Fractions, operations with fractions. Operation with common fractions

Fractions are ordinary and decimal. When the student learns about the existence of the latter, he begins at every opportunity to translate everything that is possible into decimal form, even if this is not required.

Oddly enough, the preferences of high school students and students change, because it is easier to perform many arithmetic operations with ordinary fractions. And the values that graduates deal with can sometimes be simply impossible to convert to a decimal form without loss. As a result, both types of fractions are, one way or another, adapted to the case and have their own advantages and disadvantages. Let's see how to work with them.

Definition

Fractions are the same shares. If there are ten slices in an orange, and you were given one, then you have 1/10 of the fruit in your hand. With such a notation, as in the previous sentence, the fraction will be called an ordinary fraction. If you write the same as 0.1 - decimal. Both options are equal, but have their own advantages. The first option is more convenient for multiplication and division, the second - for addition, subtraction, and in a number of other cases.

How to convert a fraction to another form

Suppose you have a common fraction and you want to convert it to a decimal. What do I need to do?

By the way, you need to decide in advance that not any number can be written in decimal form without problems. Sometimes you have to round the result, losing a certain number of decimal places, and in many areas - for example, in the exact sciences - this is a completely unaffordable luxury. At the same time, actions with decimal and ordinary fractions in the 5th grade make it possible to carry out such a transfer from one type to another without interference, at least as a training.

If from the denominator, by multiplying or dividing by an integer, you can get a value that is a multiple of 10, the transfer will pass without any difficulties: ¾ turns into 0.75, 13/20 - into 0.65.

The inverse procedure is even easier, since you can always get an ordinary fraction from a decimal fraction without loss in accuracy. For example, 0.2 becomes 1/5 and 0.08 becomes 4/25.

Internal conversions

Before performing joint actions with ordinary fractions, you need to prepare the numbers for possible mathematical operations.

First of all, you need to bring all the fractions in the example to one general form. They must be either ordinary or decimal. Immediately make a reservation that multiplication and division are more convenient to perform with the first.

In preparing the numbers for further actions, you will be helped by a rule known as and used both in the early years of studying the subject, and in higher mathematics, which is studied at universities.

Fraction properties

Suppose you have some value. Let's say 2/3. What happens if you multiply the numerator and denominator by 3? Get 6/9. What if it's a million? 2000000/3000000. But wait, because the number does not change qualitatively at all - 2/3 remain equal to 2000000/3000000. Only the form changes, not the content. The same thing happens when both parts are divided by the same value. This is the main property of the fraction, which will repeatedly help you perform actions with decimal and ordinary fractions on tests and exams.

Multiplying the numerator and denominator by the same number is called expanding a fraction, and dividing is called reducing. I must say that crossing out the same numbers at the top and bottom when multiplying and dividing fractions is a surprisingly pleasant procedure (as part of a math lesson, of course). It seems that the answer is already close and the example is practically solved.

Improper fractions

An improper fraction is one in which the numerator is greater than or equal to the denominator. In other words, if a whole part can be distinguished from it, it falls under this definition.

If such a number (greater than or equal to one) is represented as an ordinary fraction, it will be called improper. And if the numerator is less than the denominator - correct. Both types are equally convenient in the implementation of possible actions with ordinary fractions. They can be freely multiplied and divided, added and subtracted.

If at the same time an integer part is selected and at the same time there is a remainder in the form of a fraction, the resulting number will be called mixed. In the future, you will encounter various ways of combining such structures with variables, as well as solving equations where this knowledge is required.

Arithmetic operations

If everything is clear with the basic property of a fraction, then how to behave when multiplying fractions? Actions with ordinary fractions in the 5th grade involve all kinds of arithmetic operations that are performed in two different ways.

Multiplication and division are very easy. In the first case, the numerators and denominators of two fractions are simply multiplied. In the second - the same, only crosswise. Thus, the numerator of the first fraction is multiplied by the denominator of the second, and vice versa.

To perform addition and subtraction, you need to perform an additional action - bring all the components of the expression to a common denominator. This means that the lower parts of the fractions must be changed to the same value - a multiple of both available denominators. For example, for 2 and 5 it will be 10. For 3 and 6 - 6. But then what to do with the top? We cannot leave it as it was if we changed the bottom one. According to the basic property of a fraction, we multiply the numerator by the same number as the denominator. This operation must be performed on each of the numbers that we will be adding or subtracting. However, such actions with ordinary fractions in the 6th grade are already performed “on the machine”, and difficulties arise only at the initial stage of studying the topic.

