How to multiply by ordinary fractions. Rules of multiplication of fractions for the number

Multiplication and division of fractions.

Attention!
This topic has additional
Materials in a special section 555.
For those who are strongly "not very ..."
And for those who are "very ...")

This operation is much more nicer addition-subtraction! Because it's easier. I remind you: To multiply the fraction on the fraction, you need to multiply the numerators (it will be the resultant) and the denominators (this will be the denominator). I.e:

For example:

Everything is extremely simple. And please do not look for a common denominator! Do not need him here ...

To divide the fraction for the fraction, you need to flip over second(This is important!) Fraction and multiply them, i.e.:

For example:

If multiplication or division with integers and fractions was caught - nothing terrible. As with the addition, we make a fraction with a unit in the denominator - and forward! For example:

In high schools, it is often necessary to deal with three-story (or even four-storey!) Droks. For example:

How to bring this fraction to a decent mind? Yes, very simple! Use division in two points:

But do not forget about the order of division! Unlike multiplication, it is very important here! Of course, 4: 2, or 2: 4 We are not confused. But in the three-story fraction it is easy to make a mistake. Note, for example:

In the first case (expression on the left):

In the second (expression on the right):

Do you feel the difference? 4 and 1/9!

And what is the order of division? Or brackets, or (as here) the length of horizontal lines. Develop the eye meter. And if there are no brackets, nor dash, like:

then divide-multiply in a few, left to right!

And still very simple and important reception. In actions with degrees, he oh, how can I come in handy! We divide the unit to any fraction, for example, by 13/15:

The fraction turned over! And it always happens. When dividing 1 to any fraction, as a result, we get the same fraction only inverted.

That's all the actions with fractions. The thing is quite simple, but the mistakes gives more than enough. Please note the practical advice, and their (errors) will be less!

Practical Tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a harsh need! All calculations on the exam make as a full task, focusing and clearly. It is better to write two extra lines in the draft, than to accumulate when calculating the mind.

2. In the examples with different species fractions - go to ordinary fractions.

3. All fractions cut until it stops.

4. Multi-storey fractional expressions We are reduced to ordinary using division in two points (follow the division order!).

5. Unit of fraction divide in mind, just turning the fraction.

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

Remember - the correct answer, the resulting from the second (even more - the third) times - not considered! Such is a harsh life.

So, we decide in the exam mode ! This is already prepared for the exam, by the way. We solve the example, check, solve the following. They decided everything - they checked again from the first to last. Only later We look at the answers.

Calculate:

Did you cut?

We are looking for answers that coincide with yours. I specifically recorded them in disarray, away from the temptation, so to speak ... So they are answered, the point with the comma is recorded.

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

And now we make conclusions. If everything happened - I am glad for you! Elementary calculations with fractions - not your problem! You can do more serious things. If not...

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

If you like this site ...

By the way, I have another couple of interesting sites for you.)

It can be accessed in solving examples and find out your level. Testing with instant check. Learn - with interest!)

You can get acquainted with features and derivatives.

) And the denominator on the denominator (we get a denominator of the work).

Formula multiplication fractions:

For example:

Before proceeding with multiplication of numerals and denominators, it is necessary to check the possibility of cutting the fraction. If it turns out to shorten the fraction, then you will be easier to carry out calculations.

Division of ordinary fraction on the fraction.

Division fractions with the participation of a natural number.

It's not as scary as it seems. As in the case of adding, we translate an integer in the fraction with a unit in the denominator. For example:

Multiplying mixed fractions.

Rules of multiplication of fractions (mixed):

we transform mixed fractions into the wrong;
reduce the numerals and denominators of fractions;
reducing the fraction;
if you got the wrong fraction, we transform the wrong fraction into a mixed one.

Note! To multiply mixed fraction To another mixed fraction, you need to begin, lead them to the mind of the wrong fractions, and then multiply by the rule of multiplication of ordinary fractions.

