How to perform actions with ordinary fractions. Math: actions with fractions

1.Adding and subtracting fractions with the same denominator

When adding fractions with the same denominators, the numerators add up, and

When subtracting fractions with the same denominator, the numerator of the second fraction is subtracted from the numerator of the first fraction, and the denominator is left the same.

Examples: a) ; b)

2.Adding and subtracting fractions with different denominators

To add (subtract) fractions with different denominators, you need:

reduce these fractions to the lowest common denominator

add (subtract) the resulting fractions (as in paragraph 1)

Examples: a)
; b)

3.Addition and subtraction of mixed numbers

To add mixed numbers, you need:

reduce the fractional parts of these numbers to the lowest common denominator;

separately perform the addition of whole parts and separately fractional parts. If, when adding the fractional parts, you get an incorrect fraction, select the whole part from this fraction and add it to the resulting whole part.

Examples: a)
; b)

To subtract mixed numbers, you need to:

reduce the fractional parts of these numbers to the lowest common denominator; if the fractional part of the reduced one is less than the fractional part of the subtracted one, turn it into an irregular fraction, decreasing the whole part by one;

separately perform the subtraction of whole parts and separately fractional parts.

Examples: a)
; b)

4 multiplication of fractions

a) To multiply a fraction by natural number , you need to multiply its numerator by this number, and leave the denominator unchanged

Examples:

b) To multiply a fraction by a fraction, necessary:

1) write the product of the numerators in the numerator, and the product of the denominators in the denominator;

2) perform a reduction (if possible);

3) perform multiplication

Examples: a)
; b)

c) In order to multiply mixed numbers, you need to write them in the form of improper fractions, and then use the rule for multiplying fractions.

Examples:

5 division of fractions

To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor

Actions with fractions. In this article, we will analyze examples, everything is detailed with explanations. We will consider ordinary fractions. In the future, we will analyze decimal places. I recommend that you watch it all and study it sequentially.

1. Sum of fractions, difference of fractions.

Rule: when adding fractions with equal denominators, the result is a fraction - the denominator of which remains the same, and its numerator will be equal to the sum of the numerators of the fractions.

Rule: when calculating the difference of fractions with the same denominators, we get a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.

Formal notation of the sum and difference of fractions with equal denominators:

Examples (1):

It is clear that when ordinary fractions are given, then everything is simple, but if mixed? Nothing complicated ...

Option 1- you can convert them into ordinary ones and then calculate them.

Option 2- you can separately "work" with the integer and fractional parts.

Examples (2):

Yet:

What if the difference of two mixed fractions is given and the numerator of the first fraction is less than the numerator of the second? You can also act in two ways.

Examples (3):

* Translated into ordinary fractions, calculated the difference, converted the resulting incorrect fraction into a mixed one.

* Divided into whole and fractional parts, got a triple, then presented 3 as the sum of 2 and 1, whereby the unit was presented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) a unit and represent it as a fraction with the denominator we need, then we can subtract another from this fraction.

Another example:

Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, you can always translate them into incorrect ones, then perform the necessary action. After that, if as a result we get an incorrect fraction, we convert it to a mixed one.

Above, we looked at examples with fractions that have equal denominators. What if the denominators are different? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the main property of a fraction is used.

Let's look at some simple examples:

In these examples, we immediately see how one of the fractions can be transformed to get equal denominators.

If we designate ways of reducing fractions to one denominator, then this one will be called METHOD ONE.

That is, right away when "evaluating" the fraction, you need to estimate whether this approach will work - we check whether the larger denominator is divided by the smaller one. And if it is divided, then we perform the transformation - we multiply the numerator and the denominator so that the denominators of both fractions become equal.

Now look at these examples:

This approach is not applicable to them. There are also ways to bring fractions to a common denominator, consider them.

Method SECOND.

We multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first:

* In fact, we bring fractions to the form when the denominators become equal. Next, we use the rule for adding shirts with equal denominators.

