Different acts with fractions can be performed, for example, the addition of fractions. Addition of fractions can be divided into several types. In each form of the fraction of the fractions of its rules and the algorithm of action. Consider in detail every type of addition.

Addition of fractions with the same denominators.

On the example, let's see how to fold a fraction with a common denominator.

Tourists went on a hike from point A to point E. On the first day, they passed from point A to B or \\ (\\ FRAC (1) (5) \\) from the entire path. On the second day, they passed from point B to D or \\ (\\ FRAC (2) (5) \\) from the entire path. What distance did they go from the beginning of the way to point d?

To find the distance from point A to point D you need to add fractions \\ (\\ FRAC (1) (5) + \\ FRAC (2) (5) \\).

The addition of fractions with the same denominators is that the number of these frains are needed, and the denominator will remain the same.

\\ (\\ FRAC (1) (5) + \\ FRAC (2) (5) \u003d \\ FRAC (1 + 2) (5) \u003d \\ FRAC (3) (5) \\)

In the alleged form, the amount of fractions with the same denominants will look like this:

\\ (\\ BF \\ FRAC (A) (C) + \\ FRAC (B) (C) \u003d \\ FRAC (A + B) (C) \\)

Answer: Tourists passed \\ (\\ FRAC (3) (5) \\) the entire path.

Addition of fractions with different denominators.

Consider an example:

It is necessary to add two fractions \\ (\\ FRAC (3) (4) \\) and \\ (\\ FRAC (2) (7) \\).

To fold the fractions with different denominators, you must first findAnd then take advantage of the rules for fractions with the same denominators.

For denominators 4 and 7, the total denominator will be number 28. The first fraction \\ (\\ FRAC (3) (4) \\) must be multiplied by 7. The second fraction \\ (\\ FRAC (2) (7) \\) must be multiplied by 4.

\\ (\\ FRAC (3) (4) + \\ FRAC (2) (7) \u003d \\ FRAC (3 \\ Times \\ Color (Red) (7) + 2 \\ Times \\ Color (Red) (4)) (4 \\ In alpusaly, we get such a formula:

\\ (\\ BF \\ FRAC (A) (B) + \\ FRAC (C) (D) \u003d \\ FRAC (A \\ Times D + C \\ Times B) (B \\ Times D) \\)

Addition of mixed numbers or mixed fractions.

Addition occurs under the law of addition.

In mixed fractions we fold entire parts with integers and fractional parts with fractional.

If fractional parts of mixed numbers have the same denominators, the numerals are folded, and the denominator remains the same.

Mixed mixed numbers \\ (3 \\ FRAC (6) (11) \\) and \\ (1 \\ FRAC (3) (11) \\).

\\ (3 \\ FRAC (6) (11) + 1 \\ FRAC (3) (11) \u003d (\\ Color (RED) (3) + \\ Color (Blue) (\\ FRAC (6) (11))) + ( \\ Color (Red) (1) + \\ Color (Blue) (\\ FRAC (3) (11)) \u003d (\\ Color (RED) (3) + \\ Color (Red) (1) + (\\ Color ( Blue) (\\ FRAC (6) (11)) + \\ Color (Blue) (\\ FRAC (3) (11))) \u003d \\ Color (Red) (4) + (\\ Color (BLUE) (\\ FRAC (6 + 3) (11))) \u003d \\ Color (Red) (4) + \\ Color (BLUE) (\\ FRAC (9) (11)) \u003d \\ Color (Red) (4) \\ Color (Blue) (\\ FRAC (9) (11)) \\)

If fractional parts of mixed numbers have different denominators, then we find a common denominator.

Perform the addition of mixed numbers \\ (7 \\ FRAC (1) (8) \\) and \\ (2 \\ FRAC (1) (6) \\).

The denominator is different, so it is necessary to find a common denominator, it is equal to 24. Multiply the first fraction \\ (7 \\ FRAC (1) (8) \\) to an additional factor 3, and the second fraction \\ (2 \\ FRAC (1) (6) \\) on 4.

\\ (7 \\ FRAC (1) (8) + 2 \\ FRAC (1) (6) \u003d 7 \\ FRAC (1 \\ Times \\ Color (Red) (3)) (8 \\ Times \\ Color (RED) (3) ) \u003d 2 \\ FRAC (1 \\ Times \\ Color (Red) (4)) (6 \\ Times \\ Color (Red) (4)) \u003d 7 \\ FRAC (3) (24) + 2 \\ FRAC (4) (24 ) \u003d 9 \\ FRAC (7) (24) \\)

Questions on the topic:
How to fold fractions?
Answer: First you need to decide which type of expression: the fractions have the same denominators, different denominators or mixed fractions. Depending on the type of expression, we turn to the solution algorithm.

How to solve a fraction with different denominators?
Answer: It is necessary to find a common denominator, and then according to the rule of fraction with the same denominators.

How to solve mixed fractions?
Answer: We fold entire parts with integers and fractional parts with fractional.

Example number 1:
Can the amount of two as a result of obtaining the right fraction? Wrong fraction? Give examples.

\\ (\\ FRAC (2) (7) + \\ FRAC (3) (7) \u003d \\ FRAC (2 + 3) (7) \u003d \\ FRAC (5) (7) \\)

The fraction \\ (\\ FRAC (5) (7) \\) is the correct fraction, it is the result of the sum of two correct fractions \\ (\\ FRAC (2) (7) \\) and \\ (\\ FRAC (3) (7) \\).

\\ (\\ FRAC (2) (5) + \\ FRAC (8) (9) \u003d \\ FRAC (2 \\ Times 9 + 8 \\ Times 5) (5 \\ TIMES 9) \u003d \\ FRAC (18 + 40) (45) \u003d \\ FRAC (58) (45) \\)

Fraction \\ (\\ FRAC (58) (45) \\) is incorrect fractionsIt turned out as a result of the sum of the correct fractions \\ (\\ FRAC (2) (5) \\) and \\ (\\ FRAC (8) (9) \\).

Answer: On both questions, the answer is yes.

Example number 2:
Fold the fractions: a) \\ (\\ FRAC (3) (11) + \\ FRAC (5) (11) \\) b) \\ (\\ FRAC (1) (3) + \\ FRAC (2) (9) \\).

a) \\ (\\ FRAC (3) (11) + \\ FRAC (5) (11) \u003d \\ FRAC (3 + 5) (11) \u003d \\ FRAC (8) (11) \\)

b) \\ (\\ FRAC (1) (3) + \\ FRAC (2) (9) \u003d \\ FRAC (1 \\ Times \\ Color (Red) (3)) (3 \\ Times \\ Color (RED) (3)) + \\ FRAC (2) (9) \u003d \\ FRAC (3) (9) + \\ FRAC (2) (9) \u003d \\ FRAC (5) (9) \\)

Example number 3:
Write down mixed fraction in the form of the sum of the natural number and proper fraction: a) \\ (1 \\ FRAC (9) (47) \\) b) \\ (5 \\ FRAC (1) (3) \\)

a) \\ (1 \\ FRAC (9) (47) \u003d 1 + \\ FRAC (9) (47) \\)

b) \\ (5 \\ FRAC (1) (3) \u003d 5 + \\ FRAC (1) (3) \\)