The second method of multiplication of the fraction on a natural number.

It happens more convenient to use the second way of multiplication. ordinary fraci by number.

Note! For multiplication of fractions on natural number A denominator is needed to divide the number, and the numerator is left unchanged.

From the above, the example is clear that this option is more convenient for use when the denoter of the fraction is divided without a residue on a natural number.

Multi-storey fractions.

In high school classes, three-story (or more) fractions are found. Example:

To bring such a fraction to the usual mind, use division after 2 points:

Note!In dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.

Note, eg:

When dividing units on any fraction, the result will the same fraction, only inverted:

Practical tips when multiplying and dividing fractions:

1. The most important in working with fractional expressions is accuracy and attentiveness. All calculations do carefully and gently, concentrately and clearly. Better write down a few unnecessary lines in the drafts, than getting confused in the calculations in the mind.

2. In tasks with different types of fractions - go to the species of ordinary fractions.

3. All fractions reducing until it is impossible to cut.

4. Multi-storey fractional expressions are in the form of ordinary, using the division after 2 points.

5. Unit of fraction divide in mind, just turning the fraction.

Multiplication of ordinary fractions

Consider an example.

Suppose on a plate lies $ \\ FRAC (1) (3) $ part of the apple. It is necessary to find $ \\ FRAC (1) (2) $ part from it. The necessary part is the result of the multiplication of the fractions of $ \\ FRAC (1) (3) $ and $ \\ FRAC (1) (2) $. The result of multiplication of two ordinary fractions is an ordinary fraction.

Multiplying two ordinary fractions

The rule of multiplication of ordinary fractions:

The result of the multiplication of the fraction on the fraction is the fraction, the numerator of which is equal to the product of numerals of multiplying fractions, and the denominator is equal to the product of the denominators:

Example 1.

Perform multiplication of ordinary fractions $ \\ FRAC (3) (7) $ and $ \\ FRAC (5) (11) $.

Decision.

We use the rule of multiplication of ordinary fractions:

\\ [\\ FRAC (3) (7) \\ CDOT \\ FRAC (5) (11) \u003d \\ FRAC (3 \\ CDOT 5) (7 \\ CDOT 11) \u003d \\ FRAC (15) (77) \\]

Answer: $ \\ FRAC (15) (77) $

If, as a result of the multiplication of fractions, a reduced or improper fraction is obtained, then it is necessary to simplify it.

Example 2.

Perform multiplication of fractions $ \\ FRAC (3) (8) $ and $ \\ FRAC (1) (9) $.

Decision.

We use the rule of multiplication of ordinary fractions:

\\ [\\ FRAC (3) (8) \\ CDOT \\ FRAC (1) (9) \u003d \\ FRAC (3 \\ CDOT 1) (8 \\ CDOT 9) \u003d \\ FRAC (3) (72) \\]

As a result, they received a reduction fraction (on the basis of division by $ 3 $. The numerator and denominator of the fraci divide by $ 3 $, we get:

\\ [\\ FRAC (3) (72) \u003d \\ FRAC (3: 3) (72: 3) \u003d \\ FRAC (1) (24) \\]

Summary:

\\ [\\ FRAC (3) (8) \\ CDOT \\ FRAC (1) (9) \u003d \\ FRAC (3 \\ CDOT 1) (8 \\ CDot 9) \u003d \\ FRAC (3) (72) \u003d \\ FRAC (1) (24) \\]

Answer: $ \\ FRAC (1) (24). $

When multiplying fractions, cuts and denominator can be reduced to their work. In this case, the numerator and denominator of the fraraty declined to simple factors, after which the repeated multipliers are reduced and the result is.

Example 3.

Calculate the work of the fractions $ \\ FRAC (6) (75) $ and $ \\ FRAC (15) (24) $.

Decision.

