One of the most important sciences, the application of which can be seen in disciplines such as chemistry, physics, and even biology, is mathematics. The study of this science allows you to develop some mental qualities, improve abstract thinking and the ability to concentrate. One of the topics that deserve special attention in the course "Mathematics" is the addition and subtraction of fractions. For many students, its study is difficult. Perhaps our article will help to better understand this topic.
How to subtract fractions whose denominators are the same
Fractions are the same numbers with which you can perform various actions. Their difference from integers is the presence of the denominator. That is why when performing actions with fractions, it is necessary to study some of their features and rules. The simplest case is the subtraction of ordinary fractions whose denominators are represented as the same number. Perform this action is not difficult if you know a simple rule:
- In order to subtract the second from one fraction, it is necessary to subtract the numerator of the subtracted fraction from the numerator of the fraction to be reduced. This number is written in the numerator of the difference, and the denominator is the same: k / m - b / m = (kb) / m.
Examples of subtraction of fractions whose denominators are the same
Let's see how it looks on an example:
7/19 - 3/19 = (7 - 3) / 19 = 4/19.
From the numerator of the reduced fraction "7" we subtract the numerator of the subtracted fraction "3", we get "4". We write this number in the numerator of the answer, and put in the denominator the same number that was in the denominators of the first and second fractions - “19”.
The picture below shows some more similar examples.
Consider a more complex example where fractions with the same denominators are subtracted:
29/47 - 3/47 - 8/47 - 2/47 - 7/47 = (29 - 3 - 8 - 2 - 7) / 47 = 9/47.
From the numerator of the reduced fraction “29”, subtracting in turn the numerators of all subsequent fractions - “3”, “8”, “2”, “7”. As a result, we get the result “9”, which is written in the numerator of the answer, and in the denominator we write the number that is in the denominators of all these fractions - “47”.
Addition of fractions having the same denominator
Addition and subtraction of ordinary fractions is carried out according to the same principle.
- In order to add fractions whose denominators are the same, you need to add the numerators. The resulting number is the numerator of the sum, and the denominator will remain the same: k / m + b / m = (k + b) / m.
Let's see how it looks on an example:
1/4 + 2/4 = 3/4.
To the numerator of the first term fraction - "1" - add the numerator of the second term fraction - "2". The result - “3” - is written in the numerator of the sum, and the denominator is the same as that present in the fractions, “4”.
Fractions with different denominators and their subtraction
We have already considered the action with fractions that have the same denominator. As you can see, knowing simple rules, solving such examples is quite easy. But what if you need to perform an action with fractions that have different denominators? Many high school students are embarrassed by such examples. But here, if you know the principle of solution, examples will no longer be difficult for you. There is also a rule here, without which the solution of such fractions is simply impossible.
We will talk more about how to do this.
Fraction property
In order to bring several fractions to the same denominator, you need to use the main property of the fraction in the solution: after dividing or multiplying the numerator and denominator by the same number, you get a fraction equal to this one.
So, for example, the fraction 2/3 can have such denominators as “6”, “9”, “12”, etc., that is, it can be in the form of any number that is a multiple of “3”. After we multiply the numerator and denominator by “2”, we get the fraction 4/6. After we multiply the numerator and denominator of the initial fraction by “3”, we get 6/9, and if we perform the same action with the number “4”, we get 8/12. One equality can be written as follows:
2/3 = 4/6 = 6/9 = 8/12 ...
How to reduce multiple fractions to the same denominator
Consider how to reduce multiple fractions to the same denominator. For example, take the fractions shown in the picture below. First you need to determine what number can be the denominator for all of them. To facilitate, we decompose the existing denominators into factors.
The denominator of the fraction 1/2 and fraction 2/3 cannot be factorized. The denominator 7/9 has two factors 7/9 = 7 / (3 x 3), the denominator of the fraction 5/6 = 5 / (2 x 3). Now it is necessary to determine which factors will be the smallest for all these four fractions. Since the first fraction in the denominator has the number “2”, it means that it must be present in all denominators, in the fraction 7/9 there are two triples, which means that they must also be present in the denominator. Given the above, we determine that the denominator consists of three factors: 3, 2, 3 and is equal to 3 x 2 x 3 = 18.
