Addition of fractions: definitions, rules and examples of tasks

One of the most difficult to understand studentare 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 to them. Therefore, when setting forth the material, teachers often resort to analogies and explain the subtraction and addition of fractions literally on the fingers. Although no lesson in school mathematics is complete without rules and definitions.

Basic concepts

Before you start any action withfractions, it is desirable to learn a few basic definitions and rules. Initially it is important to understand what a fraction is. By it is meant a number that represents one or several fractions of a unit. For example, if a loaf is cut into 8 pieces and 3 slices of them are put into a plate, then 3/8 will be a fraction. And 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 as 0.375, it will already be a decimal fraction.

In addition, simple fractions are divided intoright, wrong and mixed. The first include all those whose numerator is less than the denominator. If, on the contrary, the denominator is less than the numerator, it will already be an improper fraction. If before the correct is an integer, they talk about mixed numbers. Thus, the fraction 1/2 is correct, and 7/2 is not. And if you write it in this form: 3¹/₂then it will become mixed.

To make it easier to understand what isaddition of fractions, and with ease of its implementation, it is important to remember the main 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 one to perform simple actions with ordinary and other fractions. In fact, this means that 1/15 and 3/45 are essentially the same number.

Addition of fractions with the same denominators

Doing this does not usually causegreat difficulty. The addition of fractions in this case very much resembles a similar action with integers. The denominator remains unchanged, and the numerators simply fold together. For example, if you want to add fractions of 2/7 and 3/7, then the solution of the school problem in the notebook will be like this:

2/7 + 3/7 = (2 + 3) / 7 = 5/7.

Moreover, this addition of fractions can be explainedon a simple example. Take an ordinary apple and cut, for example, into 8 pieces. Lay out 3 parts separately first, and then add another 2 to them. As a result, 5/8 whole apples will lie in the cup. The arithmetic problem itself is written as shown below:

3/8 + 2/8 = (3 + 2) / 8 = 5/8.

Add fractions with different denominators

But often there are more difficult tasks, whereyou need to add each other, for example, 5/9 and 3/5. This is where the first difficulties arise in the actions with fractions. Indeed, the addition of such numbers will require additional knowledge. Now it is fully required to recall their main property. To add fractions from the example, first they need to be reduced to one common denominator. To do this, you simply multiply 9 and 5 together, the numerator "5" multiplied by 5, and "3", respectively, by 9. Thus, such fractions are already added: 25/45 and 27/45. Now it only remains 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 = 1⁷/₄₅.

Но сложение дробей с такими знаменателями не always requires a simple multiplication of numbers below 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 remembering the rule for finding the smallest common multiple (abbreviated NOC) of two numbers.

Under it understand the smallest common factor of twointegers. To find it, decompose each into prime factors. Now write out those of them that are 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 simple numbers 3 and 5, and six - two and three. So, the LCM for them will be 5 x 3 x 2 = 30. Now, dividing 30 by the denominator of the first fraction, we get a multiplier for its numerator - 2. And for the second fraction this will be number 5. Thus, it remains to add 8/30 ordinary fractions and 5/30 and get the answer 13/30. Everything is extremely simple. In the notebook, this task should be written as:

4/15 + 1/6 = (4 x 2) / (15 x 2) + (1 x 5) / (6 x 5) = 8/30 + 5/30 = 13/30.

LCM (15, 6) = 30.

Addition of mixed numbers

Now, knowing all the basic techniques in the addition of simple fractions, you can try your hand at more complex examples. And it will be mixed numbers, which are understood as a fraction of this type: 2²/₃. Here before the correct fraction the whole part is written out. And many are confused when committing actions with such numbers. In fact, the same rules work here.

To add together the mixed numbersseparate the whole parts and the correct fractions. And then they summarize these 2 results. In practice, everything is much simpler, it is worth only a little exercise. For example, in the problem you need to add these mixed numbers: 1¹/₃ and 4²/₅. To do this, first add up 1 and 4 -will be 5. Then summarize 1/3 and 2/5, using the techniques of reducing to the lowest common denominator. The solution will be 11/15. And the final answer is 5¹¹/₁₅. In a school notebook it will look much shorter:

1¹/₃ + 4²/₅ = (1 + 4) + (1/3 + 2/5) = 5 + 5/15 + 6/15 = 5 + 11/15 = 5¹¹/₁₅.

Decimals

In addition to ordinary fractions, there are decimal.They, by the way, are much more common in life. For example, the price in a store often looks like this: 20.3 rubles. This is the same fraction. Of course, such folding is much easier than ordinary ones. In principle, you just need to add 2 ordinary numbers, the main thing is to put a comma in the right place. This is where the difficulties arise.

For example, you want to add the decimal fractions of 2.5 and 0.56. To do this correctly, you need to finish zero at the first at the end, and everything will be fine.

2.50 + 0.56 = 3.06.

It is important to know that any decimal fraction can be converted to a simple one, but not any simple fraction can be written as a decimal. So, from our example 2,5 = 2¹/₂ and 0.56 = 14/25. But such a fraction, as 1/6, will only be approximately equal to 0.16667. The same situation will be with other similar numbers - 2/7, 1/9 and so on.

Conclusion

Many schoolchildren, not understanding the practical sideAction fractions, refer to this topic carelessly. However, in higher grades, this basic knowledge will allow you to click like nuts complex examples with logarithms and finding derivatives. That is why it is worth once well to understand actions with fractions in order not to bite your elbows in annoyance. After all, it is unlikely that a teacher in high school will return to this topic already covered. Any high school student should be able to perform such exercises.