We use the multiplication formula of ordinary fractions:

\\ [\\ FRAC (6) (75) \\ CDOT \\ FRAC (15) (24) \u003d \\ FRAC (6 \\ CDOT 15) (75 \\ CDOT 24) \\]

Obviously, there are numbers in the numerator and denominator, which can be in pairs of $ 2 $, $ 3 $ and $ 5 $. Spread the numerator and denominator for simple factors and will make a reduction:

\\ [\\ FRAC (6 \\ CDOT 15) (75 \\ CDOT 24) \u003d \\ FRAC (2 \\ CDOT 3 \\ CDOT 3 \\ CDOT 5) (3 \\ CDOT 5 \\ CDOT 5 \\ CDOT 2 \\ CDOT 2 \\ CDOT 2 \\ CDOT 3) \u003d \\ FRAC (1) (5 \\ CDOT 2 \\ CDOT 2) \u003d \\ FRAC (1) (20) \\]

Answer: $ \\ FRAC (1) (20). $

With multiplication of fractions, a transfer law can be applied:

Multiplication of ordinary fraction on a natural number

The rule of multiplication of ordinary fraction on a natural number:

The result of multiplication of the fraction on a natural number is the fraction in which the numerator is equal to the product of the multiply fraction on the natural number, and the denominator is equal to the denominator of the multiply

where $ \\ FRAC (a) (b) $ is an ordinary fraction, $ n $ is a natural number.

Example 4.

Perform the multiplication of the fraction $ \\ FRAC (3) (17) $ $ 4 $.

Decision.

We use the rule of multiplication of an ordinary fraction on a natural number:

\\ [\\ FRAC (3) (17) \\ CDOT 4 \u003d \\ FRAC (3 \\ CDOT 4) (17) \u003d \\ FRAC (12) (17) \\]

Answer: $ \\ FRAC (12) (17). $

Do not forget to check the result of multiplication of the fraction or incorrect fraction.

Example 5.

Multiply the fraction of $ \\ FRAC (7) (15) $ by number $ 3 $.

Decision.

We use the formula for multiplying the fraction on a natural number:

\\ [\\ FRAC (7) (15) \\ CDOT 3 \u003d \\ FRAC (7 \\ CDOT 3) (15) \u003d \\ FRAC (21) (15) \\]

On the basis of division by the number $ 3 $), it can be determined that the resulting fraction can be reduced:

\\ [\\ FRAC (21) (15) \u003d \\ FRAC (21: 3) (15: 3) \u003d \\ FRAC (7) (5) \\]

As a result, they got the wrong fraction. We highlight the whole part:

\\ [\\ FRAC (7) (5) \u003d 1 \\ FRAC (2) (5) \\]

Summary:

\\ [\\ FRAC (7) (15) \\ CDOT 3 \u003d \\ FRAC (7 \\ CDOT 3) (15) \u003d \\ FRAC (21) (15) \u003d \\ FRAC (7) (5) \u003d 1 \\ FRAC (2) (five)\\]

Reduced the fraction could also be replaced by numbers in a numerator and denominator on their decomposition into simple multipliers. In this case, the decision could be recorded like this:

\\ [\\ FRAC (7) (15) \\ CDOT 3 \u003d \\ FRAC (7 \\ CDOT 3) (15) \u003d \\ FRAC (7 \\ CDOT 3) (3 \\ CDOT 5) \u003d \\ FRAC (7) (5) \u003d 1 \\ FRAC (2) (5) \\]

Answer: $ 1 \\ FRAC (2) (5). $

When multiplying the fraction on a natural number, a Movement Law can be used:

Division of ordinary fractions

The division operation is back to multiplication and its result is a fraction for which you need to multiply a known fraction to get famous work Two fractions.