Example number 4:
Calculate the amount: a) \\ (8 \\ FRAC (5) (7) + 2 \\ FRAC (1) (7) \\) b) \\ (2 \\ FRAC (9) (13) + \\ FRAC (2) (13) \\) B) \\ (7 \\ FRAC (2) (5) + 3 \\ FRAC (4) (15) \\)

a) \\ (8 \\ FRAC (5) (7) + 2 \\ FRAC (1) (7) \u003d (8 + 2) + (\\ FRAC (5) (7) + \\ FRAC (1) (7)) \u003d 10 + \\ FRAC (6) (7) \u003d 10 \\ FRAC (6) (7) \\)

b) \\ (2 \\ FRAC (9) (13) + \\ FRAC (2) (13) \u003d 2 + (\\ FRAC (9) (13) + \\ FRAC (2) (13)) \u003d 2 \\ FRAC (11 )(13) \\)

c) \\ (7 \\ FRAC (2) (5) + 3 \\ FRAC (4) (15) \u003d 7 \\ FRAC (2 \\ Times 3) (5 \\ Times 3) + 3 \\ FRAC (4) (15) \u003d 7 \\ FRAC (6) (15) + 3 \\ FRAC (4) (15) \u003d (7 + 3) + (\\ FRAC (6) (15) + \\ FRAC (4) (15)) \u003d 10 + \\ FRAC (10) (15) \u003d 10 \\ FRAC (10) (15) \u003d 10 \\ FRAC (2) (3) \\)

Task number 1:
For lunchs, eaten \\ (\\ FRAC (8) (11) \\) from the cake, and in the evening they were ate \\ (\\ FRAC (3) (11) \\). What do you think the cake completely eaten or not?

Decision:
An denominator of the fraction is 11, it indicates how many parts divided the cake. At lunch, there were 8 pieces of cake from 11. For dinner, 3 pieces of cake from 11 were eaten. Moving 8 + 3 \u003d 11, ate pieces of cake from 11, that is, the entire cake.

\\ (\\ FRAC (8) (11) + \\ FRAC (3) (11) \u003d \\ FRAC (11) (11) \u003d 1 \\)

Answer: All cake ate.

To the fraction to add an integer, it is enough to perform a number of actions, and rather counting.

For example, you have 7 - an integer, it must be added to the fraction 1/2.

We act as follows:

• 7 Multiply to the denominator (2), it turns out 14,
• to 14 add the upper part (1), 15,
• and we substitute the denominator.
• as a result, it turns out 15/2.

In such a simple way, you can add integers to fractional.

And in order to allocate an integer from the fraction, it is necessary to divide the numerator to the denominator, and the residue - and there will be a fraction.

The addition operation to the correct ordinary fraction of an integer is not difficult and sometimes consists of simply in the formation of a mixed fraction in which whole part It is put to the left of the fractional part, for example, such a fraction will be mixed:

However, more likely, when adding a fraction of an integer, an incorrect fraction is obtained, which the numerator turns out to be greater than the denominator. This operation is performed like this: an integer is represented as an incorrect fraction with the same denominator as the adaptable fraction and then simply fold the numerals of both fractions. For example, it will look like this:

5+1/8 = 5*8/8+1/8 = 40/8+1/8 = 41/8

In my opinion it is very simple.

For example, we have fraction 1/4 (this is the same as 0.25, that is, a quarter of an integer).

And to this quarter, you can add any integer, for example. three with a quarter:

3.25. Or in the fraraty it is expressed as: 3 1/4

Here, according to the sample of this example, any fractions with any integers can be folded.

You need to build an integer in the fraction with the denominator 10 (6/10). Next, bring the existing fraction to common denominator 10 (35 \u003d 610). Well, and perform the operation as with conventional fractions 610 + 610 \u003d 1210 TOTAL 12.

It can be done in two ways.

one). The fraction can be translated into an integer and make addition. For example, 1/2 is 0.5; 1/4 equals 0.25; 2/5 is 0.4 and so on.

We take an integer 5, to which you need to add a fraction 4/5. We transform the fraction: 4/5 is 4 divided by 5 and get 0.8. Adds 0.8 to 5 and we get 5.8 or 5 4/5.

2). The second method: 5 + 4/5 \u003d 29/5 \u003d 5 4/5.

Addition of fractions Simple mathematical action, example, you need to fold an integer 3 and fraction 1/7. To fold these two numbers, you must have one denominator, so you must multiply on seven and divide on this figure, then you get 21/7 + 1/7, the denominator is one, fold 21 and 1, the answer is 22/7 .

Just take and add an integer to this fraction. It is necessary to 6 + 1/2 \u003d 6 1/2. Well, if it is a decimal fraction, it can be for example so 6 + 1.2 \u003d 7.2.

To fold the fraction and an integer, you need to add fractional and write them to an integer, in the form of a complex number, for example, with the addition of an ordinary fraction with an integer, we obtain: 1/2 +3 \u003d 3 1/2; When adding decimal fractions: 0.5 +3 \u003d 3.5.

The fraction itself is not an integer, by the fact that it does not reach it, but therefore it is not necessary to translate an integer in this fraction. Therefore, the whole number remains as a whole and fully demonstrates the full denomination, and the fraction of it plots, and shows how much this whole number is not enough before adding the next full score.

Academic example.

10 + 7/3 \u003d 10 integers and 7/3.

Unless of course there are whole, then they are summed up with integers.

12 + 5 7/9 \u003d 17 and 7/9.

It depends on what an integer and what fraction.

If a both allegations positive, You should assign this fraction to an integer. It turns out a mixed number. Moreover, there may be 2 cases.

Case 1.

• The crushing is correct, i.e. Number less denominator. Then the mixed number received after the attribution and will be the answer.

4/9 + 10 \u003d 10 4/9 (ten whole four ninth).



Case 2.

• The fraction is wrong, i.e. Numerator more denominator. Then a small transformation is required. Incorrect fraction should be turned into a mixed number, in other words, allocate the whole part. This is done like this:



After that, to an integer, you need to add a whole part of the incorrect fraction and to the resulting amount to attribute its fractional part. In the same way, an integer is added to the mixed number.

1) 11/4 + 5 \u003d 2 3/4 + 5 \u003d 7 3/4 (7 three fourth).

2) 5 1/2 + 6 \u003d 11 1/2 (11 integer one second).

If one of the components or both negative, add additions by the rules of addition of numbers with different or identical signs. An integer is represented as the ratio of this number and 1, and then the numerator, and the denominator is multiplied by the number equal to the denominator of the fraction, to which an integer is added.

3) 1/5 + (-2) \u003d 1/5 + -2/1 \u003d 1/5 + -10/5 \u003d -9/5 \u003d -1 4/5 (minus 1 whole four fifters).

4) -13/3 + (-4) \u003d -13/3 + -4/1 \u003d -13/3 + -12/3 \u003d -25/3 \u003d -8 1/3 (minus 8 of integer one third).

Comment.

After dating S. negative numbers, when studying actions with them, grades grade 6 should understand that a negative fraction add a positive integer the same thing that deduct fraction from a natural number. This action is known to be done like this:



In fact, in order to make a fraction and a whole number, you just need to simply bring the existing integer to fractional, and it is simpler simple. You just need to take a denomoter (existing in the example) and make it a number of integer, multiplying it on this denominator and dividing it, here is an example:

2+2/3 = 2*3/3+2/3 = 6/3+2/3 = 8/3

Your child brought homework From school, and you do not know how to solve it? Then this mini lesson for you!

