One of the most important sciences, the application of which can be seen in such disciplines as chemistry, physics and even biology, is mathematics. Studying this science allows us 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. Many students study it 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 in the presence of a denominator. That is why when you perform operations with fractions, you need to study some of their features and rules. The simplest case is the subtraction of ordinary fractions, the denominators of which are represented in the form of the same number. Do this action will not be difficult, if you know a simple rule:
- In order to subtract the second from a single fraction, it is necessary to subtract the numerator of the fraction to be subtracted from the numerator of the decreasing fraction. This number is written in the numerator of the difference, and the denominator is left the same: k / m - b / m = (kb) / m.
Examples of subtraction of fractions whose denominators are the same
Consider how it looks like this:
7/19 - 3/19 = (7 - 3) / 19 = 4/19.
From the numerator of the decrement "7" we subtract the numerator of the subtrahend fraction "3", we get "4". We write this number in the numerator of the answer, and put the same number in the denominator that was in the denominators of the first and second fractions - "19".
In the picture below, a few more similar examples are given.
Let us consider a more complicated example, where we subtract fractions with the same denominators:
29/47 - 3/47 - 8/47 - 2/47 - 7/47 = (29 - 3 - 8 - 2 - 7) / 47 = 9/47.
From the numerator of the decremented fraction "29", the numerators of all subsequent fractions are "by turns" - "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 down the number that is in the denominators of all these fractions - "47".
Addition of fractions having the same denominator
The addition and subtraction of ordinary fractions is carried out according to the same principle.
- In order to add fractions, the denominators of which are the same, numerators must be added. The resulting number is the numerator of the sum, and the denominator remains the same: k / m + b / m = (k + b) / m.
Consider how it looks like this:
1/4 + 2/4 = 3/4.
To the numerator of the first term of the fraction - "1" - add the numerator of the second term of the fraction - "2". The result is "3" - we write down the sum to the numerator, and the denominator leaves the same one that was present in the fractions - "4".
Fractions with different denominators and their subtraction
The action with fractions that have the same denominator, we have already considered. As you can see, knowing the simple rules, it is quite easy to solve such examples. But what if you need to perform an action with fractions that have different denominators? Many secondary school students are confronted with such examples. But here, if you know the principle of solution, examples will not be difficult for you. There is also a rule, without which the solution of such fractions is simply impossible.
To subtract fractions with different denominators, they must be reduced to the same lowest denominator.
We'll talk more about how to do this.
Property of fractions
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 the denominator by the same number, a fraction equal to this will be obtained.
For example, a fraction of 2/3 may have such denominators as "6", "9", "12", etc., that is, it can have the form of any number that is a multiple of "3". After the numerator and denominator are multiplied by "2", a fraction of 4/6 is obtained. After the numerator and denominator of the original fraction are multiplied by "3", we get 6/9, and if the analogous action is performed with the digit "4", we get 8/12. One equality can be written as:
2/3 = 4/6 = 6/9 = 8/12 ...
How to bring several fractions to the same denominator
Consider how to bring several fractions to the same denominator. For example, take the fractions shown in the picture below. To begin with, it is necessary to determine what number can become a denominator for all of them. For simplicity, we expand the denominators into multipliers.
The denominator of the fraction 1/2 and the fraction 2/3 of the factor can not be decomposed. The denominator 7/9 has two factors 7/9 = 7 / (3 x 3), the denominator of the fraction is 5/6 = 5 / (2 x 3). Now it is necessary to determine which factors will be the least for all these four fractions. Since in the first fraction in the denominator there is a number "2", it means that it must be present in all the denominators, in fraction 7/9 there are two triples, hence they also must both be present in the denominator. Considering 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. In its denominator there is a "2", but there is not a single digit "3", but there must be two. For this, we multiply the denominator by two triples, but, according to the property of the fraction, we and the numerator must be multiplied 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 deuce:
2/3 = (2 x 3 x 2) / (3 x 3 x 2) = 12/18.
- 7/9 or 7 / (3 x 3) - the denominator does not have two:
7/9 = (7 x 2) / (9 x 2) = 14/18.
- 5/6 or 5 / (2 x 3) - the denominator does not have a triple:
5/6 = (5 x 3) / (6 x 3) = 15/18.
All 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 of subtraction of fractions having the same denominator, which has already been discussed.
Consider this for the 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 common 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 (common multiple), we divide by the denominator of that fraction, in which it is necessary to determine additional factors.
- 90 divided by 15. The resulting "6" will be a multiplier for 3/15.
- 90 divided by 18. The resulting "5" will be a multiplier for 4/18.
The next step in our decision is to reduce each fraction to the denominator of "90".
How it is done, we 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, we add fractions having different denominators.
Subtraction and addition of fractions having integer parts
We subtracted the fractions and added them in detail. But how to make a subtraction if the fraction has a whole part? Again, we use several rules:
- All fractions, having a whole part, are transferred to the wrong ones. In simple words, remove the whole part. To do this, multiply the integer by the denominator of the fraction, add the resulting product to the numerator. The number that will be obtained after these actions is the numerator of the irregular fraction. The denominator remains unchanged.
- If the fractions have different denominators, they should be brought to the same.
- Add or subtract with the same denominators.
- When you receive an irregular fraction, select the integer part.
There is another way with which you can add and subtract fractions with integer parts. For this, actions are performed separately with integer parts, and separately actions with fractions, and the results are written together.
The above example consists of fractions that have the same denominator. In the event that the denominators are different, they must be brought to the same, and then perform the actions, as shown in the example.
Subtracting fractions from an integer
Another of the varieties of actions with fractions is the case when the fraction must be subtracted from the natural number. At first glance, this example seems difficult to solve. However, everything here is quite simple. To solve it, it is necessary to translate an integer into a fraction, and with such a denominator that exists in the fraction to be subtracted. Next, we perform a subtraction analogous to subtraction with the same denominators. On an example it looks like this:
7 - 4/9 = (7 x 9) / 9 - 4/9 = 53/9 - 4/9 = 49/9.
The subtraction of fractions (class 6) given in this article is the basis for solving more complex examples that are considered in subsequent classes. Knowledge of this topic is used subsequently to solve functions, derivatives and so on. Therefore, it is very important to understand and understand the actions with fractions considered above.