Division of two ordinary fractions

The division rule of ordinary fractions:Obviously, the numerator and denominator of the resulting fraction can be decomposed on simple factors and reduce:

\\ [\\ FRAC (8 \\ CDOT 35) (15 \\ CDOT 12) \u003d \\ FRAC (2 \\ CDOT 2 \\ CDOT 2 \\ CDOT 5 \\ CDOT 7) (3 \\ CDOT 5 \\ CDOT 2 \\ CDOT 2 \\ CDOT 3) \u003d \\ FRAC (2 \\ CDOT 7) (3 \\ Cdot 3) \u003d \\ FRAC (14) (9) \\]

As a result, the wrong fraction was obtained from which we allocate the whole part:

\\ [\\ FRAC (14) (9) \u003d 1 \\ FRAC (5) (9) \\]

Answer: $ 1 \\ FRAC (5) (9). $

§ 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.

Take the segment AB (Fig. 17), we will take it for one and divide by 5 equal parts, Then part of the speakers of this segment will be equal to 1/5 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 those at first glance various actionslike finding the amount equal numbers And finding a fracted number, 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 experience has shown 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 The interest was called money, which 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 firewall birch firewood accounted for 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 the total number of children in the camp was taken over 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 according to the rules for dividing 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 home. 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, in the future we will not talk about the reverse frauds, but about 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:

Pay special attention to 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.

Walk already these rakes! 🙂

Multiplication and division of fractions.

Attention!
This topic has additional
materials B. Special section 555.
For those who are not very "not very. "
And for those who "very much. ")

Everything is extremely simple. And please do not look for a common denominator! Do not need him here ...

To divide the fraction for the fraction, you need to flip over second (This is important!) Fraction and multiply them, i.e.:

If multiplication or division with integers and fractions was caught - nothing terrible. As with the addition, we make a fraction with a unit in the denominator - and forward! For example:

In high schools, it is often necessary to deal with three-story (or even four-storey!) Droks. For example:

How to bring this fraction to a decent mind? Yes, very simple! Use division in two points:

In the first case (expression on the left):

In the second (expression on the right):

Do you feel the difference? 4 and 1/9!

And what is the order of division? Or brackets, or (as here) the length of horizontal lines. Develop the eye meter. And if there are no brackets, nor dash, like:

then divide-multiply in a few, left to right!

And a very simple and important technique. In actions with degrees, he oh, how can I come in handy! We divide the unit to any fraction, for example, by 13/15:

The fraction turned over! And it always happens. When dividing 1 to any fraction, as a result, we get the same fraction only inverted.

That's all the actions with fractions. The thing is quite simple, but the mistakes gives more than enough. Please note the practical advice, and their (errors) will be less!

2. In the examples with different types of fractions - we turn to ordinary fractions.

3. All fractions cut until it stops.

4. Multi-storey fractional expressions are reduced to ordinary, using division in two points (follow the order of division!).

Remember - the correct answer, the resulting from the second (even more - the third) times - not considered! Such is a harsh life.

We are looking for answers that coincide with yours. I recorded them in disorder, away from the temptation, so to speak. Here they are answers, the point with the comma is recorded.

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

And now we make conclusions. If everything happened - I am glad for you! Elementary calculations with fractions - not your problem! You can do more serious things. If not.

So you have one of two problems. Or both at once.) Lack of knowledge and (or) Inattention. But. it resolved Problems.

In a special section 555, "fractions" dismantled all these (and not only!) Examples. With detailed explanations that, why and how. Such a disclaimer helps with a shortage of knowledge and skills!

Yes, and on the second issue there is something.) Quite practical advice, how to become careful. Yes Yes! Advice that can apply everyone.

In addition to knowledge and care, a certain automatism is needed for success. Where to take it? I hear a heavy sigh ... Yes, only in practice, there is no place more.

You can use 321start.ru for workout. There in the option "Try" there are 10 examples for everyone. With instant check. For registered users - 34 examples from simple to harsh. It is only for fractions.

If you like this site.

By the way, I have another couple of interesting sites for you.)

Here you can practice in solving examples and learn your level. Testing with instant check. Learn - with interest!)

And here you can get acquainted with functions and derivatives.

Rule 1.