How to fold decimal fractions

Decimals more conveniently folded in a column. To perform addition decimal fractions, you need to stick to one simple rule:

• The discharge must be under the discharge, the comma dilated.

As you see on the example, entire units are in each other, the discharge of the tenths and hundredths is located in each other. Now we add numbers, not paying attention to the comma. What to do with a comma? The comma is transferred to the place where it was in the discharge of integers.

Addition of fractions with equal denominators

To accumulate with a common denominator, it is necessary to save the denominator without changing, find the amount of the numerals and get a fraction that will be a total amount.

Addition of fractions with different denominators by finding a common multiple

The first thing to pay attention is to the denominators. Disclaimers are different, do not share one another, are whether simple numbers. To begin with, we need to lead to one common denominator, for this there are several ways:

• 1/3 + 3/4 \u003d 13/12, To solve this example, we need to find the smallest common multiple number (NOC), which will be divided into 2 denominator. To indicate the smallest multiple number A and B - NOC (A; B). In this example, the NOC (3; 4) \u003d 12. Check: 12: 3 \u003d 4; 12: 4 \u003d 3.
• I turn the multipliers and add the numbers obtained, we get 13/12 - the wrong fraction.

• In order to translate the wrong fraction in the correct, divide the numerator to the denominator, we obtain an integer 1, the residue 1 is a numerator and 12 - denominator.

Addition of fractions by multiplication cross on a cross

For the folding of fractions with different denominants, there is another way according to the formula "Cross to the Cross". This is a guaranteed way to level the denominators, for this you need numerals multiply with an denomoter of one fraction and back. If you are only at the initial stage of learning fractions, this method is the easiest and accurate, how to get a sure result when adding fractions with different denominators.

§ 87. Addition of fractions.

Addition of fractions has a lot of similarities with the addition of integers. Addition of fractions There is an action that consists in the fact that several data numbers (terms) are connected in one number (amount) containing all units and shares of the components of the components.

We will consistently consider three cases:

1. Addition of fractions with the same denominators.
2. Addition of fractions with different denominators.
3. Addition of mixed numbers.

1. Addition of fractions with the same denominators.

Consider an example: 1/5 + 2/5.

We take the segment AB (Fig. 17), we will take it per unit and divide into 5 equal parts, then part of the speakers of this segment will be equal to 1/5 of the segment AB, and part of the same CD segment will be 2/5 AB.

From the drawing it can be seen that if you take a section of AD, it will be equal to 3/5 AV; But the segment AD is just the sum of the segments of the AC and CD. So you can write:

1 / 5 + 2 / 5 = 3 / 5

Considering the data of the components and the amount received, we see that the amount of the amount turned out from the addition of the number of components, and the denominator remained unchanged.

From here we get the following rule: to fold the fractions with the same denominators, it is necessary to fold their numerals and leave the same denominator.

Consider an example:

2. Addition of fractions with different denominators.

Folding the fractions: 3/4 + 3/8 Previously need to lead to the smallest common denominator:

Intermediate link 6/8 + 3/8 could not be written; We wrote it here for greater clarity.

Thus, to fold the fractions with different denominators, you must first lead them to the smallest common denominator, fold their numerals and sign a common denominator.

Consider an example (additional multipliers will write over the appropriate fractions):

3. Addition of mixed numbers.

Moving the numbers: 2 3/8 + 3 5/6.

We first give fractional parts of our numbers to a common denominator and rewrite them again:

Now add conscientious and fractional parts:

§ 88. Subtraction of fractions.

Subtraction of fractions is determined in the same way as the subtraction of integers. This is the action with which this sum of the two components and one of them finds the other term. Consider successively three cases:

1. Subtraction of fractions with the same denominators.
2. Subtraction of fractions with different denominators.
3. Subtraction of mixed numbers.

1. Subtraction of fractions with the same denominators.

Consider an example:

13 / 15 - 4 / 15

Take the segment AB (Fig. 18), we will take it for a unit and divide into 15 equal parts; Then part of the speakers of this segment will be 1/15 from AB, and part of the AD of the same segment will correspond to 13/15 AB. I will postpone another segment ED equal to 4/15 AB.

We need to subtract out of 13/15 fraction 4/15. In the drawing, it means that from the segment AD, you need to take away the segment ED. As a result, it will remain a segment AE, which is 9/15 segment AB. So we can write:

The example made by us shows that the difference numerator turned out from subtracting the numerators, and the denominator remained the same.

Therefore, to make the subtraction of fractions with the same denominators, the needed the numerator submitted from the numerator of the reduced and leave the former denominator.

2. Subtraction of fractions with different denominators.

Example. 3/4 - 5/8

Pre-give these fractions to the smallest general denominator:

Intermediate link 6/8 - 5/8 is written here for greater clarity, but you can continue to skip it.

Thus, in order to subtract fraction from the fraction, you must first lead them to the smallest common denominator, then from the numerator of the reduced deductible numerator subtractable and under their difference to sign the general denominator.

Consider an example:

3. Subtraction of mixed numbers.

Example. 10 3/4 - 7 2/3.

We give fractional parts of the reduced and submitted to the smallest general denominator:

We have deducted a whole of the whole and fraction from the fraction. But there are cases when the fractional part of the subtracted part of the fractional part is reduced. In such cases, it is necessary to take one unit from the integer part of the reduced, to crush it into the shares, in which fractional part is pronounced, and add to the fractional part of the decreased. And then subtraction will be performed in the same way as in the previous example:

§ 89. Multiplication of fractions.

When studying multiplication of fractions, we will consider the following questions:

1. Multiplying the fraction for an integer.
2. Finding the fraction of this number.
3. Multiplying an integer on the fraction.
4. Multiplication of the fraction on the fraction.
5. Multiplying mixed numbers.
6. Concept of interest.
7. Finding percent of this number. Consider them consistently.

1. Multiplying the fraction for an integer.

The multiplication of the fraction for an integer is the same meaning as the multiplication of an integer one. Multiply the fraction (multiplier) to an integer (multiplier) - it means to draw up the amount of the same terms, in which each term equal to the multiplier, and the number of components is equal to the factor.

So, if you need 1/9 to multiply by 7, then this can be done like this:

We easily obtained the result, since the action was made to the addition of fractions with the same denominators. Hence,

Consideration of this action shows that the multiplication of the fraction for an integer is equivalent to an increase in this fraction at as many times as the number is contained in a number of numbers. And since the increase in the fraction is achieved or by increasing its number

or by reducing its denominator , we can either multiply the numerator to the whole, or divide the denominator to it if this division is possible.

From here we receive the rule:

To multiply the fraction for an integer, you need to multiply by the number of the numerator and leave the same denominator or, if possible, divide the denominator to this number, leaving the numerator without changing.

During multiplication, abbreviations are possible, for example:

2. Finding the fraction of this number.There are many tasks, when solving which you have to find, or calculate, part of this number. The difference between these tasks from others is that they are given a number of any items or units of measurement and is required to find a part of this number, which is also indicated by a certain fraction. To facilitate understanding, we first give examples of such tasks, and then introduce the way to solve them.

Task 1.I had 60 rubles; 1/3 of this money I spent on the purchase of books. How much did the book cost?