Comparison

If two fractions have the same denominator, then the one with the larger numerator will be larger. If the upper parts are the same, then the one with the smaller denominator will be larger. It should be borne in mind that such successful situations for comparison rarely occur. Most likely, both the upper and lower parts of the expressions will not match. Then you need to remember about the possible actions with ordinary fractions and use the technique used in addition and subtraction. In addition, remember that if we are talking about negative numbers, then the larger fraction in modulus will be smaller.

Advantages of common fractions

It happens that teachers tell children one phrase, the content of which can be expressed as follows: the more information is given when formulating the task, the easier the solution will be. Does it sound weird? But really: with a large number of known values, you can use almost any formula, but if only a couple of numbers are provided, additional reflections may be required, you will have to remember and prove theorems, give arguments in favor of your rightness ...

Why are we doing this? Moreover, ordinary fractions, for all their cumbersomeness, can greatly simplify the life of a student, allowing you to reduce entire lines of values \u200b\u200bwhen multiplying and dividing, and when calculating the sum and difference, take out common arguments and, again, reduce them.

When it is required to perform joint actions with ordinary and decimal fractions, transformations are carried out in favor of the first: how do you translate 3/17 into decimal form? Only with loss of information, not otherwise. But 0.1 can be represented as 1/10, and then as 17/170. And then the two resulting numbers can be added or subtracted: 30/170 + 17/170 = 47/170.

Why are decimals useful?

If actions with ordinary fractions are more convenient to carry out, then writing everything down with their help is extremely inconvenient, decimals have a significant advantage here. Compare: 1748/10000 and 0.1748. It is the same value presented in two different versions. Of course, the second way is easier!

In addition, decimals are easier to represent because all the data has a common base that differs only by orders of magnitude. Let's say we can easily recognize a 30% discount and even evaluate it as significant. Will you immediately understand which is more - 30% or 137/379? Thus, decimal fractions provide standardization of calculations.

In high school, students solve quadratic equations. It is already extremely problematic to perform actions with ordinary fractions here, since the formula for calculating the values \u200b\u200bof the variable contains the square root of the sum. In the presence of a fraction that is not reducible to a decimal, the solution becomes so complicated that it becomes almost impossible to calculate the exact answer without a calculator.

So, each way of representing fractions has its own advantages in the appropriate context.

Forms of entry

There are two ways to write actions with ordinary fractions: through a horizontal line, into two “tiers”, and through a slash (aka “slash”) - into a line. When a student writes in a notebook, the first option is usually more convenient, and therefore more common. The distribution of a number of numbers into cells contributes to the development of attentiveness in calculations and transformations. When writing to a string, you can inadvertently confuse the order of actions, lose any data - that is, make a mistake.

Quite often in our time there is a need to print numbers on a computer. You can separate fractions with a traditional horizontal bar using a function in Microsoft Word 2010 and later. The fact is that in these versions of the software there is an option called "formula". It displays a rectangular transformable field within which you can combine any mathematical symbols, make up both two- and “four-story” fractions. In the denominator and numerator, you can use brackets, operation signs. As a result, you will be able to write down any joint actions with ordinary and decimal fractions in the traditional form, that is, the way they teach you to do it at school.

If you use the standard Notepad text editor, then all fractional expressions will need to be written through a slash. Unfortunately, there is no other way here.

Conclusion

So we have considered all the basic actions with ordinary fractions, which, it turns out, are not so many.

If at first it may seem that this is a complex section of mathematics, then this is only a temporary impression - remember, once you thought so about the multiplication table, and even earlier - about the usual copybooks and counting from one to ten.

It is important to understand that fractions are used everywhere in everyday life. You will deal with money and engineering calculations, information technology and musical literacy, and everywhere - everywhere! - fractional numbers will appear. Therefore, do not be lazy and study this topic thoroughly - especially since it is not so difficult.

Multiplication and division of fractions.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

This operation is much nicer than addition-subtraction! Because it's easier. I remind you: to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). I.e:

For example:

Everything is extremely simple. And please don't look for a common denominator! Don't need it here...

To divide a fraction by a fraction, you need to flip second(this is important!) fraction and multiply them, i.e.:

For example:

If multiplication or division with integers and fractions is caught, it's okay. As with addition, we make a fraction from a whole number with a unit in the denominator - and go! For example:

In high school, you often have to deal with three-story (or even four-story!) fractions. For example:

How to bring this fraction to a decent form? Yes, very easy! Use division through two points:

But don't forget about the division order! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But in a three-story fraction it is easy to make a mistake. Please note, for example:

In the first case (expression on the left):

In the second (expression on the right):

Feel the difference? 4 and 1/9!

What is the order of division? Or brackets, or (as here) the length of horizontal dashes. Develop an eye. And if there are no brackets or dashes, like:

then divide-multiply in order, left to right!

