We encounter fractions in life much earlier than their study in school. If you cut a whole apple in half, then we get a piece of fruit - ½. We cut it again - it will be ¼. This is the fraction. And everything seemed to be simple. For an adult. For a child (and they begin to study this topic at the end of elementary school), abstract mathematical concepts are still frighteningly incomprehensible, and the teacher should clearly explain what the correct fraction and the wrong, ordinary and decimal are, what operations can be performed with them and, most importantly, why all this is needed.
What are the fractions
Acquaintance with a new topic at school begins with ordinary fractions. They are easy to recognize by the horizontal line dividing the two numbers - above and below. The top is called the numerator, the bottom is the denominator. There is also a lowercase version of writing incorrect and regular fractions - through a slash, for example: ½, 4/9, 384/183. This option is used when the height of the line is limited and there is no way to apply a “two-story” recording form. Why? Yes, because it is more convenient. Later we will be convinced of this.
In addition to ordinary, there are also decimal fractions. Distinguishing them is very simple: if in one case a horizontal or slanted line is used, then in another it is a comma separating sequences of numbers. Let's see an example: 2.9; 163.34; 1,953. We intentionally used a semicolon as a separator to delimit numbers. The first of them will read like this: "two point nine."
New concepts
Let's return to ordinary fractions. They come in two forms.
The definition of the correct fraction is as follows: this is a fraction whose numerator is less than the denominator. Why is it important? We will see now!
You have several apples, halved. In total - 5 parts. How do you say: do you have two and a half or five second apples? Of course, the first option sounds more natural, and when talking with friends, we will use it. But if you need to calculate how much fruit everyone will get, if there are five people in the company, we will write the number 5/2 and divide it by 5 - from the point of view of mathematics this will be more obvious.
So, for the name of the right and wrong fractions, the rule is this: if in the fraction you can select the integer part (14/5, 2/1, 173/16, 3/3), then it is incorrect. If this cannot be done, as is the case with ½, 13/16, 9/10, it will be correct.
The main property of the fraction
If the numerator and denominator of a fraction are simultaneously multiplied or divided by the same number, its value will not change. Imagine: they cut a cake into 4 equal parts and gave you one. They cut the same cake into eight pieces and gave you two. Is it all the same? After all, ¼ and 2/8 are one and the same!
Abbreviation
The authors of problems and examples in textbooks in mathematics often seek to confuse students by offering cumbersome fractions in writing that can actually be reduced. Here is an example of the correct fraction: 167/334, which, it would seem, looks very "scary". But actually we can write it as ½. The number 334 is divided by 167 without a remainder - having done this operation, we get 2.
Mixed numbers
The wrong fraction can be represented as a mixed number. This is when the whole part is brought forward and recorded at the horizontal line level. In fact, the expression takes the form of a sum: 11/2 = 5 + ½; 13/6 = 2 + 1/6 and so on.
To take out the integer part, you need to divide the numerator by the denominator. The remainder of the division is written above, above the line, and the integer part is before the expression. Thus, we get two structural parts: whole units + regular fraction.
You can carry out the inverse operation - for this you need to multiply the integer by the denominator and add the resulting value to the numerator. Nothing complicated.
Multiplication and Division
Oddly enough, to multiply fractions is easier than to add. All that is required is to extend the horizontal line: (2/3) * (3/5) = 2 * 3/3 * 5 = 2/5.
With division, everything is also simple: you need to multiply the fractions crosswise: (7/8) / (14/15) = 7 * 15/8 * 14 = 15/16.
Fraction Addition
What to do if it is necessary to add or subtract fractions, and in the denominator they have different numbers? To do the same as with multiplication, it will not work - here you should understand the definition of the correct fraction and its essence. It is necessary to bring the terms to a common denominator, that is, the same numbers should appear in the lower part of both fractions.
To do this, you should use the main property of the fraction: multiply both parts by the same number. For example, 2/5 + 1/10 = (2 * 2) / (5 * 2) + 1/10 = 5/10 = ½.