Task 2. The train must pass the distance between the cities of A and B, equal to 300 km. He has already passed 2/3 of this distance. How much is this kilometers?

Task 3.In the village of 400 houses, of which 3/4 bricks, the rest are wooden. How much is all brick houses?

Here are some of those numerous tasks to find a part of the number with which we have to meet. They are usually called tasks to find the fraction of this number.

Solution of problem 1. Out of 60 rubles. I spent on books 1/3; It means that for finding the cost of books you need a number 60 to divide by 3:

Solution of task 2.The meaning of the task is to find 2/3 from 300 km. I calculate first 1/3 from 300; This is achieved by dividing 300 km on 3:

300: 3 \u003d 100 (this is 1/3 of 300).

To find two thirds from 300, you need to enlargely enlarged twice, i.e. multiply by 2:

100 x 2 \u003d 200 (this is 2/3 from 300).

Task solution 3.Here you need to determine the number of brick houses that make up 3/4 from 400. Find 1/4 from 400 first,

400: 4 \u003d 100 (this is 1/4 from 400).

To calculate three quarters from 400, the received private need to be increased in three times, i.e. multiply by 3:

100 x 3 \u003d 300 (this is 3/4 from 400).

Based on solving these tasks, we can derive the following rule:

To find the fraction value from a given number, you need to divide this number to the denomoter of the fraction and the received private multiplied to its numerator.

3. Multiplying an integer on the fraction.

Previously (§ 26) It was found that multiplication of integers should be understood as the addition of the same terms (5 x 4 \u003d 5 + 5 + 5 + 5 \u003d 20). In this paragraph (paragraph 1), it was found that multiplying the fraction for an integer - this means finding the amount of the same terms equal to this fraction.

In both cases, multiplication was in finding the amount of the same terms.

Now we go to the multiplication of an integer on the fraction. Here we will meet with such, for example, multiplication: 9 2/3. It is obvious that the former definition of multiplication is not suitable for this case. This is seen from the fact that we cannot replace such multiplication with the addition of equal numbers.

By virtue of this, we will have to give a new definition of multiplication, i.e., in other words, to answer the question that you should intelligible under multiplication by fraction, as you need to understand this action.

The meaning of multiplying an integer for a fraction is found out of the following definition: multiply a whole number (multiplier) to the fraction (multiplier) - it means to find this fraction of the multiplier.

It is, to multiply 9 by 2/3 - it means to find 2/3 of nine units. In the previous paragraph, such tasks were solved; Therefore, it is easy to imagine that we will result in 6.

But now an interesting and important question arises: why are the various actions such at first glance, as finding the amount of equal numbers and finding a number of numbers, in arithmetic is called the same word "multiplication"?

It happens because the former action (the repetition of the number of several times) and a new action (finding a fracted number) give an answer to homogeneous questions. So we proceed here from the considerations that homogeneous questions or tasks are solved by the same action.

To understand this, consider the following task: "1 m Sukna costs 50 rubles. How much will it cost 4 m of such a cloth? "

This task is solved by multiplying the number of rubles (50) by the number of meters (4), i.e. 50 x 4 \u003d 200 (rub.).

Take the same task, but in it the amount of cloth will be expressed by a fractional number: "1 m Sukna costs 50 rubles. How much will it cost 3/4 m of such a cloth? "

This task also needs to be solved by multiplying the number of rubles (50) by the number of meters (3/4).

It is possible and several times, without changing the meaning of the problem, to change the numbers in it, for example, take 9/10 m or 2 3/10 m and so on.

Since these tasks have the same content and differ only in numbers, then we call the actions used in solving them, the same word - multiplication.

How is the multiplication of an integer on the fraction?

Take the numbers found in the last task:

According to the definition, we must find 3/4 of 50. We will first find 1/4 from 50, and then 3/4.

1/4 numbers 50 is 50/4;

3/4 numbers 50 make up.

Hence.

Consider another example: 12 5/8 \u003d?

1/8 numbers 12 is 12/8,

5/99 numbers 12 are made up.

Hence,

From here we receive the rule:

To multiply an integer on the fraction, you need to multiply the integer on the fluster numerator and this product is made by a numerator, and the denominator sign the denominator of this fraction.

We write this rule using letters:

To make this rule, it should be completely understood, it should be remembered that the fraction can be considered as a private. Therefore, the rule found is useful to compare with the rule of multiplication of the number on the private, which was set out in § 38

It must be remembered that before performing multiplication, you should do (if possible) abbreviation, eg:

4. Multiplication of the fraction on the fraction. The multiplication of the fraction on the fraction is the same meaning as the multiplication of an integer on the fraction, that is, when the fraction is multiplying, the fraction is necessary from the first fraction (multiplier) to find a fraction facing the multiplier.

It is, multiplying 3/4 to 1/2 (half) - it means to find half from 3/4.

How is the multiplication of the fraction on the fraction?

Take example: 3/4 multiply by 5/7. This means that you need to find 5/7 from 3/4. Find at first 1/7 from 3/4, and then 5/7

1/7 Numbers 3/4 will express:

5/7 numbers 3/4 are expressed like this:

In this way,

Another example: 5/8 multiply by 4/9.

1/9 Numbers 5/8 is

4/9 Numbers 5/8 are made up.

In this way,

From consideration of these examples, you can withdraw the following rule:

To multiply the fraction for the fraction, you need to multiply the numerator to the numerator, and the denominator is to the denominator and the first product to make a numerator, and the second is the denominator.

This rule B. general You can write like this:

When multiplying, it is necessary to do (if possible) reduction. Consider examples:

5. Multiplying mixed numbers. Since mixed numbers can easily be replaced by incorrect fractions, then this circumstance is usually used when multiplying mixed numbers. This means that in cases where the multiplier, or the multiplier, or both of the factory are expressed by mixed numbers, they are replaced by incorrect fractions. Move, for example, mixed numbers: 2 1/2 and 3 1/5. We turn each of them into the wrong fraction and then we will multiply the resulting fractions according to the rule of the fraction for the fraction:

Rule. In order to multiply the mixed numbers, you need to pre-turn them into the wrong fraction and then multiply by the rule of the fraction for the fraction.

Note. If one of the factors is an integer, then multiplication can be performed on the basis of the distribution law like this:

6. Concept of interest. When solving problems and when performing various practical calculations, we use all sorts of fractions. But it should be borne in mind that many values \u200b\u200ballow not any, but the natural divisions for them. For example, you can take one hundredth (1/100) of the ruble, it will be a penny, two hundredths are 2 cop., Three hundredths - 3 kopecks. You can take 1/10 rubles, it will be "10 kopecks, or a grivenk. You can take a quarter of the ruble, i.e. 25 kopecks, half of the ruble, i.e. 50 kopecks. (Filter). But practically do not take, for example , 2/7 rubles because the ruble on the seventh shares is not divided.

The unit measurement unit, i.e. kilogram, makes primarily decimal divisions, for example 1/10 kg, or 100 g. And such a lobe of a kilogram, as 1/6, 1/11, 1/11, are uncommon.

In general, our (metric) measures are decimal and admit decimal units.

However, it should be noted that it is extremely useful and convenient in a wide variety of cases to use the same (monotonous) method of division of values. Many years of experience He showed that such a well-justified division is the "hundreds-" division. Consider several examples related to the most diverse regions of human practice.