Example:

* This method can be called universal, and it always works. The only drawback is that after the calculations, you may get a fraction that will need to be further reduced.

Let's consider an example:

It can be seen that the numerator and denominator are divisible by 5:

Method THIRD.

Find the least common multiple (LCM) of the denominators. This will be the common denominator. What is this number? It is the smallest natural number that is divisible by each of the numbers.

Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, they are divisible by 30, 60, 90 .... The smallest 30. The question is - how to determine this least common multiple?

There is a clear algorithm, but often it can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15), no algorithm is needed, we took large numbers (4 and 15) and doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, for example 51 and 119.

Algorithm. In order to determine the least common multiple of several numbers, you must:

- decompose each of the numbers into PRIMARY factors

- write out the decomposition of the MOST of them

- multiply it by the MISSING factors of other numbers

Let's look at some examples:

50 and 60 => 50 = 2∙5∙5 60 = 2∙2∙3∙5

the expansion of a larger number is missing one five

=> LCM (50.60) = 2 ∙ 2 ∙ 3 ​​∙ 5 ∙ 5 = 300

48 and 72 => 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3

the expansion of a larger number is missing two and three

=> LCM (48.72) = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3 = 144

* The least common multiple of two primes is equal to their product

Question! And why is it useful to find the least common multiple, because you can use the second method and simply cancel the resulting fraction? Yes, you can, but it's not always convenient. Look what the denominator for the numbers 48 and 72 will be if you simply multiply them 48 ∙ 72 = 3456. Agree that it is more pleasant to work with smaller numbers.

Let's look at some examples:

*51 = 3∙17 119 = 7∙17

the expansion of a larger number is missing a triple

=> LCM (51,119) = 3 ∙ 7 ∙ 17

Now let's apply the first method:

* Look what the difference is in the calculations, in the first case there is a minimum of them, and in the second you need to work separately on a piece of paper, and even the fraction that you received must be reduced. Finding the LCM simplifies the job considerably.

More examples:

* In the second example, it is already clear that smallest number which is divisible by 40 and 60 is 120.

TOTAL! GENERAL CALCULATION ALGORITHM!

- we reduce fractions to ordinary ones, if there is an integer part.

- we bring the fractions to a common denominator (first we look at whether one denominator is divided by another, if it is divided, then we multiply the numerator and denominator of this other fraction; if it is not divided, we act through the other methods indicated above).

- having received fractions with equal denominators, we perform actions (addition, subtraction).

- if necessary, we reduce the result.

- if necessary, select the whole part.

2. Product of fractions.

The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:

Examples:

This article provides a general look at fractions. Here we will formulate and justify the rules of addition, subtraction, multiplication, division and exponentiation of general fractions A / B, where A and B are some numbers, numerical expressions or expressions with variables. As usual, we will supply the material with explanatory examples with detailed descriptions of solutions.

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General rules for performing actions with numeric fractions

Let's agree by general numeric fractions to mean fractions in which the numerator and / or denominator can be represented not only by natural numbers, but also by other numbers or numerical expressions. For clarity, we will give several examples of such fractions:, .

We know the rules by which they are executed. According to the same rules, you can perform actions with general fractions:

Rationale for the rules

To substantiate the validity of the rules for performing actions with general numeric fractions, one can start from the following points:

• the fractional bar is essentially a division sign,
• division by some nonzero number can be viewed as multiplication by the reciprocal of the divisor (this immediately explains the rule division of fractions),
• properties of actions with real numbers,
• and its generalized understanding,

They allow you to carry out the following transformations, justifying the rules for addition, subtraction of fractions with the same and different denominators, as well as the rule for multiplying fractions:

Examples of

We will give examples of performing actions with general fractions according to the rules learned in the previous paragraph. Let's say right away that usually after performing actions with fractions, the resulting fraction requires simplification, and the process of simplifying a fraction is often more complicated than performing the previous actions. We will not dwell on the simplification of fractions (the corresponding transformations are discussed in the article on the conversion of fractions), so as not to be distracted from the topic of interest to us.

