However, B can never be zero, since it is a mathematical expression that does not exist since a number cannot be divided by zero. If we think about it logically, we see that two pears cannot be divided by zero apples, because basically it is something unreal, just as it happens in mathematical language. In fact, if you try to divide a number by zero on a scientific calculator, the result will be an error. Therefore, for a fraction to be considered as something real, 0 can never be the denominator.
The fractions themselves are distinguished between different types that serve to name the different expressions that can be found. On the one hand, we have simple fractions, which are those that express the main mathematical property described above. Later, these same simple fractions can be divided between proper fractions and improper fractions. In both cases, the numbers must always be positive. A proper fraction will be one in which the denominator will always be less than the denominator. On the other hand, in improper fractions, the numerator will always have a value greater than that of the numerator.
The main types of fractions
On the other hand, we can also find what is called a mixed fraction, which is one that can mix whole numbers with fractional numbers. This will be much better understood with an example. Let’s say that we have the fraction 30/20. As we know, fractions can be simplified by equivalent fractions. Therefore, 3/2 is the same as 30/20. Finally, 3/2 is exactly the same as 1 and 1/2, which would be one and a half units of any type of object. In addition, we can also express inverse fractions that are those in which the numerator and denominator are reversed. If we have 2/3, its inverse fraction would be 3/2.
On the other hand, the compound fraction is one in which in its mathematical expression we find different types of fractions, and integers at the same time. In this sense, it is necessary to master both types of mathematical expressions since they serve to understand and apply the necessary calculations to achieve the results we are expecting. Undoubtedly, understanding the classification into which the fractions are divided will be very useful when applying the process.
Benefits of adding or subtracting fractions
Still, there are numerous benefits that we can gain from the knowledge that adding or subtracting fractions can have. Some of the most prominent are the following:
- Pass math. Learning to add and subtract fractions is one of the essential requirements to be able to pass mathematics in Primary Education. In this sense, it is absolutely necessary to master this type of operations.
- Continue progressing in the calculation. Progressing adequately when calculating involves being able to calculate additions and subtractions of fractions in an agile and adequate way. Otherwise, we will not be able to progress in this type of field.
- Develop the brain. The brain usually has different types of intelligence, and to develop one requires some type of practice such as calculating the addition and subtraction of fractions. In this way, the mathematical part of the brain will be more adequate.
- Establish social relationships. Mathematics is a field that generates a huge number of people and a huge hobby. Therefore, if you like mathematics, it can be a field where you can develop a good way to strengthen social relationships with people who share the same interests.
How to add or subtract fractions?
Once all the types of fractions have been introduced, it’s time to teach you how to add or subtract fractions without any difficulty and in the simplest and most intuitive way possible. The goal is that you can perform this type of calculation without making mistakes. Let’s start with the addition of fractions and with something that can also be applied to subtraction, although we will see it twice in case you just need some type of operation. In order to add or subtract fractions, it is necessary that they be equivalent expressions, and therefore, it can be operated as if they were natural numbers. In this case, the first step to take is what is known as the least common multiple. It is used to get equivalent fractions quickly and easily.
The least common multiple is the largest number that makes up the different names. Let’s say that we have to add 1/3 and 2/6. In this case, the least common multiple of both would be six. The most basic way to perform a least common multiple is by doing a factor decomposition. The number is decomposed by the smallest number that is multiple and ends up taking those larger factors or those that are not repeated. In this example, six incorporates two pairs of threes while three only one. Therefore, we are left with six. Thus, six will be the denominator of the resulting fraction before carrying out the addition. The next step is to divide the two denominators and multiply the result by the numerator. In the case of 2/6 it stays the same because the denominator does not change.
On the other hand, in the case of 2/3 we have to divide six by three and multiply it by two (divide by the denominator and multiply by the numerator). The result ends up being 4/6. Finally, the last step before performing the addition of fractions is to place everything well located. 4/6 + 2/6 ends up being 6/6. Obviously, to be able to express it in a simpler way we can simplify the fractions and place a 1. Let’s not forget that in real life fractions are still expressions that serve to represent portions of different objects. Therefore, you have to be able to transfer real language to mathematical expressions such as fractions. This is the first step. We cannot only know how to operate and not correctly represent each of the expressions.
Subtracting fractions ends up being the same procedure
With the subtractions you end up following the same process that we have seen previously. In this case, we will see the other possible case that we can find with the least common multiple. If the denominators are different, you basically have to multiply. If we have to subtract 2/3 – 4/2, the denominator will end up being six. Once we have the common denominator we must basically apply the same operation that we have introduced previously. We will divide the six by each of the previous denominators and multiply them by the numerators. In this sense, the equivalent fractions will end up being 4/6 – 12/6.
The final result will end up being -8/6. It is possible to simplify this type of fraction and therefore leave a number that ends up being easier to read and interpret in case of some type of problem. Therefore, -8/6 can remain at -4/3 as the final result of this subtraction of fractions. It is important in both cases to understand what is the procedure to follow starting by leaving the equivalent expressions. Therefore, we will carry out the least common multiple to carry out this first step. Once we have the expressions prepared, we will perform an operation with real numbers as if it were a normal addition or subtraction. Finally, we will have the result that we must express as final and put an end to the process.
Adding and subtracting fractions is an absolutely necessary step if you want to achieve the expected objectives. We must be able to interpret the language that we habitually use to transform it into mathematical expressions and from then on carry out the procedure that we have already explained above. Do not forget that if you want to progress in your mathematical knowledge it is absolutely necessary to learn fractions and do exercises, the different types and above all how to operate with them. Basically because equations and logarithms maintain fractions, and even derivatives and integrals as well. So, that he studies and prepares the fractions syllabus well.