1. The price of books dropped to 12/100 former prices.

Example. The former price of the book is 10 rubles. She dropped on 1 ruble. 20 cop

2. Savings tickets are paid during the year to depositors of 2/100 amounts, which is put on the savings.

Example. On the cash desk, 500 rubles were laid, income from this amount per year is 10 rubles.

3. The number of graduates of one school amounted to 5/100 of the total number of students.

PRI MERS Only 1,200 students studied at school, of which 60 people graduated from school.

The hundredth of the number is called a percentage.

The word "percentage" is borrowed from the Latin language and its root "Cent" means a hundred. Together with the pretext (Pro Centum), this word denotes "for a hundred". The meaning of such an expression follows from the circumstance that initially in ancient Rome percentages were called money, who paid the debtor to the lender "for each hundred". The word "cent" hears in such all familiar words: centner (one hundred kilograms), the centimeter (Santimeter says).

For example, instead of saying that the plant for the past month gave marriage 1/100 from all the products developed by him, we will talk like this: the plant for the past month gave one percentage of marriage. Instead of talking: the plant has developed products for 4/8 more than the planned plan, we will say: the plant has exceeded 4 percent plan.

The above examples can be expressed otherwise:

1. The price of books decreased by 12 percent of the former price.

2. Savings cash offices pay depositors for a year 2 percent with the amount put on the savings.

3. The number of graduates of one school was 5 percent of the number of all school students.

To reduce the letter, it is accepted instead of the word "percentage" to write an icon%.

However, it is necessary to remember that in the calculations the% icon is usually not written, it can be recorded in the condition of the problem and in the final result. When performing computation, you need to write a fraction with a denominator 100 instead of an integer with this icon.

You need to be able to replace an integer with the specified bad character with a denominator 100:

Back, you need to get used to it instead of a fraction with a denominator 100 write an integer with the specified icon:

7. Finding percent of this number.

Task 1. The school received 200 cu. M Firewood, and birch firewood was 30%. How many birch firewood?

The meaning of this task is that the birch firewood was only part of those firewood that were taken to school, and this part is expressed by the fraction of 30/100. So, we have the task of finding the fraction from the number. To solve it, we must multiply by 30/100 (the tasks for finding a fraction of numbers are solved by multiplying the number by fraction.).

So, 30% of 200 equals 60.

Fraction 30/100, which occurred in this task, admits a reduction to 10. It would be possible to fulfill this reduction from the very beginning; The solution to the task would not change.

Task 2. There were 300 children in various ages in the camp. Children of the 11 years accounted for 21%, children of the 12 years accounted for 61% and, finally, 13-year-old children were 18%. How many children had each age in the camp?

In this task you need to perform three calculations, i.e., consistently find the number of children 11 years old, then 12 years and, finally, 13 years.

So, here it will be necessary to find the fraction three times. Let's do it:

1) How many children were 11 years old?

2) How many were 12-year-old children?

3) How many children were 13 years old?

After solving the problem, it is useful to fold the numbers found; The amount must be 300:

63 + 183 + 54 = 300

It should also be paid to the fact that the amount of interest, data in the condition of the problem is 100:

21% + 61% + 18% = 100%

This suggests that total number Children who were in the camp were taken 100%.

3 a d and h 3.The worker received 1,200 rubles per month. Of these, 65% he spent on food, 6% - on an apartment and heating, 4% on gas, electricity and radio, 10% - for cultural needs and 15% - savings. How much money is spent on the need specified in the task?

To solve this problem, you need 5 times to find the fraction from the number 1 200. We will do it.

1) How much money is spent on food? The task says that this consumption is 65% of the total earnings, i.e. 65/100 from the number 1 200. Make a calculation:

2) How much money is paid for an apartment with heating? Arguing like the previous one, we will come to the following calculation:

3) How much money was paid for gas, electricity and radio?

4) How much money is spent on cultural needs?

5) How much money is a worker saving?

To check it is useful to add numbers found in these 5 questions. The amount should be 1,200 rubles. All the earnings are accepted for 100%, which is easy to check by folding the number of interest, the data on the problem of the task.

We solved three tasks. Despite the fact that in these challenges it was about various things (delivery of firewood for school, the number of children of various ages, the cost of worker), they were solved by the same way. This happened because in all tasks it was necessary to find a few percent of these numbers.

§ 90. Division of fractions.

When studying dividing fractions, we will consider the following questions:

1. Delegation of a whole number.
2. Decision fraction for an integer
3. division of an integer on the fraction.
4. Dividing the fraction on the fraction.
5. division of mixed numbers.
6. Finding the number on this fraction.
7. Finding a number by its percentage.

Consider them consistently.

1. Delegation of a whole number.

As was indicated in the department of integers, the division is called the action that, according to this product, two nobles (divisible) and one of these factors (divider) is found another factory.

The division of an integer on the whole we considered in the department of integers. We met there two cases of divisions: division without a residue, or "alarm" (150: 10 \u003d 15), and division with the residue (100: 9 \u003d 11 and 1 in the residue). We can, therefore, to say that in the area of \u200b\u200bintegers, the exact division is not always possible, because the divisible is not always a piece of divider by an integer. After the introduction of multiplication by the fraction, we can have every case of dividing integers to be considered possible (only division to zero is eliminated).

For example, divided 7 by 12 - it means to find such a number, the product of which to 12 would be 7. Such a fraction is 7/12 because 7/12 12 \u003d 7. Another example: 14: 25 \u003d 14/25, because 14/25 25 \u003d 14.

Thus, to divide the integer on the whole, it is necessary to draw up a fraction, the numerator of which is equal to the division, and the denominator is a divider.

2. Dividing the fraction for an integer.

Split the shot 6/7 by 3. According to the above definition of division, we have a product (6/7) and one of the factors (3); It is required to find such a second factor, which from multiplication by 3 would give this work 6/7. Obviously, he should be three times less than this work. So, the task assigned to us was to reduce 6/7 fraction 3 times.

We already know that the decrease in the fraction can be performed or by reducing its numerator, or by increasing its denominator. Therefore, you can write:

In this case, the numerator 6 is divided into 3, so the numerator should be reduced by 3 times.

Take another example: 5/8 divided by 2. Here Nizer 5 is not divided by 2, it means that it will have to multiply the denominator:

Based on this, you can express the rule: to divide the fraction for an integer, you need to divide the smaller of the fraction (if possible), leaving the same denominator, or multiply by this number of the denomoter, leaving the same numerator.

3. division of an integer on the fraction.

Let it be required to divide 5 per 1/2, that is, to find such a number that, after multiplication by 1/2, will give a product 5. Obviously, this number must be greater than 5, since 1/2 is the correct fraction, but when multiplying the number For the correct fraction, the work should be less than the multiple. To make it clearer, we write our actions as follows: 5: 1/2 \u003d h. , So x 1/2 \u003d 5.

We have to find such a number h. which, being multiplied by 1/2 gave 5. Since multiplying a number of 1/2 is to find 1/2 of this number, then, therefore, 1/2 unknown number h. equal to 5, and all the numbers h. twice as much, that is, 5 2 \u003d 10.

Thus, 5: 1/2 \u003d 5 2 \u003d 10

Check:

Consider another example. Let it be required to divide 6 to 2/3. Let's try to first find the desired result using the drawing (Fig. 19).