Let's start with examples of adding and subtracting numeric fractions with the same denominator. First, add the fractions and. Obviously, the denominators are equal. According to the corresponding rule, we write down a fraction, the numerator of which is equal to the sum of the numerators of the original fractions, and the denominator is left the same, we have. The addition is completed, it remains to simplify the resulting fraction: ... So, .

It was possible to conduct a solution differently: first, make the transition to ordinary fractions, and then carry out the addition. With this approach, we have .

Now let's subtract from the fraction fraction ... The denominators of fractions are equal, therefore, we act according to the rule of subtraction of fractions with the same denominators:

Let's move on to examples of addition and subtraction of fractions with different denominators. The main difficulty here lies in bringing the fractions to a common denominator. For general fractions, this is a rather extensive topic, we will analyze it in detail in a separate article. common denominator of fractions... Now let's limit ourselves to a couple general recommendations since in this moment we are more interested in the technique of performing actions with fractions.

In general, the process is similar to reducing common fractions to a common denominator. That is, the denominators are represented in the form of products, then all the factors are taken from the denominator of the first fraction and the missing factors from the denominator of the second fraction are added to them.

When the denominators of the added or subtracted fractions do not have common factors, then it is logical to take their product as the common denominator. Let's give an example.

Let's say we need to add fractions and 1/2. Here, as a common denominator, it is logical to take the product of the denominators of the original fractions, that is,. In this case, the additional factor for the first fraction will be 2. After multiplying the numerator and denominator by it, the fraction will take the form. And for the second fraction, the additional factor is the expression. With its help, the fraction 1/2 is reduced to the form. It remains to add the resulting fractions with the same denominators. Here is a summary of the entire solution:

In the case of general fractions, we are no longer talking about the lowest common denominator, to which ordinary fractions are usually reduced. Although in this matter it is still desirable to strive for some minimalism. By this we want to say that you should not take the product of the denominators of the original fractions as a common denominator. For example, it is not at all necessary to take the common denominator of fractions and the product ... Here, we can take as a common denominator.

We turn to examples of multiplication of general fractions. Let's multiply fractions and. The rule for performing this action instructs us to write down a fraction, the numerator of which is the product of the numerators of the original fractions, and the denominator is the product of the denominators. We have ... Here, as in many other cases when multiplying fractions, you can cancel the fraction: .

The rule for dividing fractions allows you to go from division to multiplication by the reciprocal. Here you need to remember that in order to get the inverse of the given fraction, you need to rearrange the numerator and denominator of this fraction. Here's an example of going from general division of numeric fractions to multiplication: ... It remains to perform the multiplication and simplify the resulting fraction (if necessary, see the transformation of irrational expressions):

Concluding the information of this paragraph, we recall that any number or numerical expression can be represented as a fraction with a denominator of 1, therefore, addition, subtraction, multiplication and division of a number and a fraction can be considered as performing the corresponding action with fractions, one of which has a unit in the denominator ... For example, replacing in the expression root of three fractions, we will go from multiplying a fraction by a number to multiplying two fractions: .

Performing actions on fractions containing variables

The rules from the first part of this article are also applied to perform actions with fractions that contain variables. Let us justify the first of them - the rule of addition and subtraction of fractions with the same denominators, the rest are proved in absolutely the same way.

Let us prove that for any expressions A, C, and D (D is not identically zero) the equality on its range of admissible values ​​of variables.

Let's take some set of variables from the ODV. Let for these values ​​of the variables the expressions A, C and D take the values ​​a 0, c 0 and d 0. Then the substitution of the values ​​of the variables from the selected set into the expression turns it into the sum (difference) of numeric fractions with the same denominators of the form, which, according to the rule of addition (subtraction) of numeric fractions with the same denominators, is equal to. But substitution of the values ​​of variables from the selected set into the expression converts it to the same fraction. This means that for the selected set of values ​​of variables from the LDZ, the values ​​of the expressions and are equal. It is clear that the values ​​of these expressions will be equal for any other set of values ​​of the variables from the ODZ, which means that the expressions and are identically equal, that is, the equality being proved is true .

