Once we know how to add and solve operations with natural numbers such as subtraction, multiplication or division, among others, the next step when it comes to further advancing our mathematical knowledge is learning to add fractions. It is practically the same as the operations with natural numbers, although it is a bit more complicated because there are some intermediate steps that must be carried out to correctly complete the operations.
For this reason, it is necessary to know how to carry out some mathematical procedures that we will see below, such as the least common multiple or the greatest common factor to balance the denominators and numerators and, therefore, be able to add fractions correctly and solve fraction exercises.
What are fractions?
At a mathematical level, a fraction could be defined as the division of a quantity by another quantity. At a more colloquial level, fractions are used to express portions of a quantity. Let’s imagine a cake or a pizza, which are meals that are divided into portions. If we want to express half, we will express it with the fraction 1/2, which is 0.5. On the other hand, if the case is to express a quarter, we will express the fraction as 1/4, which is equivalent to 0.25. The main components of a fraction would be the numerator, the denominator and the line that serves to divide the two parts.
The numerator is the number that is expressed in the upper part of the fraction, later we would find the line that is used to divide the two parts and finally the number below the line is defined as the denominator. At a mathematical level, fractions can also be defined as A/B, where the numerator is an integer, while B can never be 0 since the characteristics of traditional fractions are real and rational numbers. When we divide something by zero we would never obtain a rational number as a result since you cannot have two parts of anything, for example.
Fractions can also be represented graphically. In most cases, a number of squares is drawn that is completely equivalent to the denominator and those squares are colored according to the quantity that we have expressed in the numerator. In addition, we can also define different types of fractions according to their typology and their way of expressing themselves at a mathematical level. If the two numbers are portions or integers we would find the simplest type of all. On the other hand, within simple fractions we find proper fractions when the numerator is less than the denominator, whereas if it is the other way around, and the numerator is greater than the denominator, the fraction would become improper.
On the other hand, mixed fractions would be the mixture between a whole number and an improper fraction. Let’s see a clear example. We start from the fraction 30/20. To simplify it, we can remove the two zeros by dividing both the numerator and denominator by 10. So we’re left with 3/2. This fraction is similar to 1 multiplier times 1/2, which would be when we have a pizza and a half, for example. Finally, inverse fractions are those in which the numerator and denominator are reversed. In this sense, 1/2 becomes 2/1 or 3/2 becomes 2/3. The compound fractions are those in which we are dividing fractions and finally the decimal fractions are all those that can end up being expressed as a percentage.
Beyond all this, we can also call equivalent fractions as all those in which there is an equivalence between two different fractions that end up having exactly the same value between them. Undoubtedly, there are multiple types of fractions totally different from each other and with different meanings.
Benefits of knowing how to add fractions
However, there are multiple benefits of knowing how to add fractions for our day to day, both personally and professionally. The most prominent are the following:
- Know how to express portions. Operating with natural numbers is essential for multiple jobs, but the next step is to perform exactly the same type of operations, albeit with fractions, two numbers divided by each other.
- Being able to pass math. In addition, fractions are one of the important requirements to be able to pass mathematics when we are in Primary Education. Therefore, if we are not able to understand and know how to operate with them, we will not pass the subject.
- Develop a professional career. The development of a career related to mathematics requires us to know how to operate with fractions. It is an intermediate step for many, but essential for all those who want to dedicate themselves to being mathematics teachers.
- Advance in the knowledge of mathematics. Finally, there are many operations that are really more complicated than fractions, such as equations or logarithms. However, if we are not able to operate with fractions we will not be able to face the next challenge.
For this reason, we must put our batteries with the fractions to be able to consolidate a base before continuing in our knowledge with mathematics.
How to add fractions?
When adding fractions we will focus on two main methods. On the one hand, you will learn to do it manually since it is necessary for you to understand the entire process for your day to day. However, technology also plays a really important role in the mathematical world. For this reason, today we can already find multiple scientific calculators that perform fraction operations for us quickly and effectively. To begin we are going to explain how to solve sums of fractions in a simple and intuitive way.
The first step you should know is that no sum can be solved without first having a common denominator. In this sense, if we find fractions that have a different denominator, we have to establish the least common multiple, that is, find the number that encompasses all the factors. Let’s say we have to add 1/5 + 1/2. Therefore, the first step is to establish that previously established number. In this example, the answer is 10 because it is the only number that encompasses both 5 and 2. In these cases, it is easy because cross multiplication is established. But for example, if we find some power of any of the factors, the smallest factor is left out of the operation.
Once we have 10 as a common denominator, the next step is to find an equivalent numerator. Therefore, in this case we must divide the 10 by the denominator and multiply it by the numerator. Thus, we will obtain 2/10 + 5/10. Once we have the fractions ready to operate, basically what we will have to do is add the numerators as if they were two natural numbers. In this sense, the final result will be 7/10. In some cases, interpretation may be required. Let’s go with another example. Let’s imagine that I have eaten a quarter of the cake and my brother has eaten half. What is the total amount of cake that we have eaten between the two of us?
The calculator, as another resource to solve fractions
First, we must express the statement in fraction format. Therefore, the sum that we must operate will be 1/4 + 1/2. In this case, as we have explained previously, the greatest common multiple is a power that includes the other factor 4 is equal to two squared and, therefore, the two is already included. In this case, the common denominator will be four. Since the denominator is the same as that of another fraction, 1/4 stays absolutely the same. In the other case, 1/2 must be modified. We divide 4 by the denominator and multiply it by the numerator, resulting in 2/4. Therefore, the final result ends up being 3/4, equivalent to 75% of the pie.
It is also important to indicate, especially for those students who are taking it at school, that in some exercises it will be necessary to simplify the result. Imagine that we have 4/8 and, therefore, we have powers that allow us to divide the two parts of the fraction as the numerator or the denominator. In this case, it is 4 since it is possible to divide 4 and 8 by that number. Therefore, 1/2 would be exactly the same as 4/8. In fact, if we look for the decimal number of both we will get 0.5. Therefore, they are the so-called equivalent fractions that we talked about earlier.
However, despite the fact that we recommend that you learn to add fractions, there are many other resources that we have at our disposal to carry out this operation. Scientific calculators are one of our best allies. With them it is already possible to carry out these operations, being the Casio one of the best options that we have at our disposal. To do it we must introduce the numerator, then look for the fraction sign that is represented in the form of an inverted L and then introduce the denominator. Finally, we’ll hit the plus sign and introduce the other fraction. Pressing the equal we will obtain the final result and, therefore, it will not be necessary to look for the least common multiple and perform the addition.
Therefore, the addition of fractions and their knowledge is absolutely essential if you want to delve into the mathematical language and try to perform more complicated operations. You must learn to do it because it is something regulated in the academic program, but it is especially important to pay attention in those cases that are interesting, such as the binary system.