Fig.19.

I will depict a segment AB, equal to 6 some units, and divide each unit into 3 equal parts. In each unit, three thirds (3/3) in the entire segment of the AV 6 times more, t. E. 18/3. Connect with small brackets 18 of the obtained segments of 2; It turns out only 9 segments. So, the fraction 2/3 is contained in b units of 9 times, or, in other words, a fraction 2/3 of 9 times less than 6 whole units. Hence,

How to get this result without a drawing with the help of only calculations? We will argue this: it takes 6 divided by 2/3, i.e. it is required to answer the question how many times 2/3 are contained in 6. We first find out: how many times 1/3 is contained in 6? In a whole unit - 3 thirds, and in 6 units - 6 times more, i.e. 18 of the third; To find this number, we must multiply on 3. So, 1/3 is contained in b units 18 times, and 2/3 are contained in b not 18 times, and twice as many times, i.e. 18: 2 \u003d 9. Consequently When dividing 6 to 2/3, we performed the following actions:

From here we receive the rule of division of an integer on the fraction. To divide an integer on the fraction, it is necessary to multiply this to the denominator of this fraction and, making this product with a numerator, divide it into the numerator of this fraction.

We write a rule with the help of letters:

To make this rule, it should be completely understood, it should be remembered that the fraction can be considered as a private. Therefore, the rule found is useful to compare with the rule of division of the number on the private, which was set out in § 38. Pay attention to the fact that there was the same formula.

During division, abbreviations are possible, for example:

4. Dividing the fraction on the fraction.

Let it be required to divide 3/4 by 3/8. What will indicate the number that will result in division? It will answer the question how many times the fraction 3/8 is contained in the fraction 3/4. To sort out this issue, make a drawing (Fig. 20).

Take the segment AB, we will take it per unit, divide into 4 equal parts and note 3 parts. A speakers will be equal to 3/4 segment AV. Now we divide each of the four initial segments in half, then the segment AV is divided into 8 equal parts and each such part will be equal to 1/8 of the segment AV. By connecting arcs of 3 of these segments, then each of the segments AD and DC will be equal to 3/8 segment AB. The drawing shows that the segment equal to 3/8 is contained in a segment of 3/4, exactly 2 times; So, the result of the division can be written as:

3 / 4: 3 / 8 = 2

Consider another example. Let it be required to divide 15/16 to 3/4:

We can argue like this: you need to find such a number that, after multiplication by 3/3, will give a product equal to 15/16. We write the calculations like this:

15 / 16: 3 / 32 = h.

3 / 32 H. = 15 / 16

3/7 of an unknown number h. make up 15/16

1/4 of an unknown number h. make up

32/32 numbers h. make up.

Hence,

Thus, in order to divide the fraction on the fraction, you need a numerator of the first fraction to multiply by the denominator, and the denominator of the first fraction is multiplied by the second and the first product to make a numerator, and the second is the denominator.

We write a rule using letters:

During division, abbreviations are possible, for example:

5. division of mixed numbers.

When dividing mixed numbers, they must be previously addressed to incorrect fractions, and then divide the fractions obtained by division rules fractional numbers. Consider an example:

Reverse mixed numbers in the wrong fraction:

Now we divide:

Thus, in order to divide the mixed numbers, you need to turn them into the wrong fraction and then divide by the rules of fraction.

6. Finding the number on this fraction.

Among the various tasks on the fraction sometimes there are those in which some fraction of an unknown number is given and it is required to find this number. This type of task will be inverse to the tasks to find the fraction of this number; There was a number there and it was necessary to find some fraction from this number, there is a fraction of the number and it is necessary to find this number itself. This thought will be even clearer if we turn to solve this type of tasks.

Task 1.On the first day, the glaziers glazed 50 windows, which is 1/3 of all windows of the built house. How many windows in this house?

Decision. The task says that the glaced 50 windows make up 1/3 of all windows at home, it means that the entire windows are 3 times more, that is,

The house had 150 windows.

Task 2. The store sold 1,500 kg of flour, which is 3/8 of the total reserve of flour that has been in the store. What was the initial stock of flour in the store?

Decision. From the condition of the problem, it can be seen that the flour sold 1,500 kg is 3/8 of the total stock; So, 1/8 of this stock will be 3 times less, i.e., it is necessary to reduce it for its calculation 3 times:

1 500: 3 \u003d 500 (this is 1/8 stock).

Obviously, the whole stock will be 8 times more. Hence,

500 8 \u003d 4 000 (kg).

The initial stock of flour in the store was equal to 4,000 kg.

From consideration of this task, you can withdraw the following rule.

To find, the number for this value of its fraction, it is enough to divide this value to the fluster numerator and the result multiply to the denomoter.

We solved two challenges to find the number of this fraction. Such objectives, as it is especially clearly seen from the latter, is solved by two actions: division (when you find one part) and multiplication (when you find the entire number).

However, after we studied the division of the frains, the above tasks can be solved by one action, namely: division into fraction.

For example, the last task can be solved by one action as follows:

In the future, the task of finding the number by its fraction we will solve in one action - division.

7. Finding a number by its percentage.

These tasks will need to find a number, knowing a few percent of this number.

Task 1. At the beginning of this year I received 60 rubles in the savings checkout. income with the amount put on me on saving a year ago. How much money did I put in the savings cashier? (Cashs give depositors of 2% income per year.)

The meaning of the task is that some amount of money was put on me in a savings office and lay there. After the year I received 60 rubles from it. income, which is 2/00 of the money I put. How much money did I put?

Consequently, knowing some of these money, expressed in two ways (in rubles and fravia), we must find the whole, as long as an unknown amount. This is an ordinary task to find the number of this fraction. These tasks are solved by division:

So, 3000 rubles were put in the savings office.

Task 2. Fishermen for two weeks completed a monthly plan by 64%, preparing 512 tons of fish. What was their plan?

From the condition of the problem it is known that the fishermen performed a part of the plan. This part is 512 tons, which is 64% of the plan. How many tons of fish need to prepare according to plan, we are unknown. In finding this number and will be solving the problem.

Such tasks are solved by division:

So, according to the plan you need to prepare 800 tons of fish.

Task 3.The train went from Riga to Moscow. When he passed the 276th kilometer, one of the passengers asked the passing conductor, which part of the way they had already drove. The conductor answered this: "30% of the entire path passed." What is the distance from Riga to Moscow?

From the condition of the task it is clear that 30% of Riga to Moscow is 276 km away. We need to find all the distance between these cities, i.e., on this part, find a whole:

§ 91. Mutually reverse numbers. Replacing division by multiplication.

We take a shot 2/3 and rearrange the numerator to the site of the denominator, it turns out 3/2. We got a fraction inverse this.

In order to obtain a fraction, the inverse one, it is necessary to put its numerator to the place of the denominator, and the denominator is on the square of the numerator. By this way, we can get a fraction inverse any fraction. For example:

3/4, inverse 4/3; 5/6, reverse 6/5

Two fractions possessing the property that the numerator first is the denominator of the second, and the denominator is the first number is called mutually reverse.