To multiply the fraction on a natural number, it is necessary to multiply its numerator to this number, and the denominator is left unchanged.

Rule 2.

To multiply the fraction for the fraction, it is necessary:

1. Find the product of numerators and the product of denominators of these frains

2. The first product is recorded by a numerator, and the second is the denominator.

Rule 3.

In order to perform multiplication of mixed numbers, they must be recorded in the form of incorrect fractions, and then use the rules of multiplication of fractions.

Rule 4.

To divide one fraction to another, it is necessary to multiply by the number, the reverse divider.

Example 1.

Calculate

Example 2.

Calculate

Example 3.

Calculate

Example 4.

Calculate

Mathematics. Other materials

The erection of the number into a rational degree. (

The erection of a natural degree. (

Generalized interval method in solving algebraic inequalities (by Kolchanov A.V.)

Method of replacing multipliers in solving algebraic inequalities (by Kolchanov A.V.)

Signs of divisibility (Lunga Alena)

Check yourself on 'multiplication and division of ordinary fractions'

Multiplication of fractions

Multiplication of ordinary fractions Consider in several possible options.

Multiplication of ordinary fraction

This is the easiest case in which you need to use the following. rules of multiplication of fractions.

To multiply fraction, it is necessary:

the numerator of the first fraction is to multiply the second fraction on the numerator and their work is to write to the numerator of the new fraction;

the denominator of the first fraction is multiplied by the denominator of the second fraction and their work to write to the denominator of the new fraction;

Before multiplying the numerals and denominators, check whether it is impossible to cut the fraction. Reducing fractions in calculations will greatly facilitate your calculations.

Multiplication of fractions on a natural number

To fraction multiply to a natural number It is necessary to multiply the fluster to this number, and the denomoter of the fraction is unchanged.

If, as a result of multiplication, it turned out the wrong fraction, do not forget to turn it into mixed number, that is, to highlight the whole part.

Multiplying mixed numbers

In order to multiply mixed numbers, you must first turn them into the wrong fraction and then multiply according to the rule of multiplication of ordinary fractions.

Another way to multiply the fraction on the natural number

Sometimes when calculating it is more convenient to use another way of multiplying the ordinary fraction.

To multiply the fraction on a natural number, a denoter of a fraction is to divide into this number, and the numerator is left for the same.

As can be seen from the example, this option is more convenient to use the rule if the shooter is divided without a residual on a natural number.

Division fraction

How to divide the fraction on the number faster? We will examine the theory, make a conclusion and see examples as dividing the fraction on the number can be performed according to a new short rule.

Usually, the division of the fraction is performed according to the rules of division of fractions. The first number (fraction) is multiplied by the reverse one. Since the second number is the integer inverse to it - the fraction, the numerator of which is equal to one, and the denominator is a given number. Schematically dividing the fraction on the natural number looks like this:

From here we conclude:

to split the fraction by the number, it is necessary to multiply the denominator to this number, and the numerator leave the same. The rule can be formulated even in short:

when dividing the fraction, the number goes to the denominator.

Perform the division of the fracted by the number:

To divide the fraction to the number, the numerator will rewrite unchanged, and the denominator will multiply on this number. Reducing 6 and 3 to 3.

When dividing the fraction, the number of the numerator is rewritten, and the denominator multiplies with this number. Reduce 16 and 24 to 8.

When dividing the fraction, the number goes to the denominator, so the numerator is left the same, and the denominator multiplies to the divider. Reduce 21 and 35 to 7.

Multiplication and division of fractions

Last time we learned to fold and deduct the fraction (see the lesson "Addition and subtraction of fractions"). The most difficult moment in the actions was to bring fractions to the general denominator.

Now it's time to deal with multiplication and division. Good news It is that these operations are performed even easier than addition and subtraction. To begin with, consider the simplest case when there are two positive fractions without a selected part.