How to choose, to which denominator lead the terms? This should be the minimum number that is a multiple of both numbers in the denominators of fractions: for 1/3 and 1/9 it will be 9; for ½ and 1/7 - 14, because a smaller value divisible by 2 and 7 without a remainder does not exist.
Using
What are the wrong fractions for? After all, it is much more convenient to immediately single out the whole part, get a mixed number - and it's up to the end! It turns out that if you want to perform the multiplication or division of two fractions, it is more profitable to use the wrong ones.
Take the following example: (2 + 3/17) / (37/68).
It would seem that there is nothing to reduce at all. But what if we write the result of addition in the first brackets in the form of an incorrect fraction? See: (37/17) / (37/68)
Now everything falls into place! We write an example in such a way that everything becomes obvious: (37 * 68) / (17 * 37).
Reduce 37 in the numerator and denominator and, finally, divide the upper and lower parts by 17. Do you remember the basic rule for correct and incorrect fractions? We can multiply and divide them by any number if we do it simultaneously for the numerator and denominator.
So, we get the answer: 4. The example looked complicated, and the answer contains only one digit. In mathematics, this happens so often. The main thing is not to be afraid and follow simple rules.
Common mistakes
When carrying out actions with fractions, a student can easily make one of the popular mistakes. Usually they occur due to inattention, and sometimes due to the fact that the studied material has not yet been deposited in the head as it should.
Often the sum of numbers in the numerator makes one want to reduce its individual components. Suppose, in the example: (13 + 2) / 13, written without parentheses (with a horizontal line), many students, by inexperience, cross out 13 from above and below. But this can not be done in any case, because this is a gross mistake! If instead of the addition there was a multiplication sign, we would get the number 2 in the answer. But when performing addition, no operations with one of the terms are permissible, only with the whole sum.
Still children often are mistaken at division of fractions. We take two regular irreducible fractions and divide them into each other: (5/6) / (25/33). The student can mix up and write the resulting expression as (5 * 25) / (6 * 33). But this would happen when multiplying, and in our case everything will be somewhat different: (5 * 33) / (6 * 25). We shorten what is possible and see 11/10 in the answer. The resulting incorrect fraction is written as a decimal - 1.1.
Parentheses
Remember that in any mathematical expression, the order of actions is determined by the priority of the operation signs and the presence of brackets. Other things being equal, the countdown of the sequence of actions occurs from left to right. This is also true for fractions - the expression in the numerator or denominator is calculated strictly according to this rule.
After all, what is the correct fraction? This is the result of dividing one number by another. If they are not completely divided, a fraction is obtained - that’s all.
How to record a fraction on a computer
Since standard tools do not always allow you to create a fraction consisting of two "tiers", students sometimes go to different tricks. For example, copy the numerators and denominators into the Paint editor and glue them together, drawing a horizontal line between them. Of course, there is a simpler option, which, by the way, provides a ton of additional features that will become useful to you in the future.
Open Microsoft Word. One of the panels at the top of the screen is called the "Insert" - click it. On the right, on the side where the closing and minimizing icons are located, there is a Formula button. This is exactly what we need!
If you use this function, a rectangular area will appear on the screen, in which you can use any mathematical signs that are not on the keyboard, as well as write fractions in a classical form. That is, separating the numerator and denominator with a horizontal bar. You may even be surprised that such a correct fraction is so easy to record.
Learn math
If you study in grades 5-6, then soon knowledge of mathematics (including the ability to work with fractions!) Will be required in many school subjects. In almost any task in physics, when measuring the mass of substances in chemistry, in geometry and trigonometry, fractions cannot be dispensed with. Soon you will learn how to compute everything in your mind without even writing down expressions on paper, but more and more complex examples will appear. Therefore, learn what the correct fraction is and how to work with it, do not lag behind the curriculum, do your homework in a timely manner, and then you will succeed.