Consider the first fraction - 1/2. Its denominator has “2”, but there is not a single digit “3”, but there should be two. To do this, we multiply the denominator by two triples, but, according to the fraction property, we and the numerator must multiply by two triples:
1/2 = (1 x 3 x 3) / (2 x 3 x 3) = 9/18.
Similarly, we perform actions with the remaining fractions.
- 2/3 - the denominator lacks one triple and one two:
2/3 = (2 x 3 x 2) / (3 x 3 x 2) = 12/18. - 7/9 or 7 / (3 x 3) - the denominator lacks a deuce:
7/9 = (7 x 2) / (9 x 2) = 14/18. - 5/6 or 5 / (2 x 3) - the triple is missing in the denominator:
5/6 = (5 x 3) / (6 x 3) = 15/18.
Together, it looks like this:
How to subtract and add fractions having different denominators
As mentioned above, in order to add or subtract fractions having different denominators, they must be brought to the same denominator, and then use the rules for subtracting fractions having the same denominator, which was already described.
Consider this as an example: 4/18 - 3/15.
Find the multiple of numbers 18 and 15:
- The number 18 consists of 3 x 2 x 3.
- The number 15 consists of 5 x 3.
- The total multiple will consist of the following factors 5 x 3 x 3 x 2 = 90.
After the denominator is found, it is necessary to calculate a factor that will be different for each fraction, that is, the number by which it will be necessary to multiply not only the denominator, but also the numerator. For this, the number that we found (the common multiple) is divided by the denominator of the fraction for which additional factors need to be determined.
- 90 divided by 15. The resulting number "6" will be a factor for 3/15.
- 90 divided by 18. The resulting number "5" will be a factor for 4/18.
The next step in our decision is to bring each fraction to the denominator of "90".
How this is done, we have already said. Consider how this is written in the example:
(4 x 5) / (18 x 5) - (3 x 6) / (15 x 6) = 20/90 - 18/90 = 2/90 = 1/45.
If fractions with small numbers, then you can determine the common denominator, as in the example shown in the picture below.
Similarly produced and the addition of fractions having different denominators.
Subtraction and addition of fractions having integer parts
Subtraction of fractions and their addition we have already examined in detail. But how to subtract if the fraction has an integer part? Again, let's use a few rules:
- All fractions having an integer part should be converted to incorrect ones. In simple words, remove the whole part. To do this, multiply the integer part number by the denominator of the fraction, add the resulting product to the numerator. The number that comes after these actions is the numerator of the wrong fraction. The denominator remains unchanged.
- If fractions have different denominators, you should bring them to the same.
- Perform addition or subtraction with the same denominators.
- When receiving the wrong fraction, select the whole part.
There is another way by which you can add and subtract fractions with integer parts. For this, separate actions with integer parts and separate actions with fractions are performed, and the results are recorded together.
The given example consists of fractions that have the same denominator. In the case when the denominators are different, they must be brought to the same, and then perform actions, as shown in the example.
Subtraction of fractions from an integer
Another of the varieties of actions with fractions is the case when the fraction must be subtracted from a natural number. At first glance, such an example seems difficult to solve. However, everything is pretty simple here. To solve it, it is necessary to translate an integer into a fraction, and with such a denominator, which is available in the deductible fraction. Next, we perform a subtraction similar to the subtraction with the same denominators. For example, it looks like this:
7 - 4/9 = (7 x 9) / 9 - 4/9 = 53/9 - 4/9 = 49/9.
Subtraction of fractions given in this article (Grade 6) is the basis for solving more complex examples, which are considered in subsequent classes. The knowledge of this topic is subsequently used to solve functions, derivatives, and so on. Therefore, it is very important to understand and understand the actions with fractions discussed above.