One of the most difficult for a student to understand is different actions with simple fractions. This is due to the fact that it is still difficult for children to think abstractly, and fractions, in fact, look exactly like that for them. Therefore, when setting out material, teachers often resort to analogies and explain the subtraction and addition of fractions literally on the fingers. Although no rules in school mathematics are complete without rules and definitions.
Basic concepts
Before you begin any
action with fractions , it is advisable to learn a few basic definitions and rules. Initially, it is important to understand what fraction is. By it is meant a number representing one or more fractions of a unit. For example, if you cut a loaf into 8 parts and put 3 slices of them on a plate, then 3/8 will be a fraction. Moreover, in such a spelling it will be a simple fraction, where the number above the line is the numerator, and below it is the denominator. But if you write it down as 0.375, it will already be a decimal fraction.
In addition, simple fractions are divided into regular, incorrect and mixed. The first group includes all those whose numerator is less than the denominator. If, on the contrary, the denominator is smaller than the numerator, it will already be the wrong fraction. If the correct one is an integer, they talk about mixed numbers. Thus, the fraction 1/2 is correct, but the fraction 7/2 is not. And if you write it in this form: 3 1/2 , then it will become mixed.
To make it easier to understand what is the addition of fractions, and to easily perform it, it is important to remember the basic property of the fraction. Its essence is as follows. If the numerator and denominator are multiplied by the same number, then the fraction will not change. It is this property that allows you to perform simple actions with ordinary and other fractions. In fact, this means that 1/15 and 3/45 are essentially the same number.
Adding fractions with identical denominators
Performing this action usually does not cause much difficulty. The addition of fractions in this case very much resembles a similar action with integers. The denominator remains unchanged, and the numerators simply add up to each other. For example, if you need to add fractions 2/7 and 3/7, then the solution to the school problem in a notebook will be like this:
2/7 + 3/7 = (2 + 3) / 7 = 5/7.
Moreover, this addition of fractions can be explained by a simple example. Take an ordinary apple and cut, for example, into 8 parts. Put 3 parts separately first, and then add 2 more to them. And as a result, 5/8 of the whole apple will lie in the cup. The arithmetic task itself is written as shown below:
3/8 + 2/8 = (3 + 2) / 8 = 5/8.
Adding fractions with different denominators
But often there are more complicated tasks, where you need to add together, for example, 5/9 and 3/5. This is where the first difficulties arise in actions with fractions. After all, the addition of such numbers will require additional knowledge. Now fully need to recall their main property. To add fractions from an example, first you need to bring them to one common denominator. To do this, you just need to multiply 9 and 5 with each other, the numerator "5" multiply by 5, and "3", respectively, by 9. Thus, such fractions are already added: 25/45 and 27/45. Now it remains only to add the numerators and get the answer 52/45. On a piece of paper, an example would look like this:
5/9 + 3/5 = (5 x 5) / (9 x 5) + (3 x 9) / (5 x 9) = 25/45 + 27/45 = (25 + 27) / 45 = 52 / 45 = 1 7/45 .
But adding fractions with such denominators does not always require a simple multiplication of the numbers under the line. First look for the lowest common denominator. For example, as for fractions 2/3 and 5/6. For them it will be the number 6. But the answer is not always obvious. In this case, it is worth recalling the rule of finding the smallest common multiple (abbreviated NOC) of two numbers.
By it is meant the smallest common factor of two integers. To find it, they decompose each into simple factors. Now they write out those that are included at least once in each number. Multiply them among themselves and get the same denominator. In fact, everything looks a little easier.
For example, you want to add fractions 4/15 and 1/6. So, 15 is obtained by multiplying the simple digits 3 and 5, and six - two and three. So, the NOC for them will be 5 x 3 x 2 = 30. Now, dividing 30 by the denominator of the first fraction, we get the factor for its numerator - 2. And for the second fraction it will be 5. Thus, it remains to add the ordinary fractions 8/30 and 5/30 and get a response 13/30. Everything is extremely simple. In the notebook, this task should be written as follows:
4/15 + 1/6 = (4 x 2) / (15 x 2) + (1 x 5) / (6 x 5) = 8/30 + 5/30 = 13/30.
NOC (15, 6) = 30.
Addition of mixed numbers
Now, knowing all the basic techniques in adding simple fractions, you can try your hand at more complex examples. And these will be mixed numbers, by which they understand a fraction of this kind: 2
2/3 . Here, the whole part is written out before the correct fraction. And many are confused when performing actions with such numbers. In fact, the same rules work here.
To add mixed numbers to each other, integer parts and regular fractions are separately added. And then these 2 results are summarized. In practice, everything is much simpler, you just need to practice a little. For example, in the problem it is required to add such mixed numbers: 1 1/3 and 4 2/5 . To do this, first add up 1 and 4 - it turns out 5. Then add 1/3 and 2/5 using the techniques of reducing to the lowest common denominator. The decision will be 11/15. And the final answer is 5 11/15 . In a school notebook this will look much shorter:
1 1/3 + 4 2/5 = (1 + 4) + (1/3 + 2/5) = 5 + 5/15 + 6/15 = 5 + 11/15 = 5 11/15.
Decimal
In addition to ordinary fractions, there are also decimal. By the way, they are much more common in life. For example, the price in the store often looks like this: 20.3 rubles. This is the very fraction. Of course, these are much easier to stack than ordinary ones. In principle, you just need to add 2 ordinary numbers, most importantly, put a comma in the right place. This is where difficulties arise.
For example, you need to add such decimal fractions of 2.5 and 0.56. To do this correctly, you need to add zero to the first one at the end, and everything will be in order.
2.50 + 0.56 = 3.06.
It is important to know that any decimal fraction can be converted to a simple fraction, but not any simple fraction can be written as a decimal. So, from our example, 2.5 = 2 1/2 / and 0.56 = 14/25. But such a fraction as 1/6 will only be approximately 0.16667. The same situation will be with other similar numbers - 2/7, 1/9 and so on.
Conclusion
Many students, not understanding the practical side of the action with fractions, relate to this topic through their sleeves. However, in the older classes, this basic knowledge will allow you to click complicated examples with logarithms and finding derivatives as nuts. Therefore, it’s worth it once to understand well the actions with fractions, so that later you do not bite your elbows with frustration. After all, it is unlikely that a teacher in high school will return to this already covered topic. Any high school student should be able to perform such exercises.