To multiply two fractions, it is necessary to multiply their numerals and denominators. The first number will be the numerator of the new fraction, and the second is the denominator.

To split two fractions, you need to multiply the first fraction to the "inverted" second.

From the definition it follows that the division of fractions is reduced to multiplication. To "flip" the fraction, it is enough to change the numerator and denominator in places. Therefore, we will consider the whole lesson mostly multiplying.

As a result of multiplication, it may occur (and often it really occurs) a shortage of fraction - it, of course, must be reduced. If after all the cuts, the fraction was incorrect, it should be allocated to the whole part. But what exactly will not be when multiplying, it is to bring to a common denominator: no methods of "cross-elder", the greatest multipliers and the smallest common multiples.

A task. Find the value of the expression:

By definition, we have:

Multiplication of fractions with a whole part and negative fractions

If in fractions is present whole partThey need to be translated into the wrong - and only then multiply according to the schemes above.

If there is a minus in a denoter in a denoter or before it, it can be reached out of multiplication or completely removed according to the following rules:

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

Until now, these rules have met only when adding and subtracting negative fractionsWhen it was necessary to get rid of the whole part. For the work, they can be generalized to "burn" several minuses at once:

I draw out the minuses in pairs until they disappear completely. In extreme cases, one minus can survive - the one who did not find a couple;
If there are no minuses, the operation is completed - you can proceed to multiplication. If the last minus does not cross out, since he did not find a couple, we endure it outside the multiplication. It turns out a negative fraction.

All fractions are translated into the wrong, and then we endure the minuses outside the multiplication. What remains, multiply by the usual rules. We get:

Once again I remind you that the minus, which stands before the fraction with the whole part highlighted, belongs to the whole fraction, and not only to its whole part (this applies to the last two examples).

Also pay attention to negative numbers: When multiplying, they are in brackets. This is done in order to separate the minuses from the multiplication signs and make the entire record more accurate.

Reduction of fractions "On the fly"

Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction more to multiplication. After all, essentially, the numerals and denominants of fractions are ordinary multipliers, and therefore they can be cut using the main property of the fraction. Take a look at the examples:

In all examples, the numbers that were subjected to reduction were marked, and what remained from them.

Please note: in the first case, the multipliers decreased completely. There are few units in their place, which, generally speaking, you can not write. In the second example, it was not possible to achieve a complete reduction, but the total volume of computation was still decreased.

However, in no case do not use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you want to cut. Here, look:

So you can not do!

An error occurs due to the fact that when adding the fraction in the numerator, the amount appears, and not the product of numbers. Therefore, it is impossible to apply the main property of the fraction because in this property. we are talking It is about multiplication of numbers.

Other grounds for cutting fractions simply does not exist, so correct solution The previous task looks like this:

As you can see, the correct answer was not so beautiful. In general, be careful.

Division fractions.

Decision fraction on a natural number.

Examples of fission fractions on a natural number

Division of a natural number of fraction.

Examples of natural-numbered division

Division of ordinary fractions.

Examples of division of ordinary fractions

Division of mixed numbers.

convert mixed fractions to incorrect;
multiply the first fraction on the fraction, inverse second;
reduce the resulting fraction;
if it turned out the wrong fraction to convert an irregular fraction into a mixed one.

Examples of dividing mixed numbers

1 1 2: 2 2 3 \u003d 1 · 2 + 1 2: 2 · 3 + 2 3 \u003d 3 2: 8 3 \u003d 3 2 · 3 8 \u003d 3 · 3 2 · 8 \u003d 9 16

2 1 7: 3 5 \u003d 2 · 7 + 1 7: 3 5 \u003d 15 7: 3 5 \u003d 15 7 · 5 3 \u003d 15 · 5 7 · 3 \u003d 5 · 5 7 \u003d 25 7 \u003d 7 · 3 + 4 7 \u003d 3 4 7

Any obscene comments will be removed, and their authors are blacklisted!

