In the article, we will show how to solve fractions with simple clear examples. Let's understand what a fraction is and consider solving fractions!

concept fractions is introduced into the course of mathematics starting from the 6th grade of secondary school.

Fractions look like: ±X / Y, where Y is the denominator, it tells how many parts the whole was divided into, and X is the numerator, it tells how many such parts were taken. For clarity, let's take an example with a cake:

In the first case, the cake was cut equally and one half was taken, i.e. 1/2. In the second case, the cake was cut into 7 parts, from which 4 parts were taken, i.e. 4/7.

If the part of dividing one number by another is not a whole number, it is written as a fraction.

For example, the expression 4:2 \u003d 2 gives an integer, but 4:7 is not completely divisible, so this expression is written as a fraction 4/7.

In other words fraction is an expression that denotes the division of two numbers or expressions, and which is written with a slash.

If the numerator is less than the denominator, the fraction is correct, if vice versa, it is incorrect. A fraction can contain an integer.

For example, 5 whole 3/4.

This entry means that in order to get the whole 6, one part of four is not enough.

If you want to remember how to solve fractions for 6th grade you need to understand that solving fractions basically comes down to understanding a few simple things.

• A fraction is essentially an expression for a fraction. That is, a numerical expression of what part a given value is from one whole. For example, the fraction 3/5 expresses that if we divide something whole into 5 parts and the number of parts or parts of this whole is three.
• A fraction can be less than 1, for example 1/2 (or essentially half), then it is correct. If the fraction is greater than 1, for example 3/2 (three halves or one and a half), then it is incorrect and to simplify the solution, it is better for us to select the whole part 3/2= 1 whole 1/2.
• Fractions are the same numbers as 1, 3, 10, and even 100, only the numbers are not whole, but fractional. With them, you can perform all the same operations as with numbers. Counting fractions is not more difficult, and further we will show this with specific examples.

How to solve fractions. Examples.

A variety of arithmetic operations are applicable to fractions.

Bringing a fraction to a common denominator

For example, you need to compare the fractions 3/4 and 4/5.

To solve the problem, we first find the lowest common denominator, i.e. the smallest number that is divisible without remainder by each of the denominators of the fractions

Least common denominator(4.5) = 20

Then the denominator of both fractions is reduced to the lowest common denominator

Answer: 15/20

Addition and subtraction of fractions

If it is necessary to calculate the sum of two fractions, they are first brought to a common denominator, then the numerators are added, while the denominator remains unchanged. The difference of fractions is considered in a similar way, the only difference is that the numerators are subtracted.

For example, you need to find the sum of fractions 1/2 and 1/3

Now find the difference between the fractions 1/2 and 1/4

Multiplication and division of fractions

Here the solution of fractions is simple, everything is quite simple here:

• Multiplication - numerators and denominators of fractions are multiplied among themselves;
• Division - first we get a fraction, the reciprocal of the second fraction, i.e. swap its numerator and denominator, after which we multiply the resulting fractions.

For example:

On this about how to solve fractions, all. If you have any questions about solving fractions, something is not clear, then write in the comments and we will answer you.

If you are a teacher, then it is possible to download a presentation for an elementary school (http://school-box.ru/nachalnaya-shkola/prezentazii-po-matematike.html) which will come in handy.

This article is a general look at operations with fractions. Here we formulate and justify the rules for addition, subtraction, multiplication, division and raising to the power of fractions of the general form 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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Rules for performing operations with numerical fractions of a general form

Let's agree that general numerical fractions are 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, here are a few examples of such fractions: .

We know the rules by which . By the same rules, you can perform operations with fractions of a general form:

Rationale for the rules

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

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

They allow you to carry out the following transformations that justify the rules for adding, subtracting fractions with the same and different denominators, as well as the rule for multiplying fractions:

Examples

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

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

It was possible to carry out the decision in a different way: first, make the transition to ordinary fractions, and then carry out addition. With this approach, we have .

Now subtract from the fraction fraction . The denominators of fractions are equal, therefore, we act according to the rule for subtracting fractions with the same denominators:

Let's move on to examples of adding and subtracting fractions with different denominators. Here the main difficulty lies in bringing the fractions to a common denominator. For fractions of a general form, this is a rather extensive topic, we will analyze it in detail in a separate article. reducing fractions to a common denominator. For now, we will limit ourselves to a couple of general recommendations, since at the moment we are more interested in the technique of performing actions with fractions.