Examples of adding and subtracting fractions with variables

When the denominators of the added or subtracted fractions are the same, then everything is quite simple - the numerators are added or subtracted, and the denominator remains the same. It is clear that the fraction obtained after this is simplified if necessary and possible.

Note that sometimes the denominators of fractions differ only at first glance, but in fact they are identically equal expressions, such as, and, or and. And sometimes it is enough to simplify the original fractions in order for their identical denominators to appear.

Example.

, b) , v) .

Solution.

a) We need to subtract fractions with the same denominators. According to the corresponding rule, we leave the denominator unchanged and subtract the numerators, we have ... Action completed. But you can still expand the parentheses in the numerator and give similar terms: .

b) Obviously, the denominators of the added fractions are the same. Therefore, add up the numerators, and leave the denominator the same:. The addition is complete. But it is easy to see that the resulting fraction can be canceled. Indeed, the numerator of the resulting fraction can be convoluted by the formula of the square of the sum as (lgx + 2) 2 (see formulas for abbreviated multiplication), thus, the following transformations take place: .

c) Fractions in the sum have different denominators. But, having transformed one of the fractions, you can proceed to the addition of fractions with the same denominators. We will show two solutions.

The first way. The denominator of the first fraction can be factorized using the difference of squares formula, and then cancel this fraction: ... Thus, . It still does not hurt to get rid of irrationality in the denominator of the fraction: .

Second way. Multiplying the numerator and denominator of the second fraction by (this expression does not vanish for any value of the variable x from the ODZ for the original expression) allows you to achieve two goals at once: to get rid of irrationality and go to the addition of fractions with the same denominators. We have

Answer:

a) , b) , v) .

The last example led us to the question of reducing fractions to a common denominator. There we almost accidentally came to the same denominators, simplifying one of the added fractions. But in most cases, when adding and subtracting fractions with different denominators, you have to purposefully bring the fractions to a common denominator. For this, the denominators of fractions are usually represented in the form of products, all factors are taken from the denominator of the first fraction and the missing factors from the denominator of the second fraction are added to them.

Example.

Perform actions with fractions: a) , b), c) .

Solution.

a) There is no need to do anything with the denominators of the fractions. As a common denominator, we take the product ... In this case, the expression is an additional factor for the first fraction, and the number 3 for the second fraction. These additional factors bring the fractions to a common denominator, which later allows us to perform the action we need, we have

b) In this example, the denominators are already represented as products, and no additional transformations are required. Obviously, the factors in the denominators differ only in the exponents, therefore, as a common denominator, we take the product of the factors with the largest exponents, that is, ... Then the additional factor for the first fraction will be x 4, and for the second - ln (x + 1). We are now ready to perform subtraction of fractions:

c) A c this case first, let's work with the denominators of the fractions. The formulas the difference of squares and the square of the sum allow you to go from the original sum to the expression ... Now it is clear that these fractions can be reduced to a common denominator ... With this approach, the solution will look like this:

Answer:

a)

b)

v)

Examples of multiplying fractions with variables

Multiplication of fractions gives a fraction, the numerator of which is the product of the numerators of the original fractions, and the denominator is the product of the denominators. Here, as you can see, everything is familiar and simple, and we can only add that the fraction obtained as a result of performing this action is often cancellable. In these cases, it is reduced, if, of course, it is necessary and justified.

In mathematics, various types of numbers have been studied since their inception. There are many sets and subsets of numbers. Among them, there are integers, rational, irrational, natural, even, odd, complex and fractional. Today we will analyze information about the last set - fractional numbers.

Defining fractions

Fractions are numbers made up of whole parts and fractions of one. Just like integers, there is an infinite number of fractions between two integers. In mathematics, actions with fractions are performed as with integers and natural numbers. It's pretty simple and can be learned in a couple of lessons.