Welcome to Onlinemschool.
My name is Dovgik Mikhail Viktorovich. I am the owner and author of this site, I wrote all theoretical material, and also developed online exercises and calculators that you can use to study mathematics.

Fraction. Multiplication and division of fractions.

Multiplying ordinary fraction for fraction.

In order to multiply ordinary fractions, you need to multiply the numerator to the numerator (we obtain the product numerator) and the denominator to the denominator (we get a denominator of the work).

Formula multiplication fractions:

Note! Here you do not need to look for a common denominator !!

Division of ordinary fraction on the fraction.

The division of an ordinary fraction on the fraction is happening: turning the second fraction (i.e., change the numerator and denominator in places) and then the fractions are folded.

Formula of division of ordinary fractions:

Multiplication of the fraction on the natural number.

Note! When multiplying the fraction on a natural number, the fluster numerator is multiplied by our natural number, and the denoter of the fraction is left for the same. If the result of the work was incorrect fraction, then be sure to highlight the whole part, turning the wrong fraction into the mixed.

Division fractions with the participation of a natural number.

It's not as scary as it seems. As in the case of adding, we translate an integer in the fraction with a unit in the denominator. For example:

Multiplying mixed fractions.

Rules of multiplication of fractions (mixed):

we transform mixed fractions into the wrong;
reduce the numerals and denominators of fractions;
reducing the fraction;
if you got the wrong fraction, we transform the wrong fraction into a mixed one.

Note! To multiply the mixed fraction on another mixed fraction, you need to begin, lead them to the mind of the wrong fractions, and then multiply by the rule of multiplication of ordinary fractions.

The second method of multiplication of the fraction on a natural number.

It is more convenient to use the second way of multiplying an ordinary fraction for a number.

Note! To multiply the fraction on a natural number, a denominator of a fraction is to divide into this number, and the numerator is left unchanged.

From the above, the example is clear that this option is more convenient for use when the denoter of the fraction is divided without a residue on a natural number.

Multi-storey fractions.

In high school classes, three-story (or more) fractions are found. Example:

To bring such a fraction to the usual mind, use division after 2 points:

Note! In dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.

Note, eg:

When dividing units on any fraction, the result will the same fraction, only inverted:

Practical tips when multiplying and dividing fractions:

2. In tasks with different types of fractions - go to the species of ordinary fractions.

3. All fractions reducing until it is impossible to cut.

4. Multi-storey fractional expressions are in the form of ordinary, using the division after 2 points.

Nearby and not the advanced song "Spring Tango" (time comes - birds from the south arrive) - music. Valery Milyaev laid himself, misunderstood, mischiven, in the sense that I did not guessed, all the verbs with not separately wrote, I did not know about the prefilment. It happens, […]
The page was not found in the third final reading a package of government documents was adopted, providing for the creation of special administrative districts (SAR). Due to the exit from the European Union, the United Kingdom will not be included in the European VAT zone and [...]
The combined investigative committee will appear in the fall, the joint investigative committee will appear in the fall. The consequence of all the power structures will be gathered under one roof with the fourth attempt in the fall of 2014, according to Izvestia, President Vladimir Putin [...]
Patent for the algorithm as a patent on the algorithm looks like a patent for the algorithm preparation is preparing technical descriptions methods of storage, processing, and transmission, signals and / or data It is usually not submitted for the purposes of patenting special difficulties, and [...]
What is important to know about the new bill on pensions on December 12, 1993, the Constitution of the Russian Federation (taking into account the amendments made by the laws of the Russian Federation amendments to the Constitution of the Russian Federation 30.12.2008 N 6-FKZ, from 30.12.2008 N 7-FKZ, from [...]
Chastushki Pro Pension Woman Cool For Jubileele Men For Jubileele Men - Choir for Jubilee Women - Dedication to Women's Pensioners Comic Will Interesting Competitions for Pensioners Host: Dear Friends! For a minute of attention! Sensation! Only [...]