Now we think what kind of fraction will be reverse for 1/2. Obviously, it will be 2/1, or simply 2. I find out the fraction, inverse this, we got an integer. And this case is not a single; On the contrary, for all fractions with a numerator 1 (unit) inverse will be integers, for example:

1/3, inverse 3; 1/5, reverse 5

Since, when finding back fractions, we met with integers, then in the future we will not talk about the reverse frauds, but about the reverse numbers.

We find out how to write a number inverse to an integer. For fractions, it is solved simply: a denominator needs to put the number of the numerator. In this way, you can get the opposite number for an integer, since any whole number can be meant denominator 1. So, the number, inverse 7, will be 1/7, because 7 \u003d 7/1; For the number 10, the reverse will be 1/10, since 10 \u003d 10/1

This thought can be expressed differently: the number inverse to this number is obtained from dividing the unit to this number.. Such an assertion is fair not only for integers, but also for fractions. In fact, if you want to write a number, inverse fraction 5/9, then we can take 1 and divide it by 5/9, i.e.

Now we specify one property Mutually reverse numbers, which will be useful for us: the product of mutually reverse numbers is equal to one. Indeed:

Using this property, we can find the reverse numbers as follows. Let it be necessary to find the number inverse 8.

Denote by his letter h. , then 8. h. \u003d 1, from here h. \u003d 1/8. Find another number, inverse 7/12 denote by his letter h. , then 7/12 h. \u003d 1, from here h. \u003d 1: 7/12 or h. = 12 / 7 .

We introduced here a concept of mutually reverse numbers in order to increasing the division of fractions slightly.

When we divide the number 6 to 3/5, then we carry out the following actions:

Note special attention On the expression and compare it with the specified :.

If you take an expression separately, without connection with the previous one, then it is impossible to resolve the question from which it originated: from division 6 to 3/5 or from multiplication 6 to 5/3. In both cases, the same thing turns out. So we can say that the division of one number to another can be replaced by multiplying divide into the number, reverse divider.

Examples that we give below fully confirm this conclusion.

Calculator online.
Calculating expressions with numerical fractions.
Multiplication, subtraction, division, addition and reduction of fractions with different denominators.

With this calculator online you can multiply, subtract, divide, fold and reduce numerical fractions with different denominatives.

The program works with correct, incorrect and mixed numerical fractions.

This program (Calculator online) can:
- Perform the addition of mixed fractions with different denominators
- Perform a deduction of mixed fractions with different denominators
- Perform a division of mixed fractions with different denominators
- Perform multiplication of mixed fractions with different denominators
- bring a fraction to a common denominator
- convert mixed fractions to the wrong
- Reduce the fractions

You can also enter the expression with fractions, but one single fraction.
In this case, the fraction will be reduced and the whole part is highlighted from the result.

Online calculator To calculate expressions with numerical fractions, it does not simply give a task response, it leads a detailed solution with explanations, i.e. Displays the process of finding a solution.

This program may be useful as high school students. secondary schools When preparing K. control work and exams, when checking knowledge before the exam, parents to control the solution of many problems in mathematics and algebra. Or maybe you are too expensive to hire a tutor or buy new textbooks? Or you just want to make your homework in mathematics or algebra as possible? In this case, you can also use our programs with a detailed solution.

Thus, you can conduct your own training and / or training of your younger brothers or sisters, while the level of education in the field of solved tasks increases.

If you are not familiar with the rules of entering expressions with numerical fractions, we recommend familiarizing with them.

Rules for entering expressions with numerical fractions

Only an integer can act as a numerator, denominator and a whole part of the fraction.

The denominator cannot be negative.

When entering a numeric fraction, the numerator separated from the denominator to the fission mark: /
Input: -2/3 + 7/5
Result: \\ (- \\ FRAC (2) (3) + \\ FRAC (7) (5) \\)

The whole part is separated from the fraraty ampersand sign: &
Enter: -1 & 2/3 * 5 & 8/3
Result: \\ (- 1 \\ FRAC (2) (3) \\ CDOT 5 \\ FRAC (8) (3) \\)

The division of fractions is introduced by a colon sign ::
Enter: -9 & 37/12: -3 & 5/14
Result: \\ (- 9 \\ FRAC (37) (12): \\ left (-3 \\ FRAC (5) (14) \\ RIGHT) \\)
Remember that it is impossible to share on zero!

When entering expressions with numerical fractions, you can use brackets.
Input: -2/3 * (6&1/2-5/9) : 2&1/4 + 1/3
Result: \\ (- \\ FRAC (2) (3) \\ Cdot \\ left (6 \\ FRAC (1) (2) - \\ FRAC (5) (9) \\ RIGHT): 2 \\ FRAC (1) (4) + \\ FRAC (1) (3) \\)

Enter the expression with numeric fractions.

Calculate

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A bit of theory.

Ordinary fractions. Division with the rest

If we need to divide 497 to 4, then when dividing, we will see that 497 is not divided into 4 aimed, i.e. The balance remains from division. In such cases, they say that division with the restAnd the solution is written in this form:
497: 4 \u003d 124 (1 residue).

The components of division in the left part of equality are called the same as when dividing without a residue: 497 - dividend, 4 - divider. The result of division when dividing with the residue is called incomplete private. In our case, this is the number 124. And finally, the last component that is not in the usual division - residue. In cases where the residue is not, they say that one number was divided into another without residue, or aim. It is believed that with this division, the residue is zero. In our case, the residue is 1.

The residue is always less than the divider.

Verification check can be made multiplication. If, for example, there is equality 64: 32 \u003d 2, then the check can be done like this: 64 \u003d 32 * 2.

Often in cases where the division with the residue is performed, it is convenient to use equality
a \u003d b * n + r,
Where a is divisible, B - divider, n - incomplete private, R is the residue.

Private from division natural numbers You can record in the form of a fraction.

The fraction is divisible, and the denominator is a divider.

Since the fraction numerator is divisible, and the denominator is a divider, believe that the trait fraction means the action of the division. Sometimes it is convenient to record division in the form of a fraction, without using the ":" sign.

The private from the division of natural numbers M and N can be written in the form of fractions \\ (\\ frac (m) (n) \\), where the numerator M is divisible, and the denominator P is a divider:
\\ (M: N \u003d \\ FRAC (M) (N) \\)

The following rules are true:

To obtain a fraction \\ (\\ frac (m) (n) \\), it is necessary to divide the unit to N of equal parts (fractions) and take M of such parts.

To obtain a fraction \\ (\\ FRAC (M) (N) \\), it is necessary to divide the number n.

To find a part of the whole, the number corresponding to the whole, divide the denominator and the result multiply to the fluster numerator, which expresses this part.

To find an integer in its part, it is necessary a number corresponding to this part, divide into the numerator and the result multiply to the denomoter of the fraction, which expresses this part.

If the numerator and the denominator of the fraction is multiplied by the same number (except zero), the fraction will not change:
\\ (\\ LARGE \\ FRAC (A) (B) \u003d \\ FRAC (A \\ CDOT N) (B \\ Cdot N) \\)

If the numerator, and the denomoter denomoter is divided into one and the same number (except for zero), the fraction will not change:
\\ (\\ LARGE \\ FRAC (A) (B) \u003d \\ FRAC (A: M) (B: \u200b\u200bM) \\)
This property is called the main property of the fraci.

The last two transformations are called reduction of the fraci.

If the fraction needs to be represented in the form of fractions with the same denominator, then such an action is called bringing fractions to a common denominator.