In general, the process is similar to reduction to a common denominator of ordinary fractions. That is, the denominators are presented as products, then all the factors from the denominator of the first fraction are taken 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 a common denominator. Let's take 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 fractions of a general form, we are no longer talking about the least 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 it is not necessary to immediately 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, as a common denominator, we can take .

We turn to examples of multiplication of fractions of a general form. Multiply the fractions and . The rule for performing this action tells us to write down a fraction whose numerator 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 reduce the fraction: .

The rule for dividing fractions allows you to move from division to multiplication by a reciprocal. Here you need to remember that in order to get a fraction reciprocal of a given one, you need to swap the numerator and denominator of this fraction. Here is an example of the transition from dividing general 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 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 proceed from multiplying a fraction by a number to multiplying two fractions: .

Performing operations with fractions containing variables

The rules from the first part of this article also apply to performing operations 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 exactly the same way.

Let us prove that for any expressions A , C and D (D is identically non-zero) we have the equality on its range of acceptable values ​​of variables.

Let's take some set of variables from ODZ. Let the expressions A , C and D take the values ​​a 0 , c 0 and d 0 for these values ​​of the variables. Then substituting the values ​​of 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 substituting the values ​​of the variables from the selected set into the expression turns it into the same fraction. This means that for the selected set of variable values ​​from the ODZ, the values ​​of the expressions and are equal. It is clear that the values ​​of the indicated expressions will be equal for any other set of values ​​of variables from the ODZ, which means that the expressions and are identically equal, that is, the equality being proved is true .

Examples of addition and subtraction of fractions with variables

When the denominators of the fractions being added or subtracted 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, for example, and , or and . And sometimes it is enough to simplify the initial fractions so that their identical denominators “appear”.

Example.

, b) , in) .

Solution.

a) We need to subtract fractions with the same denominators. According to the corresponding rule, we leave the denominator the same and subtract the numerators, we have . Action done. But you can still open the brackets in the numerator and bring like terms: .

b) Obviously, the denominators of the added fractions are the same. Therefore, we add the numerators, and leave the denominator the same: . Addition completed. But it is easy to see that the resulting fraction can be reduced. Indeed, the numerator of the resulting fraction can be reduced by the square of the sum as (lgx+2) 2 (see the abbreviated multiplication formulas), so the following transformations take place: .

c) Fractions in the sum have different denominators. But, by converting one of the fractions, you can proceed to adding fractions with the same denominators. We show two solutions.

First way. The denominator of the first fraction can be factored using the difference of squares formula, and then reduce this fraction: . In this way, . It doesn’t hurt to get rid of irrationality in the denominator of a fraction: .

The second way. Multiplying the numerator and denominator of the second fraction (this expression does not vanish for any values ​​of the variable x from the DPV for the original expression) allows you to achieve two goals at once: get rid of irrationality and move on to adding fractions with the same denominators. We have

Answer:

a) , b) , in) .

The last example brought us to the question of bringing 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, one has to purposefully bring the fractions to a common denominator. To do this, the denominators of fractions are usually presented as 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 additional factor for the first fraction is the expression, and for the second fraction - the number 3. These additional factors bring fractions to a common denominator, which further allows us to perform the action we need, we have

b) In this example, the denominators are already presented as products, and no additional transformations are required. Obviously, the factors in the denominators differ only in 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) . Now we are ready to subtract fractions:

c) And in this case, to begin with, we will work with the denominators of fractions. The formulas of 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)

in)

Examples of multiplying fractions with variables

Multiplying fractions gives a fraction whose numerator 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 this action is often reduced. In these cases, it is reduced, unless, of course, it is necessary and justified.

This article deals with operations on fractions. Rules for addition, subtraction, multiplication, division or exponentiation of fractions of the form A B will be formed and justified, where A and B can be numbers, numeric expressions or expressions with variables. In conclusion, examples of solutions with a detailed description will be considered.

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Rules for performing operations with numerical fractions of a general form

Numerical fractions of a general form have a numerator and a denominator, in which there are natural numbers or numerical expressions. If we consider such fractions as 3 5 , 2 , 8 4 , 1 + 2 3 4 (5 - 2) , 3 4 + 7 8 2 , 3 - 0 , 8 , 1 2 2 , π 1 - 2 3 + π , 2 0 , 5 ln 3 , then it is clear that the numerator and denominator can have not only numbers, but also expressions of a different plan.