The article presents two types

Ordinary fractions

Ordinary fractions are the integer part a and two numbers separated by the fractional bar b / c. Ordinary fractions can be extremely handy if the fractional part cannot be represented in a rational decimal notation. In addition, it is more convenient to perform arithmetic operations through the fractional bar. The upper part is called the numerator, the lower part is called the denominator.

Fractional Actions: Examples

The main property of a fraction. At multiplying the numerator and denominator by the same nonzero number results in a number equal to the given one. This property of a fraction perfectly helps to bring the denominator for addition (this will be discussed below) or to reduce the fraction, to make it more convenient for counting. a / b = a * c / b * c. For example, 36/24 = 6/4 or 9/13 = 18/26

Reducing to a common denominator. To bring the denominator of a fraction, it is necessary to represent the denominator in the form of factors, and then multiply by the missing numbers. For example, 7/15 and 12/30; 7/5 * 3 and 12/5 * 3 * 2. We see that the denominators differ by two, so we multiply the numerator and denominator of the first fraction by 2. We get: 14/30 and 12/30.

Compound fractions- ordinary fractions with a highlighted integer part. (A b / c) To represent a compound fraction as an ordinary fraction, you must multiply the number in front of the fraction by the denominator, and then add it with the numerator: (A * c + b) / c.

Arithmetic operations with fractions

It will not be superfluous to consider the well-known arithmetic operations only when working with fractional numbers.

Addition and subtraction. Adding and subtracting ordinary fractions is just as easy as adding whole numbers, except for one difficulty - the presence of a fractional bar. When adding fractions with the same denominator, it is necessary to add only the numerators of both fractions, the denominators remain unchanged. For example: 5/7 + 1/7 = (5 + 1) / 7 = 6/7

If the denominators of two fractions are different numbers, you first need to bring them to a common one (as discussed above). 1/8 + 3/2 = 1/2 * 2 * 2 + 3/2 = 1/8 + 3 * 4/2 * 4 = 1/8 + 12/8 = 13/8. Subtraction follows exactly the same principle: 8/9 - 2/3 = 8/9 - 6/9 = 2/9.

Multiplication and division. Actions with fractions by multiplication occur according to the following principle: the numerators and denominators are multiplied separately. V general view the multiplication formula looks like this: a / b * c / d = a * c / b * d. In addition, as you multiply, you can reduce the fraction by eliminating the same factors from the numerator and denominator. In other words, the numerator and denominator are divided by the same number: 4/16 = 4/4 * 4 = 1/4.

To divide one ordinary fraction by another, you need to change the numerator and denominator of the divisor and multiply the two fractions, according to the principle discussed earlier: 5/11: 25/11 = 5/11 * 11/25 = 5 * 11/11 * 25 = 1/5

Decimal fractions

Decimal fractions are the more popular and commonly used version of fractional numbers. It is easier to write them down in a line or represent them on a computer. The structure of the decimal fraction is as follows: first, the integer is written, and then, after the decimal point, the fractional part is written. At its core decimals- these are compound ordinary fractions, but their fractional part is represented by a number divided by a multiple of 10. This is where their name comes from. Operations with decimal fractions are similar to operations with integers, since they are also written in decimal notation. Also, unlike ordinary fractions, decimals can be irrational. This means that they can be endless. They are written as 7, (3). The following record is read: seven point, three tenths in the period.

Basic Decimal Operations

Addition and subtraction of decimal fractions. Performing operations with fractions is no more difficult than with whole natural numbers. The rules are absolutely similar to those used when adding or subtracting natural numbers. They can be considered as a column in the same way, however, if necessary, replace the missing places with zeros. For example: 5.5697 - 1.12. In order to perform subtraction in a column, you need to equalize the number of numbers after the decimal point: (5.5697 - 1.1200). So, the numerical value does not change and it will be possible to count in a column.

Actions with decimal fractions cannot be performed if one of them is irrational. To do this, you need to translate both numbers into fractions, and then use the techniques described earlier.