Right and incorrect fractions. Mixed numbers

You already know that the fraction can be obtained if it is divided into an integer part and take several such parts. For example, the fraction \\ (\\ FRAC (3) (4) \\) means three fourth shares of the unit. In many tasks of the previous paragraph, ordinary fractions were used to indicate a part of the whole. Common sense suggests that part should always be less than the whole, but then be with such fractions, as, for example, \\ (\\ (5) (5) \\) or \\ (\\ FRAC (8) (5) \\)? It is clear that this is not part of the unit. Probably, therefore, such fractions, who have a numerator more denominator or equal to him, called incorrect fractions. The remaining fractions, i.e., the fractions, in which the numerator is less than the denominator, is called regular fractions.

As you know, any ordinary fraction, and the correct, and incorrect, can be considered as the result of dividing the numerator to the denominator. Therefore, in mathematics, in contrast to the usual language, the term "wrong fraction" means not that we did something wrong, but only the fact that this fraction is a numerator more denominator or equal to him.

If the number consists of a whole part and fraction, then such the fractions are called mixed.

For example:
\\ (5: 3 \u003d 1 \\ FRAC (2) (3) \\): 1 is a whole part, A \\ (\\ FRAC (2) (3) \\) - fractional part.

If the fractional numerator \\ (\\ FRAC (A) (B) \\) is divided into a natural number N, then to divide this fraction on N, it is necessary to divide its numerator to this:
\\ (\\ LARGE \\ FRAC (A) (B): N \u003d \\ FRAC (A: N) (B) \\)

If the fractional numerator \\ (\\ FRAC (A) (B) \\) is not divided into a natural number n, then to divide this fraction on N, it is necessary to multiply its denominator:
\\ (\\ LARGE \\ FRAC (A) (B): N \u003d \\ FRAC (A) (BN) \\)

Note that the second rule is valid and in the case when the numerator is divided into N. Therefore, we can apply it when it is difficult to determine at first glance, the fraction numerator is divided into N or not.

Actions with fractions. Addition fractions.

With fractional numbers, as with natural numbers, arithmetic actions can be performed. Consider first adding fractions. Easy to fold the fractions with the same denominators. We find, for example, the amount \\ (\\ FRAC (2) (7) \\) and \\ (\\ FRAC (3) (7) \\). It is easy to understand that \\ (\\ FRAC (2) (7) + \\ FRAC (2) (7) \u003d \\ FRAC (5) (7) \\)

To fold the fractions with the same denominators, you need to fold their numerals, and the denominator is left for the same.

Using letters, the fraction of fractions with the same denominators can be written as follows:
\\ (\\ LARGE \\ FRAC (A) (C) + \\ FRAC (B) (C) \u003d \\ FRAC (A + B) (C) \\)

If you need to fold the fractions with different denominators, then they should be pre-lead to a common denominator. For example:
\\ (\\ LARGE \\ FRAC (2) (3) + \\ FRAC (4) (5) \u003d \\ FRAC (2 \\ CDOT 5) (3 \\ CDOT 5) + \\ FRAC (4 \\ CDOT 3) (5 \\ CDOT 3 ) \u003d \\ FRAC (10) (15) + \\ FRAC (12) (15) \u003d \\ FRAC (10 + 12) (15) \u003d \\ FRAC (22) (15) \\)

For fractions, as for natural numbers, the transitional and combination properties of addition are valid.

Addition of mixed fractions

Such records as \\ (2 \\ FRAC (2) (3) \\) are called mixed fractions. In this case, the number 2 is called whole part mixed fraction, and the number \\ (\\ FRAC (2) (3) \\) - it fractional part. The record \\ (2 \\ FRAC (2) (3) \\) is read like this: "Two and two thirds".

When dividing the number 8 to the number 3, two answers can be obtained: \\ (\\ FRAC (8) (3) \\) and \\ (2 \\ FRAC (2) (3) \\). They express one same fractional number, i.e. \\ (\\ FRAC (8) (3) \u003d 2 \\ FRAC (2) (3) \\)

Thus, the wrong fraction \\ (\\ FRAC (8) (3) \\) is represented as a mixed fraction \\ (2 \\ FRAC (2) (3) \\). In such cases, they say that from the wrong fraction allocated a whole part.

Subtraction of fractions (fractional numbers)

The subtraction of fractional numbers, as well as natural, is determined on the basis of the action of the addition: to deduct out of one number - it means to find such a number that when adding with the second gives the first. For example:
\\ (\\ FRAC (8) (9) - \\ FRAC (1) (9) \u003d \\ FRAC (7) (9) \\) Since \\ (\\ FRAC (7) (9) + \\ FRAC (1) (9 ) \u003d \\ FRAC (8) (9) \\)

The deduction rule of fractions with the same denominators is similar to the rule of addition of such fractions:
to find the difference in fractions with the same denominators, it is necessary from the numerator of the first fraction to find the numerator of the second, and the denominator is left for the same.

With the help of letters, this rule is written as follows:
\\ (\\ LARGE \\ FRAC (A) (C) - \\ FRAC (B) (C) \u003d \\ FRAC (A-B) (C) \\)

Multiplication of fractions

To multiply the fraction in the fraction, you need to multiply their numerators and denominators and the first product to record the numerator, and the second is the denominator.

With the help of letters, the shortage of fractions can be written as:
\\ (\\ LARGE \\ FRAC (A) (B) \\ CDOT \\ FRAC (C) (D) \u003d \\ FRAC (A \\ CDOT C) (B \\ Cdot D) \\)

Taking advantage of the formulated rule, multiplying the fraction on a natural number, on a mixed fraction, as well as multiply mixed fractions. To do this, it is necessary to write a natural number in the form of a fraction with a denominator 1, mixed fraction - as an incorrect fraction.

The result of multiplication should be simplified (if possible), reducing the fraction and highlighting the whole part of the incorrect fraction.

For fractions, as well as for natural numbers, a variable and combinated multiplication properties, as well as the distribution property of multiplication relative to addition.

Division of fractions

Take the fraction \\ (\\ FRAC (2) (3) \\) and "invert" it, changing the numerator and denominator in places. We obtain fraction \\ (\\ FRAC (3) (2) \\). This fraction is called inverse fractions \\ (\\ FRAC (2) (3) \\).

If we now "turn" the fraction \\ (\\ FRAC (3) (2) \\), then we obtain the initial fraction \\ (\\ FRAC (2) (3) \\). Therefore, such fractions, as \\ (\\ FRAC (2) (3) \\) and \\ (\\ FRAC (3) (2) \\) are called mutually back.

Mutually reverse are, for example, fractions \\ (\\ FRAC (6) (5) \\) and \\ (\\ FRAC (5) (6) \\), \\ (\\ FRAC (7) (18) \\) and \\ (\\ FRAC (18) (7) \\).

With the help of letters, mutually inverse fractions can be written as follows: \\ (\\ FRAC (A) (B) \\) and \\ (\\ FRAC (B) (A) \\)

It is clear that the work of mutually reverse fractions is 1. For example: \\ (\\ FRAC (2) (3) \\ CDOT \\ FRAC (3) (2) \u003d 1 \\)

Using mutually inverse fractions, it is possible to divide the frains of reduced to multiplication.

The division rule of the fraction on the fraction:
to divide one fraction to another, you need to multiply by fraction, the reverse divider.