Definition 1

There are rules by which actions are performed with ordinary fractions. It is also suitable for fractions of a general form:

• When subtracting fractions with the same denominators, only the numerators are added, and the denominator remains the same, namely: a d ± c d \u003d a ± c d, the values ​​a, c and d ≠ 0 are some numbers or numerical expressions.
• When adding or subtracting fractions with different denominators, it is necessary to reduce to a common one, and then add or subtract the resulting fractions with the same indicators. Literally, it looks like this a b ± c d = a p ± c r s , where the values ​​a , b ≠ 0 , c , d ≠ 0 , p ≠ 0 , r ≠ 0 , s ≠ 0 are real numbers, and b p = d r = s. When p = d and r = b, then a b ± c d = a d ± c d b d.
• When multiplying fractions, an action is performed with numerators, after which with denominators, then we get a b c d \u003d a c b d, where a, b ≠ 0, c, d ≠ 0 act as real numbers.
• When dividing a fraction by a fraction, we multiply the first by the second reciprocal, that is, we swap the numerator and denominator: a b: c d \u003d a b d c.

Rationale for the rules

Definition 2

There are the following mathematical points that you should rely on when calculating:

• a fractional bar means a division sign;
• division by a number is treated as a multiplication by its reciprocal;
• application of the property of actions with real numbers;
• application of the basic property of a fraction and numerical inequalities.

With their help, you can make transformations of the form:

a d ± c d = a d - 1 ± c d - 1 = a ± c d - 1 = a ± c d ; a b ± c d = a p b p ± c r d r = a p s ± c e s = a p ± c r s ; a b c d = a d b d b c b d = a d a d - 1 b c b d - 1 = = a d b c b d - 1 b d - 1 = a d b c b d b d - 1 = = (a c) (b d) - 1 = a c b d

Examples

In the previous paragraph, it was said about actions with fractions. It is after this that the fraction needs to be simplified. This topic was discussed in detail in the section on converting fractions.

First, consider the example of adding and subtracting fractions with the same denominator.

Example 1

Given fractions 8 2 , 7 and 1 2 , 7 , then according to the rule it is necessary to add the numerator and rewrite the denominator.

Solution

Then we get a fraction of the form 8 + 1 2 , 7 . After performing the addition, we get a fraction of the form 8 + 1 2 , 7 = 9 2 , 7 = 90 27 = 3 1 3 . So 8 2 , 7 + 1 2 , 7 = 8 + 1 2 , 7 = 9 2 , 7 = 90 27 = 3 1 3 .

Answer: 8 2 , 7 + 1 2 , 7 = 3 1 3

There is another way to solve. To begin with, a transition is made to the form of an ordinary fraction, after which we perform a simplification. It looks like this:

8 2 , 7 + 1 2 , 7 = 80 27 + 10 27 = 90 27 = 3 1 3

Example 2

Let us subtract from 1 - 2 3 log 2 3 log 2 5 + 1 fractions of the form 2 3 3 log 2 3 log 2 5 + 1 .

Since equal denominators are given, it means that we are calculating a fraction with the same denominator. We get that

1 - 2 3 log 2 3 log 2 5 + 1 - 2 3 3 log 2 3 log 2 5 + 1 = 1 - 2 - 2 3 3 log 2 3 log 2 5 + 1

There are examples of calculating fractions with different denominators. An important point is the reduction to a common denominator. Without this, we will not be able to perform further actions with fractions.

The process is remotely reminiscent of reduction to a common denominator. That is, a search is made for the least common divisor in the denominator, after which the missing factors are added to the fractions.

If the added fractions have no common factors, then their product can become one.

Example 3

Consider the example of adding fractions 2 3 5 + 1 and 1 2 .