Multiplication and division. Decimal multiplication is similar to natural multiplication. They can also be multiplied in a column, simply without paying attention to the comma, and then separate with a comma in the final value the same number of digits as the sum after the decimal point was in two decimal fractions. For example, 1.5 * 2.23 = 3.345. Everything is very simple, and should not be difficult if you have already mastered the multiplication of natural numbers.

Division also coincides with division of natural numbers, but with a slight deviation. To divide by a decimal number in a column, you must discard the comma in the divisor, and multiply the dividend by the number of decimal places in the divisor. Then perform division as with natural numbers. In case of incomplete division, you can add zeros to the dividend on the right, also adding a zero in the answer after the decimal point.

Examples of actions with decimal fractions. Decimal fractions are a very handy tool for calculating arithmetic. They combine the convenience of natural numbers, whole numbers and the precision of common fractions. In addition, it is quite easy to translate some fractions into others. Actions with fractions do not differ from actions with natural numbers.

1. Addition: 1.5 + 2.7 = 4.2
2. Subtraction: 3.1 - 1.6 = 1.5
3. Multiplication: 1.7 * 2.3 = 3.91
4. Division: 3.6: 0.6 = 6

In addition, decimals are suitable for representing percentages. So, 100% = 1; 60% = 0.6; and vice versa: 0.659 = 65.9%.

That's all there is to know about fractions. The article considered two types of fractions - ordinary and decimal. Both are fairly easy to calculate, and if you have completely mastered natural numbers and operations with them, you can safely start learning fractional numbers.

1º. Integers- these are numbers used for counting. The set of all natural numbers is denoted by N, i.e. N = (1, 2, 3, ...).

Fraction called a number consisting of several parts of one. Ordinary fraction is called a number of the form, where a natural number n shows how many equal parts the unit is divided into, and the natural number m shows how many such equal parts are taken. The numbers m and n are called accordingly numerator and denominator fractions.

If the numerator less than the denominator, then the ordinary fraction is called correct; if the numerator is equal to or greater than the denominator, then the fraction is called wrong... A number consisting of whole and fractional parts is called mixed number.

For example, - regular fractions, - irregular fractions, 1 - mixed number.

2º. When performing actions over ordinary fractions remember the following rules:

1)Basic property of a fraction... If the numerator and denominator of a fraction are multiplied or divided by the same natural number, you get a fraction equal to the given one.

For example, a); b) .

Division of the numerator and denominator of a fraction by their common divisor, other than one, is called reduction of fraction.

2) In order to represent the mixed number as an improper fraction, you need to multiply its whole part by the denominator of the fractional part and add the numerator of the fractional part to the resulting product, write down the resulting sum as the numerator of the fraction, and leave the denominator the same.

Similarly, any natural number can be written as an improper fraction with any denominator.

For example, a), since; b) etc.

3) To write an incorrect fraction as a mixed number (that is, select the whole part from the incorrect fraction), you need to divide the numerator by the denominator, take the quotient of the division as the whole part, the remainder as the numerator, leave the denominator the same.

For example, a), since 200: 7 = 28 (rest. 4);
b), since 20: 5 = 4 (rest 0).

4) To bring the fractions to the lowest common denominator, you need to find the least common multiple (LCM) of the denominators of these fractions (this will be their lowest common denominator), divide the lowest common denominator by the denominators of these fractions (i.e. find additional factors for the fractions) , multiply the numerator and denominator of each fraction by its additional factor.

For example, let's bring the fractions to the lowest common denominator:

630: 18 = 35, 630: 10 = 63, 630: 21 = 30.

Means, ; ; .

5) Rules for arithmetic operations with ordinary fractions:

a) Addition and subtraction of fractions with the same denominator is performed according to the rule:

b) Addition and subtraction of fractions with different denominators is performed according to rule a), having previously reduced the fractions to the lowest common denominator.

c) When adding and subtracting mixed numbers, you can turn them into improper fractions, and then perform the actions according to rules a) and b),