And another very simple and important trick. In actions with degrees, it will come in handy for you! Let's divide the unit by any fraction, for example, by 13/15:

The shot has turned over! And it always happens. When dividing 1 by any fraction, the result is the same fraction, only inverted.

That's all the actions with fractions. The thing is quite simple, but gives more than enough errors. Take note of practical advice, and there will be fewer of them (mistakes)!

Practical Tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a severe need! Do all the calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in a draft than to mess up when calculating in your head.

2. In examples with different types of fractions - go to ordinary fractions.

3. We reduce all fractions to the stop.

4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).

5. We divide the unit into a fraction in our mind, simply by turning the fraction over.

Here are the tasks you need to complete. Answers are given after all tasks. Use the materials of this topic and practical advice. Estimate how many examples you could solve correctly. The first time! Without a calculator! And draw the right conclusions...

Remember the correct answer obtained from the second (especially the third) time - does not count! Such is the harsh life.

So, solve in exam mode ! This is preparation for the exam, by the way. We solve an example, we check, we solve the following. We decided everything - we checked again from the first to the last. Only after look at the answers.

Calculate:

Did you decide?

Looking for answers that match yours. I specifically wrote them down in a mess, away from the temptation, so to speak ... Here they are, the answers, written down with a semicolon.

0; 17/22; 3/4; 2/5; 1; 25.

And now we draw conclusions. If everything worked out - happy for you! Elementary calculations with fractions are not your problem! You can do more serious things. If not...

So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

We will agree to consider that under "actions with fractions" in our lesson we will understand actions with ordinary fractions. A fraction is a fraction that has attributes such as a numerator, a fractional bar, and a denominator. This distinguishes an ordinary fraction from a decimal fraction, which is obtained from an ordinary one by reducing the denominator to a multiple of 10. A decimal fraction is written with a comma separating the integer part from the fractional one. We will talk about actions with ordinary fractions, since it is they that cause the greatest difficulties for students who have forgotten the basics of this topic, covered in the first half of the school mathematics course. At the same time, when transforming expressions in higher mathematics, it is mainly operations with ordinary fractions that are used. Some abbreviations of fractions are worth something! Decimal fractions do not cause much difficulty. So go ahead!

Two fractions and are called equal if .

For example, because

The fractions and (since ), and (since ) are also equal.

Obviously, both fractions and are equal. This means that if the numerator and denominator of a given fraction are multiplied or divided by the same natural number, then a fraction equal to the given one will be obtained:.

This property is called the basic property of a fraction.

The basic property of a fraction can be used to change the signs of the numerator and denominator of a fraction. If the numerator and denominator of the fraction are multiplied by -1, then we get. This means that the value of a fraction will not change if the signs of the numerator and denominator are changed at the same time. If you change the sign of only the numerator or only the denominator, then the fraction will change its sign:

Fraction reduction

Using the basic property of a fraction, you can replace a given fraction with another fraction equal to the given one, but with a smaller numerator and denominator. This substitution is called fraction reduction.

Let, for example, be given a fraction. The numbers 36 and 48 have the greatest common divisor 12. Then

In the general case, fraction reduction is always possible if the numerator and denominator are not coprime numbers. If the numerator and denominator are relatively prime numbers, then the fraction is called irreducible.

So, reducing a fraction means dividing the numerator and denominator of a fraction by a common factor. All of the above applies to fractional expressions containing variables.

Example 1 Reduce fraction

Decision. To factorize the numerator into factors, having previously presented the monomial - 5 xy as a sum - 2 xy - 3xy, we get

To factorize the denominator, we use the difference of squares formula:

As a result

Bringing fractions to a common denominator

Let two fractions and be given. They have different denominators: 5 and 7. Using the basic property of a fraction, you can replace these fractions with others equal to them, and such that the resulting fractions will have the same denominators. Multiplying the numerator and denominator of the fraction by 7, we get

Multiplying the numerator and denominator by 5, we get

So, the fractions are reduced to a common denominator:

But this is not the only solution to the problem: for example, these fractions can also be reduced to a common denominator of 70:

and in general to any denominator divisible by both 5 and 7.

Let's consider one more example: let's reduce the fraction and to a common denominator. Arguing as in the previous example, we get

But in this case, you can bring the fractions to a common denominator, less than the product of the denominators of these fractions. Find the least common multiple of 24 and 30: LCM(24, 30) = 120 .

Since 120:4=5, in order to write a fraction with a denominator of 120, both the numerator and the denominator must be multiplied by 5, this number is called an additional factor. Means .

Further, we get 120:30=4. Multiplying the numerator and denominator of the fraction by an additional factor of 4, we get .