Solution

In this case, the common denominator is the product of the denominators. Then we get that 2 · 3 5 + 1 . Then, when setting additional factors, we have that to the first fraction it is equal to 2, and to the second 3 5 + 1. After multiplication, the fractions are reduced to the form 4 2 3 5 + 1. The general cast 1 2 will be 3 5 + 1 2 · 3 5 + 1 . We add the resulting fractional expressions and get that

2 3 5 + 1 + 1 2 = 2 2 2 3 5 + 1 + 1 3 5 + 1 2 3 5 + 1 = = 4 2 3 5 + 1 + 3 5 + 1 2 3 5 + 1 = 4 + 3 5 + 1 2 3 5 + 1 = 5 + 3 5 2 3 5 + 1

Answer: 2 3 5 + 1 + 1 2 = 5 + 3 5 2 3 5 + 1

When we are dealing with fractions of a general form, then the least common denominator is usually not the case. It is unprofitable to take the product of numerators as a denominator. First you need to check if there is a number that is less in value than their product.

Example 4

Consider the example 1 6 2 1 5 and 1 4 2 3 5 when their product is equal to 6 2 1 5 4 2 3 5 = 24 2 4 5 . Then we take 12 · 2 3 5 as a common denominator.

Consider examples of multiplications of fractions of a general form.

Example 5

To do this, it is necessary to multiply 2 + 1 6 and 2 · 5 3 · 2 + 1.

Solution

Following the rule, it is necessary to rewrite and write the product of numerators as a denominator. We get that 2 + 1 6 2 5 3 2 + 1 2 + 1 2 5 6 3 2 + 1 . When the fraction is multiplied, reductions can be made to simplify it. Then 5 3 3 2 + 1: 10 9 3 = 5 3 3 2 + 1 9 3 10 .

Using the rule of transition from division to multiplication by a reciprocal, we get the reciprocal of the given one. To do this, the numerator and denominator are reversed. Let's look at an example:

5 3 3 2 + 1: 10 9 3 = 5 3 3 2 + 1 9 3 10

After that, they must perform multiplication and simplify the resulting fraction. If necessary, get rid of the irrationality in the denominator. We get that

5 3 3 2 + 1: 10 9 3 = 5 3 3 9 3 10 2 + 1 = 5 2 10 2 + 1 = 3 2 2 + 1 = = 3 2 - 1 2 2 + 1 2 - 1 = 3 2 - 1 2 2 2 - 1 2 = 3 2 - 1 2

Answer: 5 3 3 2 + 1: 10 9 3 = 3 2 - 1 2

This paragraph is applicable when a number or numerical expression can be represented as a fraction with a denominator equal to 1, then the operation with such a fraction is considered a separate paragraph. For example, the expression 1 6 7 4 - 1 3 shows that the root of 3 can be replaced by another 3 1 expression. Then this record will look like a multiplication of two fractions of the form 1 6 7 4 - 1 3 = 1 6 7 4 - 1 3 1 .

Performing an action with fractions containing variables

The rules discussed in the first article are applicable to operations with fractions containing variables. Consider the subtraction rule when the denominators are the same.

It is necessary to prove that A , C and D (D not equal to zero) can be any expressions, and the equality A D ± C D = A ± C D is equivalent to its range of valid values.

It is necessary to take a set of ODZ variables. Then A, C, D must take the corresponding values ​​a 0 , c 0 and d0. A substitution of the form A D ± C D results in a difference of the form a 0 d 0 ± c 0 d 0 , where, according to the addition rule, we obtain a formula of the form a 0 ± c 0 d 0 . If we substitute the expression A ± C D , then we get the same fraction of the form a 0 ± c 0 d 0 . From this we conclude that the chosen value that satisfies the ODZ, A ± C D and A D ± C D are considered equal.

For any value of the variables, these expressions will be equal, that is, they are called identically equal. This means that this expression is considered to be a provable equality of the form A D ± C D = A ± C D .

Examples of addition and subtraction of fractions with variables

When there are the same denominators, it is only necessary to add or subtract the numerators. This fraction can be simplified. Sometimes you have to work with fractions that are identically equal, but at first glance this is not noticeable, since some transformations must be performed. For example, x 2 3 x 1 3 + 1 and x 1 3 + 1 2 or 1 2 sin 2 α and sin a cos a. Most often, a simplification of the original expression is required in order to see the same denominators.

Example 6

Calculate: 1) x 2 + 1 x + x - 2 - 5 - x x + x - 2 , 2) l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) , x - 1 x - 1 + x x + 1 .

