There comes a time when the teacher begins to explain what the correct fractions are in the math class. At this moment, the student opens up a whole host of new tasks and exercises, for the implementation of which you have to "strain up". Not all students understand this topic the first time, but we will try to explain everything in a clear language. After all, in fact, there is nothing complicated and terrible here.
The meaning of the concept of "fraction"
At every step, a person encounters situations in which it is necessary to separate and connect objects and their parts. Whether we chop a log or cut a cake, choose a bank with the highest percentage of income, or even look at the time, the right fractions are waiting for us everywhere. This, in essence, is just a fraction, a fragment - the upper value shows us how many pieces we have, and the lower one - how many of them are needed to get an integer value.
A view from different angles
Before you figure out how to make the wrong fraction correct, you need to understand more fundamental issues. Namely - what are we talking about?
Consider an example from everyday life. Take the pie, cut into identical pieces - each of them will, in fact, be the right fraction, namely, a part of some whole. What happens if we put all the fragments together? One whole pie. But what if there are more parts than necessary? We combined the pieces, getting a whole pie, and also the extra ones remained!
From a mathematical point of view, we got the wrong fraction - this is when the parts in the sum give a value greater than unity. Knowing it in a problem or equation is easier than ever. The lower part - the denominator - it has less than the upper - the numerator. And if the lower number is greater than the upper, then this is the right fraction.
Using
In order for a person to want to study a subject or a specific topic, he must realize the practical value of the new information. What are the right and wrong fractions for? Where are they used? It is impossible to work with mathematical expressions without knowing fractions. And in other sciences one cannot do without such information: neither in chemistry, nor in physics, nor in economics, nor even in sociology or politics!
For example, they interviewed a group of people regarding the new candidacy of the country's president. Someone voted for one, while someone preferred the second, and on the TV screen we will see the percentages. And what is the percentage? This is the right fraction! In this case, the proportion of voters among a single set of respondents. In general, without fractions in this world - nowhere. So, you need to study them.
Mixed number
We already know what the right fraction is. And the wrong one is one in which the numerator is greater than the denominator. It turns out that we have an integer and some additional part. Why not write everything in this form? This will be called a mixed number.
Imagine: the pie is cut into four parts, and in addition to them you have one more - the fifth. If you want to share with several friends, then everything is in order - you can just give everyone a piece. But storing a pie is more convenient as a whole, isn't it? Here is the same in mathematics: it happens that it is more convenient to use the representation of a number in the form of an irregular fraction, and in other cases it can be useful to isolate the whole parts in them - this will be called a mixed number.
Take 5/2 as an example. To get a mixed number, we need to subtract the denominator from the numerator as many times as it fits there. In this case, two times, and as a result we get two integers and one second. Such a conversion is a translation of the wrong fraction into the right one. When instead of the wording “three second” we get the expression “one whole and one second”, we come to the form in the form of a mixed number.
Operations
With fractions, you can perform all the same operations as with integers: addition, subtraction, multiplication, division. Later you will learn how to raise to a power, extract square and cubic roots, take logarithms. In the meantime, you need to learn how to perform simple operations with correct and incorrect fractions.
When multiplying and dividing it is most convenient to use not mixed numbers, but the usual representation: only the numerator and denominator, without the integer part. So, we have two numbers and the sign of the operation between them - let it be such an expression: (1/2) * (2/3). And then everything, it turns out, is very simple: we multiply the upper and lower parts, and write the result through a fraction bar: (1 * 2) / (2 * 3). Reduce the deuces in the numerator and denominator, getting the answer: 1/3.
When dividing, it will be almost the same, only the second component in the expression “will turn over”: (1/2) / (2/3) = (1/2) * (3/2) = 3/4.
Amount and Difference
When adding and subtracting, you can equally well use both mixed numbers and irregular fractions (if the need arises for an appropriate choice). To do this, you need to bring the terms to a common denominator.
How can I do that? If you remember the basic property of a fraction, then you know the answer - you need to multiply both fractions by such numbers so that they have the same values at the bottom. For example, there are the following values: 1/3 and 1/7. In accordance with the rule, we multiply the correct fraction by 1/3 by 7, and 1/7 by 3. We get 7/21 and 3/21. Now the numbers can be added up without hindrance: (7 + 3) / 21 = 10/21.
But it is not always necessary to multiply by the neighboring denominator - if we had 1/4 and 1/8, it would be easier to multiply the first term by 2, and this is the end: 2/8 + 1/8 = 3/8. In the same way, the difference is calculated.
Mistakes
Schoolchildren easily understand the topic of incorrect and correct fractions. What is so complicated? If mistakes do occur, it is almost always inattentive - a common denominator is found incorrectly, for example. There is, of course, one popular mistake, and it is allowed in the equations.
There is an expression: (3/4) x = 3. It is required to find out what “X” is equal to. The mistake may be that the student is multiplying both sides of the equation by ¾ rather than dividing. And then instead of the correct answer (x = 4), the wrong one is obtained: x = 9/4. It is easy to get rid of this problem - you just need to be lazy for some time to write down the procedure for dividing the right and left parts. Then the mistake immediately catches the eye.
Entry form
Fractions can be recorded vertically, or horizontally. In the first case, it turns out something similar to a column, where from top to bottom we get: the first number, the horizontal bar, the second number. And if the line is narrow and “swinging” in height does not work, then you can write these elements in a row, for example: 1/6, 34/37. Please note that such regular fractions are already written with a slash. Otherwise, nothing has changed significantly.
There are also decimal fractions. They are convenient to use, but not any number can be represented in this form - for this it should be divided by ten without a remainder, otherwise the accuracy is lost. Look, ½ can be written in decimal, getting 0.5, and 1/3 - is no longer possible. Rather, it turns out 0.333 ... and so on ad infinitum. In mathematics, this is called "three in a period."
In a text editor
Can I record a fraction on a computer? The Word provides such an opportunity. You just need to go to the "Insert" section. There you will see the “Formula” button, when you click on it a new window will open. In it you can find both regular fractions, and many other, much more complex symbols - integrals, differentials, square roots.
You may not yet know such words, but one day you will go through math in them too. Remember that all these signs can be found in one place.
At the same time, in Notepad there is no such possibility. There you can write fractions only in a line, through an oblique line.
Conclusion
In any science, accuracy is important. Therefore, all the "pieces" must be taken into account, and for this it is necessary to understand how to work with the right and wrong fractions. Without them, the plane will not take off, and the computer will not turn on, and the dish according to the cookbook cannot be prepared, and even music cannot be written. In general, understanding this topic in mathematics is an absolutely necessary task, and most importantly - is not at all difficult. Practice doing your homework, adding, multiplying, comparing fractions. Then you will very quickly learn to do everything in your mind and will be able to move on to new interesting topics. And there are still a great many of them in mathematics.