So, these fractions are reduced to a common denominator.

The least common multiple of the denominators of these fractions is the smallest possible common denominator.

For fractional expressions that include variables, the common denominator is a polynomial that is divisible by the denominator of each fraction.

Example 2 Find the common denominator of fractions and .

Decision. The common denominator of these fractions is a polynomial, since it is divisible by both and by. However, this polynomial is not the only one that can be a common denominator of these fractions. It can also be a polynomial , and polynomial , and polynomial etc. Usually they take such a common denominator that any other common denominator is divisible by the chosen one without a remainder. Such a denominator is called the least common denominator.

In our example, the least common denominator is . Got:

;

We managed to bring fractions to the lowest common denominator. This happened by multiplying the numerator and denominator of the first fraction by , and the numerator and denominator of the second fraction by . Polynomials and are called additional factors, respectively, for the first and second fractions.

Addition and subtraction of fractions

The addition of fractions is defined as follows:

For example,

If a b = d, then

This means that to add fractions with the same denominator, it is enough to add the numerators, and leave the denominator the same. For example,

If fractions with different denominators are added, then the fractions are usually reduced to the lowest common denominator, and then the numerators are added. For example,

Now consider an example of adding fractional expressions with variables.

Example 3 Convert expression to one fraction

Decision. Let's find the least common denominator. To do this, we first factorize the denominators.

Fractions are ordinary numbers, they can also be added and subtracted. But due to the fact that they have a denominator, more complex rules are required here than for integers.

Consider the simplest case, when there are two fractions with the same denominators. Then:

To add fractions with the same denominators, add their numerators and leave the denominator unchanged.

To subtract fractions with the same denominators, it is necessary to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.

Within each expression, the denominators of the fractions are equal. By definition of addition and subtraction of fractions, we get:

As you can see, nothing complicated: just add or subtract the numerators - and that's it.

But even in such simple actions, people manage to make mistakes. Most often they forget that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.

Getting rid of the bad habit of adding denominators is quite simple. Try to do the same when subtracting. As a result, the denominator will be zero, and the fraction (suddenly!) will lose its meaning.

Therefore, remember once and for all: when adding and subtracting, the denominator does not change!

Also, many people make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus, and where - a plus.

This problem is also very easy to solve. It is enough to remember that the minus before the fraction sign can always be transferred to the numerator - and vice versa. And of course, do not forget two simple rules:

Plus times minus gives minus;
Two negatives make an affirmative.

Let's analyze all this with specific examples:

Task. Find the value of the expression:

In the first case, everything is simple, and in the second, we will add minuses to the numerators of fractions:

What if the denominators are different

You cannot directly add fractions with different denominators. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.

There are many ways to convert fractions. Three of them are discussed in the lesson " Bringing fractions to a common denominator", so we will not dwell on them here. Let's take a look at some examples:

Task. Find the value of the expression:

In the first case, we bring the fractions to a common denominator using the "cross-wise" method. In the second, we will look for the LCM. Note that 6 = 2 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are coprime. Therefore, LCM(6; 9) = 2 3 3 = 18.

What if the fraction has an integer part

I can please you: different denominators of fractions are not the greatest evil. Much more errors occur when the whole part is highlighted in the fractional terms.

Of course, for such fractions there are own addition and subtraction algorithms, but they are rather complicated and require a long study. Better use the simple diagram below:

Convert all fractions containing an integer part to improper. We get normal terms (even if with different denominators), which are calculated according to the rules discussed above;
Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
If this is all that was required in the task, we perform the inverse transformation, i.e. we get rid of the improper fraction, highlighting the integer part in it.

The rules for switching to improper fractions and highlighting the integer part are described in detail in the lesson "What is a numerical fraction". If you don't remember, be sure to repeat. Examples:

Task. Find the value of the expression:

Everything is simple here. The denominators inside each expression are equal, so it remains to convert all fractions to improper ones and count. We have:

To simplify the calculations, I skipped some obvious steps in the last examples.

A small note to the last two examples, where fractions with a highlighted integer part are subtracted. The minus before the second fraction means that it is the whole fraction that is subtracted, and not just its whole part.

Reread this sentence again, look at the examples, and think about it. This is where beginners make a lot of mistakes. They like to give such tasks at control work. You will also meet them repeatedly in the tests for this lesson, which will be published shortly.

Summary: General Scheme of Computing

In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:

If an integer part is highlighted in one or more fractions, convert these fractions to improper ones;
Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the compilers of the problems did this);
Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with the same denominators;
Reduce the result if possible. If the fraction turned out to be incorrect, select the whole part.

Remember that it is better to highlight the whole part at the very end of the task, just before writing the answer.