Solution

1. To make a calculation, you need to subtract fractions that have the same denominators. Then we get that x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + 1 - 5 - x x + x - 2 . After that, you can open the brackets with the reduction of similar terms. We get that x 2 + 1 - 5 - x x + x - 2 = x 2 + 1 - 5 + x x + x - 2 = x 2 + x - 4 x + x - 2
2. Since the denominators are the same, it remains only to add the numerators, leaving the denominator: l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) = l g 2 x + 4 + 4 x (l g x + 2)
   The addition has been completed. It can be seen that the fraction can be reduced. Its numerator can be folded using the sum square formula, then we get (l g x + 2) 2 from the abbreviated multiplication formulas. Then we get that
   l g 2 x + 4 + 2 l g x x (l g x + 2) = (l g x + 2) 2 x (l g x + 2) = l g x + 2 x
3. Given fractions of the form x - 1 x - 1 + x x + 1 with different denominators. After the transformation, you can proceed to addition.

Let's consider a two way solution.

The first method is that the denominator of the first fraction is subjected to factorization using squares, and with its subsequent reduction. We get a fraction of the form

x - 1 x - 1 = x - 1 (x - 1) x + 1 = 1 x + 1

So x - 1 x - 1 + x x + 1 = 1 x + 1 + x x + 1 = 1 + x x + 1 .

In this case, it is necessary to get rid of irrationality in the denominator.

1 + x x + 1 = 1 + x x - 1 x + 1 x - 1 = x - 1 + x x - x x - 1

The second way is to multiply the numerator and denominator of the second fraction by x - 1 . Thus, we get rid of irrationality and proceed to adding a fraction with the same denominator. Then

x - 1 x - 1 + x x + 1 = x - 1 x - 1 + x x - 1 x + 1 x - 1 = = x - 1 x - 1 + x x - x x - 1 = x - 1 + x x - x x - 1

Answer: 1) x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + x - 4 x + x - 2, 2) l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) = l g x + 2 x, 3) x - 1 x - 1 + x x + 1 = x - 1 + x x - x x - 1.

In the last example, we found that reduction to a common denominator is inevitable. To do this, you need to simplify the fractions. To add or subtract, you always need to look for a common denominator, which looks like the product of the denominators with the addition of additional factors to the numerators.

Example 7

Calculate the values ​​of fractions: 1) x 3 + 1 x 7 + 2 2, 2) x + 1 x ln 2 (x + 1) (2 x - 4) - sin x x 5 ln (x + 1) (2 x - 4) , 3) ​​1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x

Solution

1. The denominator does not require any complex calculations, so you need to choose their product of the form 3 x 7 + 2 2, then to the first fraction x 7 + 2 2 is chosen as an additional factor, and 3 to the second. When multiplying, we get a fraction of the form x 3 + 1 x 7 + 2 2 = x x 7 + 2 2 3 x 7 + 2 2 + 3 1 3 x 7 + 2 2 = = x x 7 + 2 2 + 3 3 x 7 + 2 2 = x x 7 + 2 2 x + 3 3 x 7 + 2 2
2. It can be seen that the denominators are presented as a product, which means that additional transformations are unnecessary. The common denominator will be the product of the form x 5 · ln 2 x + 1 · 2 x - 4 . From here x 4 is an additional factor to the first fraction, and ln (x + 1) to the second. Then we subtract and get:
   x + 1 x ln 2 (x + 1) 2 x - 4 - sin x x 5 ln (x + 1) 2 x - 4 = = x + 1 x 4 x 5 ln 2 (x + 1 ) 2 x - 4 - sin x ln x + 1 x 5 ln 2 (x + 1) (2 x - 4) = = x + 1 x 4 - sin x ln (x + 1) x 5 ln 2 (x + 1) (2 x - 4) = x x 4 + x 4 - sin x ln (x + 1) x 5 ln 2 (x + 1) (2 x - 4) )
3. This example makes sense when working with denominators of fractions. It is necessary to apply the formulas for the difference of squares and the square of the sum, since they will make it possible to pass to an expression of the form 1 cos x - x · cos x + x + 1 (cos x + x) 2 . It can be seen that the fractions are reduced to a common denominator. We get that cos x - x cos x + x 2 .

Then we get that

1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x = = 1 cos x - x cos x + x + 1 cos x + x 2 = = cos x + x cos x - x cos x + x 2 + cos x - x cos x - x cos x + x 2 = = cos x + x + cos x - x cos x - x cos x + x 2 = 2 cos x cos x - x cos x + x2

Answer:

1) x 3 + 1 x 7 + 2 2 = x x 7 + 2 2 x + 3 3 x 7 + 2 2, 2) x + 1 x ln 2 (x + 1) 2 x - 4 - sin x x 5 ln (x + 1) 2 x - 4 = = x x 4 + x 4 - sin x ln (x + 1) x 5 ln 2 (x + 1) ( 2 x - 4) , 3) ​​1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x = 2 cos x cos x - x cos x + x 2 .

Examples of multiplying fractions with variables

When multiplying fractions, the numerator is multiplied by the numerator and the denominator by the denominator. Then you can apply the reduction property.

Example 8

Multiply fractions x + 2 x x 2 ln x 2 ln x + 1 and 3 x 2 1 3 x + 1 - 2 sin 2 x - x.

Solution

You need to do the multiplication. We get that

x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x) = = x - 2 x 3 x 2 1 3 x + 1 - 2 x 2 ln x 2 ln x + 1 sin (2 x - x)

The number 3 is transferred to the first place for the convenience of calculations, and you can reduce the fraction by x 2, then we get an expression of the form

3 x - 2 x x 1 3 x + 1 - 2 ln x 2 ln x + 1 sin (2 x - x)

Answer: x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x) = 3 x - 2 x x 1 3 x + 1 - 2 ln x 2 ln x + 1 sin (2 x - x) .

Division

Division of fractions is similar to multiplication, since the first fraction is multiplied by the second reciprocal. If we take, for example, the fraction x + 2 x x 2 ln x 2 ln x + 1 and divide by 3 x 2 1 3 x + 1 - 2 sin 2 x - x, then this can be written as

x + 2 x x 2 ln x 2 ln x + 1: 3 x 2 1 3 x + 1 - 2 sin (2 x - x) , then replace with a product of the form x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x)

Exponentiation

Let's move on to consider the action with fractions of a general form with exponentiation. If there is a degree with a natural exponent, then the action is considered as a multiplication of identical fractions. But it is recommended to use a general approach based on the properties of powers. Any expressions A and C, where C is not identically equal to zero, and any real r on the ODZ for an expression of the form A C r, the equality A C r = A r C r is true. The result is a fraction raised to a power. For example, consider:

x 0 , 7 - π ln 3 x - 2 - 5 x + 1 2 , 5 = = x 0 , 7 - π ln 3 x - 2 - 5 2 , 5 x + 1 2 , 5

The order of operations with fractions

Actions on fractions are performed according to certain rules. In practice, we notice that an expression can contain several fractions or fractional expressions. Then it is necessary to perform all actions in a strict order: raise to a power, multiply, divide, then add and subtract. If there are brackets, the first action is performed in them.

Example 9

Calculate 1 - x cos x - 1 c o s x · 1 + 1 x .

Solution

Since we have the same denominator, then 1 - x cos x and 1 c o s x , but it is impossible to subtract according to the rule, first the actions in brackets are performed, then the multiplication, and then the addition. Then, when calculating, we get that

1 + 1 x = 1 1 + 1 x = x x + 1 x = x + 1 x

When substituting the expression into the original one, we get that 1 - x cos x - 1 cos x · x + 1 x. When multiplying fractions, we have: 1 cos x x + 1 x = x + 1 cos x x . Having made all the substitutions, we get 1 - x cos x - x + 1 cos x · x . Now you need to work with fractions that have different denominators. We get:

x 1 - x cos x x - x + 1 cos x x = x 1 - x - 1 + x cos x x = = x - x - x - 1 cos x x = - x + 1 cos x x

Answer: 1 - x cos x - 1 c o s x 1 + 1 x = - x + 1 cos x x .

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Instruction

Reduction to a common denominator.

Let fractions a/b and c/d be given.

The numerator and denominator of the first fraction is multiplied by LCM / b

The numerator and denominator of the second fraction is multiplied by LCM/d

An example is shown in the figure.

To compare fractions, they need to have a common denominator, then compare the numerators. For example, 3/4< 4/5, см. .

Addition and subtraction of fractions.

To find the sum of two ordinary fractions, they must be reduced to a common denominator, and then add the numerators, the denominator is unchanged. An example of adding fractions 1/2 and 1/3 is shown in the figure.

The difference of fractions is found in a similar way, after finding the common denominator, the numerators of the fractions are subtracted, see the figure.

When multiplying ordinary fractions, the numerators and denominators are multiplied together.

In order to divide two fractions, you need a fraction of the second fraction, i.e. change its numerator and denominator, and then multiply the resulting fractions.

Related videos

Sources:

• fractions grade 5 by example
• Basic tasks for fractions

Module represents the absolute value of the expression. Parentheses are used to designate a module. The values ​​contained in them are taken modulo. The solution of the module is to open parentheses according to certain rules and find the set of values ​​of the expression. In most cases, a module is expanded in such a way that the submodule expression takes on a series of positive and negative values, including zero. Based on these properties of the module, further equations and inequalities of the original expression are compiled and solved.

Instruction

Write down the original equation with . For it, open the module. Consider each submodule expression. Determine at what value of the unknown quantities included in it, the expression in modular brackets vanishes.

To do this, equate the submodule expression to zero and find the resulting equation. Write down the found values. In the same way, determine the values ​​of the unknown variable for each modulus in the given equation.

Draw a number line and plot the resulting values ​​on it. The values ​​of the variable in the zero module will serve as constraints in solving the modular equation.

In the original equation, you need to open the modular ones, changing the sign so that the values ​​of the variable correspond to those displayed on the number line. Solve the resulting equation. Check the found value of the variable against the restriction specified by the module. If the solution satisfies the condition, it is true. Roots that do not satisfy the restrictions should be discarded.

Similarly, expand the modules of the original expression, taking into account the sign, and calculate the roots of the resulting equation. Write down all the obtained roots that satisfy the constraint inequalities.

Fractional numbers allow you to express the exact value of a quantity in different ways. With fractions, you can perform the same mathematical operations as with integers: subtraction, addition, multiplication, and division. To learn how to decide fractions, it is necessary to remember some of their features. They depend on the type fractions, the presence of an integer part, a common denominator. Some arithmetic operations after execution require reduction of the fractional part of the result.

You will need

• - calculator

Instruction

Look carefully at the numbers. If there are decimals and irregulars among the fractions, it is sometimes more convenient to first perform actions with decimals, and then convert them to the wrong form. Can you translate fractions in this form initially, writing the value after the decimal point in the numerator and putting 10 in the denominator. If necessary, reduce the fraction by dividing the numbers above and below by one divisor. Fractions in which the whole part stands out, lead to the wrong form by multiplying it by the denominator and adding the numerator to the result. This value will become the new numerator fractions. To extract the whole part from the initially incorrect fractions, divide the numerator by the denominator. Write the whole result from fractions. And the remainder of the division becomes the new numerator, the denominator fractions while not changing. For fractions with an integer part, it is possible to perform actions separately, first for the integer and then for the fractional parts. For example, the sum of 1 2/3 and 2 ¾ can be calculated:
- Converting fractions to the wrong form:
- 1 2/3 + 2 ¾ = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12;
- Summation separately of integer and fractional parts of terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 + (8/12 + 9/12) = 3 + 17/12 = 3 + 1 5/12 = 4 5 /12.

For with values ​​below the line, find the common denominator. For example, for 5/9 and 7/12, the common denominator will be 36. For this, the numerator and denominator of the first fractions you need to multiply by 4 (it will turn out 28/36), and the second - by 3 (it will turn out 15/36). Now you can do the calculations.

If you are going to calculate the sum or difference of fractions, first write down the found common denominator under the line. Perform the necessary actions between the numerators, and write the result above the new line fractions. Thus, the new numerator will be the difference or the sum of the numerators of the original fractions.

To calculate the product of fractions, multiply the numerators of the fractions and write the result in place of the numerator of the final fractions. Do the same for the denominators. When dividing one fractions write one fraction on the other, and then multiply its numerator by the denominator of the second. At the same time, the denominator of the first fractions multiplied accordingly by the numerator of the second. At the same time, a kind of reversal of the second fractions(divider). The final fraction will be from the results of multiplying the numerators and denominators of both fractions. Easy to learn fractions, written in the condition in the form of a "four-story" fractions. If it separates two fractions, rewrite them with a ":" delimiter, and continue with normal division.

To get the final result, reduce the resulting fraction by dividing the numerator and denominator by one whole number, the largest possible in this case. In this case, there must be integer numbers above and below the line.

note

Don't do arithmetic with fractions that have different denominators. Choose a number such that when the numerator and denominator of each fraction are multiplied by it, as a result, the denominators of both fractions are equal.

Useful advice

When writing fractional numbers, the dividend is written above the line. This quantity is referred to as the numerator of a fraction. Under the line, the divisor, or denominator, of the fraction is written. For example, one and a half kilograms of rice in the form of a fraction will be written as follows: 1 ½ kg of rice. If the denominator of a fraction is 10, it is called a decimal fraction. In this case, the numerator (dividend) is written to the right of the whole part separated by a comma: 1.5 kg of rice. For the convenience of calculations, such a fraction can always be written in the wrong form: 1 2/10 kg of potatoes. To simplify, you can reduce the numerator and denominator values ​​by dividing them by a single whole number. In this example, dividing by 2 is possible. The result is 1 1/5 kg of potatoes. Make sure that the numbers you are going to do arithmetic with are in the same form.

Instruction

Click once on the "Insert" menu item, then select the "Symbol" item. This is one of the easiest ways to insert fractions to text. It consists in the following. The set of ready characters has fractions. Their number is usually small, but if you need to write ½, not 1/2 in the text, then this option will be the most optimal for you. In addition, the number of fraction characters may depend on the font. For example, for the Times New Roman font, there are slightly fewer fractions than for the same Arial. Vary fonts to find the best option when it comes to simple expressions.

Click on the menu item "Insert" and select the sub-item "Object". You will see a window with a list of possible objects to insert. Choose among them Microsoft Equation 3.0. This app will help you type fractions. And not only fractions, but also complex mathematical expressions containing various trigonometric functions and other elements. Double-click on this object with the left mouse button. You will see a window containing many symbols.

To print a fraction, select the symbol representing a fraction with an empty numerator and denominator. Click on it once with the left mouse button. An additional menu will appear, specifying the scheme of the fractions. There may be several options. Choose the most suitable for you and click on it once with the left mouse button.

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 decimals. I recommend to watch the whole and study 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 they are 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:

And if the difference of two mixed fractions is given and the numerator of the first fraction is less than the numerator of the second? It can also be done in two ways.

Examples (3):

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

* Divided into integer and fractional parts, got three, then presented 3 as the sum of 2 and 1, with the unit 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) the unit and present it as a fraction with the denominator we need, then from this fraction we can already subtract another.

Another example:

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

Above, we looked at examples with fractions that have equal denominators. What if the denominators differ? 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 the fraction is used.

Consider simple examples:

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

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

That is, immediately when “evaluating” the fraction, you need to figure out whether such an approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divided, then we perform the transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.

Now look at these examples:

This approach does not apply to them. There are other ways to reduce fractions to a common denominator, consider them.

Method SECOND.

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 of adding timid with equal denominators.

Example:

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

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? This 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, 30, 60, 90 are divisible by them .... Least 30. Question - how to determine this least common multiple?

There is a clear algorithm, but often this 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), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, such as 51 and 119.

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

- decompose each of the numbers into SIMPLE factors

- write out the decomposition of the BIGGER of them

- multiply it by the MISSING factors of other numbers

Consider examples:

50 and 60 50 = 2∙5∙5 60 = 2∙2∙3∙5

in the expansion of a larger number, one five is missing

=> LCM(50,60) = 2∙2∙3∙5∙5 = 300

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

in the expansion of a larger number, two and three are missing

=> LCM(48,72) = 2∙2∙2∙2∙3∙3 = 144

* The least common multiple of two prime numbers 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 reduce the resulting fraction? Yes, you can, but it's not always convenient. See what the denominator will be for the numbers 48 and 72 if you simply multiply them 48∙72 = 3456. Agree that it is more pleasant to work with smaller numbers.

Consider examples:

*51 = 3∙17 119 = 7∙17

in the expansion of a larger number, a triple is missing

=> LCM(51,119) = 3∙7∙17

And now we apply the first method:

* Look at the difference 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 got needs to be reduced. Finding the LCM simplifies the work considerably.

More examples:

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

TOTAL! GENERAL CALCULATION ALGORITHM!

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

- we bring the fractions to a common denominator (first we look to see if one denominator is divisible by another, if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act using 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: