Examples of multiplying simple fractions with different denominators. Rule for multiplying fractions by whole numbers

§ 87. Addition of fractions.

Adding fractions has many similarities to adding whole numbers. Addition of fractions is an action consisting in the fact that several given numbers (terms) are combined into one number (sum), containing all the units and fractions of the units of the terms.

We will consider three cases sequentially:

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

1. Addition of fractions with like denominators.

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

Let's take segment AB (Fig. 17), take it as one and divide it into 5 equal parts, then part AC of this segment will be equal to 1/5 of segment AB, and part of the same segment CD will be equal to 2/5 AB.

From the drawing it is clear that if we take the segment AD, it will be equal to 3/5 AB; but the segment AD is precisely the sum of the segments AC and CD. So we can write:

1 / 5 + 2 / 5 = 3 / 5

Considering these terms and the resulting sum, we see that the numerator of the sum was obtained by adding the numerators of the terms, and the denominator remained unchanged.

From this we get the following rule: To add fractions with the same denominators, you need to add their numerators and leave the same denominator.

Let's look at an example:

2. Addition of fractions with different denominators.

Let's add the fractions: 3 / 4 + 3 / 8 First they need to be reduced to the lowest common denominator:

The intermediate link 6/8 + 3/8 could not be written; we've written it here for clarity.

Thus, to add fractions with different denominators, you must first reduce them to the lowest common denominator, add their numerators and label the common denominator.

Let's consider an example (we will write additional factors above the corresponding fractions):

3. Addition of mixed numbers.

Let's add the numbers: 2 3/8 + 3 5/6.

Let’s first bring the fractional parts of our numbers to a common denominator and rewrite them again:

Now we add the integer and fractional parts sequentially:

§ 88. Subtraction of fractions.

Subtracting fractions is defined in the same way as subtracting whole numbers. This is an action with the help of which, given the sum of two terms and one of them, another term is found. Let us consider three cases in succession:

1. Subtracting fractions with like denominators.
2. Subtracting fractions with different denominators.
3. Subtraction of mixed numbers.

1. Subtracting fractions with like denominators.

Let's look at an example:

13 / 15 - 4 / 15

Let's take the segment AB (Fig. 18), take it as a unit and divide it into 15 equal parts; then part AC of this segment will represent 1/15 of AB, and part AD of the same segment will correspond to 13/15 AB. Let us set aside another segment ED equal to 4/15 AB.

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

The example we made shows that the numerator of the difference was obtained by subtracting the numerators, but the denominator remained the same.

Therefore, to subtract fractions with like denominators, you need to subtract the numerator of the subtrahend from the numerator of the minuend and leave the same denominator.

2. Subtracting fractions with different denominators.

Example. 3/4 - 5/8

First, let's reduce these fractions to the lowest common denominator:

The intermediate 6 / 8 - 5 / 8 is written here for clarity, but can be skipped later.

Thus, in order to subtract a fraction from a fraction, you must first reduce them to the lowest common denominator, then subtract the numerator of the minuend from the numerator of the minuend and sign the common denominator under their difference.

Let's look at an example:

3. Subtraction of mixed numbers.

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

Let us reduce the fractional parts of the minuend and subtrahend to the lowest common denominator:

We subtracted a whole from a whole and a fraction from a fraction. But there are cases when the fractional part of the subtrahend is greater than the fractional part of the minuend. In such cases, you need to take one unit from the whole part of the minuend, split it into those parts in which the fractional part is expressed, and add it to the fractional part of the minuend. And then the subtraction will be performed in the same way as in the previous example:

§ 89. Multiplication of fractions.

When studying fraction multiplication, we will consider the following questions:

1. Multiplying a fraction by a whole number.
2. Finding the fraction of a given number.
3. Multiplying a whole number by a fraction.
4. Multiplying a fraction by a fraction.
5. Multiplication of mixed numbers.
6. The concept of interest.
7. Finding the percentage of a given number. Let's consider them sequentially.

1. Multiplying a fraction by a whole number.

Multiplying a fraction by a whole number has the same meaning as multiplying a whole number by an integer. To multiply a fraction (multiplicand) by an integer (factor) means to create a sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.

This means that if you need to multiply 1/9 by 7, then it can be done like this:

We easily obtained the result, since the action was reduced to adding fractions with the same denominators. Hence,

Consideration of this action shows that multiplying a fraction by a whole number is equivalent to increasing this fraction by as many times as the number of units contained in the whole number. And since increasing a fraction is achieved either by increasing its numerator

or by reducing its denominator , then we can either multiply the numerator by an integer or divide the denominator by it, if such division is possible.

From here we get the rule:

To multiply a fraction by a whole number, you multiply the numerator by that whole number and leave the denominator the same, or, if possible, divide the denominator by that number, leaving the numerator unchanged.

When multiplying, abbreviations are possible, for example:

2. Finding the fraction of a given number. There are many problems in which you have to find, or calculate, part of a given number. The difference between these problems and others is that they give the number of some objects or units of measurement and you need to find a part of this number, which is also indicated here by a certain fraction. To facilitate understanding, we will first give examples of such problems, and then introduce a method for solving them.

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

Task 2. The train must travel a distance between cities A and B equal to 300 km. He has already covered 2/3 of this distance. How many kilometers is this?

Task 3. There are 400 houses in the village, 3/4 of them are brick, the rest are wooden. How many brick houses are there in total?

These are some of the many problems we encounter to find a part of a given number. They are usually called problems to find the fraction of a given number.

Solution to problem 1. From 60 rub. I spent 1/3 on books; This means that to find the cost of books you need to divide the number 60 by 3:

Solving problem 2. The point of the problem is that you need to find 2/3 of 300 km. Let's first calculate 1/3 of 300; this is achieved by dividing 300 km by 3:

300: 3 = 100 (that's 1/3 of 300).

To find two-thirds of 300, you need to double the resulting quotient, i.e., multiply by 2:

100 x 2 = 200 (that's 2/3 of 300).

Solving problem 3. Here you need to determine the number of brick houses that make up 3/4 of 400. Let’s first find 1/4 of 400,

400: 4 = 100 (that's 1/4 of 400).

To calculate three quarters of 400, the resulting quotient must be tripled, i.e. multiplied by 3:

100 x 3 = 300 (that's 3/4 of 400).

Based on the solution to these problems, we can derive the following rule:

To find the value of a fraction from a given number, you need to divide this number by the denominator of the fraction and multiply the resulting quotient by its numerator.

3. Multiplying a whole number by a fraction.

Earlier (§ 26) it was established that the multiplication of integers should be understood as the addition of identical terms (5 x 4 = 5+5 +5+5 = 20). In this paragraph (point 1) it was established that multiplying a fraction by an integer means finding the sum of identical terms equal to this fraction.

In both cases, multiplication consisted of finding the sum of identical terms.

Now we move on to multiplying a whole number by a fraction. Here we will encounter, for example, multiplication: 9 2 / 3. It is clear that the previous definition of multiplication does not apply to this case. This is evident from the fact that we cannot replace such multiplication by adding equal numbers.

Because of this, we will have to give a new definition of multiplication, i.e., in other words, answer the question of what should be understood by multiplication by a fraction, how this action should be understood.

The meaning of multiplying a whole number by a fraction is clear from the following definition: multiplying an integer (multiplicand) by a fraction (multiplicand) means finding this fraction of the multiplicand.

Namely, multiplying 9 by 2/3 means finding 2/3 of nine units. In the previous paragraph, such problems were solved; so it’s easy to figure out that we’ll end up with 6.

But now an interesting and important question arises: why are such seemingly different operations, such as finding the sum of equal numbers and finding the fraction of a number, called in arithmetic by the same word “multiplication”?

This happens because the previous action (repeating a number with terms several times) and the new action (finding the fraction of a number) give answers to homogeneous questions. This means that we proceed here from the considerations that homogeneous questions or tasks are solved by the same action.

To understand this, consider the following problem: “1 m of cloth costs 50 rubles. How much will 4 m of such cloth cost?

This problem is solved by multiplying the number of rubles (50) by the number of meters (4), i.e. 50 x 4 = 200 (rubles).

Let’s take the same problem, but in it the amount of cloth will be expressed as a fraction: “1 m of cloth costs 50 rubles. How much will 3/4 m of such cloth cost?”

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

You can change the numbers in it several more times, without changing the meaning of the problem, for example, take 9/10 m or 2 3/10 m, etc.

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

How do you multiply a whole number by a fraction?

Let's take the numbers encountered in the last problem:

According to the definition, we must find 3/4 of 50. Let's first find 1/4 of 50, and then 3/4.

1/4 of 50 is 50/4;

3/4 of the number 50 is .

Hence.

Let's consider another example: 12 5 / 8 =?

1/8 of the number 12 is 12/8,

5/8 of the number 12 is .

Hence,

From here we get the rule:

To multiply a whole number by a fraction, you need to multiply the whole number by the numerator of the fraction and make this product the numerator, and sign the denominator of this fraction as the denominator.

Let's write this rule using letters:

To make this rule completely clear, it should be remembered that a fraction can be considered as a quotient. Therefore, it is useful to compare the found rule with the rule for multiplying a number by a quotient, which was set out in § 38

It is important to remember that before performing multiplication, you should do (if possible) reductions, For example:

4. Multiplying a fraction by a fraction. Multiplying a fraction by a fraction has the same meaning as multiplying a whole number by a fraction, i.e., when multiplying a fraction by a fraction, you need to find the fraction that is in the factor from the first fraction (the multiplicand).

Namely, multiplying 3/4 by 1/2 (half) means finding half of 3/4.

How do you multiply a fraction by a fraction?

Let's take an example: 3/4 multiplied by 5/7. This means you need to find 5/7 of 3/4. Let's first find 1/7 of 3/4, and then 5/7

1/7 of the number 3/4 will be expressed as follows:

5/7 numbers 3/4 will be expressed as follows:

Thus,

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

1/9 of 5/8 is ,

4/9 of the number 5/8 is .

Thus,

From these examples the following rule can be deduced:

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

This rule can be written in general form as follows:

When multiplying, it is necessary to make (if possible) reductions. Let's look at examples:

5. Multiplication of mixed numbers. Since mixed numbers can easily be replaced by improper fractions, this circumstance is usually used when multiplying mixed numbers. This means that in cases where the multiplicand, or the multiplier, or both factors are expressed as mixed numbers, they are replaced by improper fractions. Let's multiply, for example, mixed numbers: 2 1/2 and 3 1/5. Let's turn each of them into an improper fraction and then multiply the resulting fractions according to the rule for multiplying a fraction by a fraction:

Rule. To multiply mixed numbers, you must first convert them into improper fractions and then multiply them according to the rule for multiplying fractions by fractions.

Note. If one of the factors is an integer, then the multiplication can be performed based on the distribution law as follows:

6. The concept of interest. When solving problems and performing various practical calculations, we use all kinds of fractions. But it must be borne in mind that many quantities allow not just any, but natural divisions for them. For example, you can take one hundredth (1/100) of a ruble, it will be a kopeck, two hundredths is 2 kopecks, three hundredths is 3 kopecks. You can take 1/10 of a ruble, it will be "10 kopecks, or a ten-kopeck piece. You can take a quarter of a ruble, i.e. 25 kopecks, half a ruble, i.e. 50 kopecks (fifty kopecks). But they practically don’t take it, for example , 2/7 of a ruble because the ruble is not divided into sevenths.

The unit of weight, i.e. the kilogram, primarily allows for decimal divisions, for example 1/10 kg, or 100 g. And such fractions of a kilogram as 1/6, 1/11, 1/13 are not common.

In general, our (metric) measures are decimal and allow decimal divisions.

However, it should be noted that it is extremely useful and convenient in a wide variety of cases to use the same (uniform) method of subdividing quantities. Many years of experience have shown that such a well-justified division is the “hundredth” division. Let us consider several examples relating to the most diverse areas of human practice.

1. The price of books has decreased by 12/100 of the previous price.

Example. The previous price of the book was 10 rubles. It decreased by 1 ruble. 20 kopecks

2. Savings banks pay depositors 2/100 of the amount deposited for savings during the year.

Example. 500 rubles are deposited in the cash register, the income from this amount for the year is 10 rubles.

3. The number of graduates from one school was 5/100 of the total number of students.

EXAMPLE There were only 1,200 students at the school, of which 60 graduated.

The hundredth part of a number is called a percentage.

The word "percent" is borrowed from Latin and its root "cent" means one hundred. Together with the preposition (pro centum), this word means “for a hundred.” The meaning of this expression follows from the fact that initially in ancient Rome interest was the name given to the money that the debtor paid to the lender “for every hundred.” The word “cent” is heard in such familiar words: centner (one hundred kilograms), centimeter (say centimeter).

For example, instead of saying that over the past month the plant produced 1/100 of all products produced by it was defective, we will say this: over the past month the plant produced one percent of defects. Instead of saying: the plant produced 4/100 more products than the established plan, we will say: the plant exceeded the plan by 4 percent.

The above examples can be expressed differently:

1. The price of books has decreased by 12 percent of the previous price.

2. Savings banks pay depositors 2 percent per year on the amount deposited in savings.

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

To shorten the letter, it is customary to write the % symbol instead of the word “percentage”.

However, you need to remember that in calculations the % sign is usually not written; it can be written in the problem statement and in the final result. When performing calculations, you need to write a fraction with a denominator of 100 instead of a whole number with this symbol.

You need to be able to replace an integer with the indicated icon with a fraction with a denominator of 100:

Conversely, you need to get used to writing an integer with the indicated symbol instead of a fraction with a denominator of 100:

7. Finding the percentage of a given number.

Task 1. The school received 200 cubic meters. m of firewood, with birch firewood accounting for 30%. How much birch firewood was there?

The meaning of this problem is that birch firewood made up only part of the firewood that was delivered to the school, and this part is expressed in the fraction 30/100. This means that we have a task to find a fraction of a number. To solve it, we must multiply 200 by 30/100 (problems of finding the fraction of a number are solved by multiplying the number by the fraction.).

This means that 30% of 200 equals 60.

The fraction 30/100 encountered in this problem can be reduced by 10. It would be possible to do this reduction from the very beginning; the solution to the problem would not have changed.

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

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

This means that here you will need to find the fraction of the number three times. Let's do it:

1) How many 11-year-old children were there?

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

3) How many 13-year-old children were there?

After solving the problem, it is useful to add the numbers found; their sum should be 300:

63 + 183 + 54 = 300

It should also be noted that the sum of the percentages given in the problem statement is 100:

21% + 61% + 18% = 100%

This suggests that the total number of children in the camp was taken as 100%.

3 a d a h a 3. The worker received 1,200 rubles per month. Of this, he spent 65% on food, 6% on apartments and heating, 4% on gas, electricity and radio, 10% on cultural needs and 15% saved. How much money was spent on the needs indicated in the problem?

To solve this problem you need to find the fraction of 1,200 5 times. Let's do this.

1) How much money was spent on food? The problem says that this expense is 65% of total earnings, i.e. 65/100 of the number 1,200. Let’s do the calculation:

2) How much money did you pay for an apartment with heating? Reasoning similarly to the previous one, we arrive at the following calculation:

3) How much money did you pay for gas, electricity and radio?

4) How much money was spent on cultural needs?

5) How much money did the worker save?

To check, it is useful to add up the numbers found in these 5 questions. The amount should be 1,200 rubles. All earnings are taken as 100%, which is easy to check by adding up the percentage numbers given in the problem statement.

We solved three problems. Despite the fact that these problems dealt with different things (delivery of firewood for the school, the number of children of different ages, the worker's expenses), they were solved in the same way. This happened because in all problems it was necessary to find several percent of given numbers.

§ 90. Division of fractions.

As we study division of fractions, we will consider the following questions:

1. Divide an integer by an integer.
2. Dividing a fraction by a whole number
3. Dividing a whole number by a fraction.
4. Dividing a fraction by a fraction.
5. Division of mixed numbers.
6. Finding a number from its given fraction.
7. Finding a number by its percentage.

Let's consider them sequentially.

1. Divide an integer by an integer.

As was indicated in the department of integers, division is the action that consists in the fact that, given the product of two factors (dividend) and one of these factors (divisor), another factor is found.

We looked at dividing an integer by an integer in the section on integers. We encountered two cases of division there: division without a remainder, or “entirely” (150: 10 = 15), and division with a remainder (100: 9 = 11 and 1 remainder). We can therefore say that in the field of integers, exact division is not always possible, because the dividend is not always the product of the divisor by the integer. After introducing multiplication by a fraction, we can consider any case of dividing integers possible (only division by zero is excluded).

For example, dividing 7 by 12 means finding a number whose product by 12 would be equal to 7. Such a number is the fraction 7 / 12 because 7 / 12 12 = 7. Another example: 14: 25 = 14 / 25, because 14 / 25 25 = 14.

Thus, to divide a whole number by a whole number, you need to create a fraction whose numerator is equal to the dividend and the denominator is equal to the divisor.

2. Dividing a fraction by a whole number.

Divide the fraction 6 / 7 by 3. According to the definition of division given above, we have here the product (6 / 7) and one of the factors (3); it is required to find a second factor that, when multiplied by 3, would give the given product 6/7. Obviously, it should be three times smaller than this product. This means that the task set before us was to reduce the fraction 6/7 by 3 times.

We already know that reducing a fraction can be done either by decreasing its numerator or by increasing its denominator. Therefore you can write:

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

Let's take another example: 5 / 8 divided by 2. Here the numerator 5 is not divisible by 2, which means that the denominator will have to be multiplied by this number:

Based on this, a rule can be made: To divide a fraction by a whole number, you need to divide the numerator of the fraction by that whole number.(if possible), leaving the same denominator, or multiply the denominator of the fraction by this number, leaving the same numerator.

3. Dividing a whole number by a fraction.

Let it be necessary to divide 5 by 1/2, i.e., find a number that, after multiplying by 1/2, will give the product 5. Obviously, this number must be greater than 5, since 1/2 is a proper fraction, and when multiplying a number the product of a proper fraction must be less than the product being multiplied. To make this clearer, let's write our actions as follows: 5: 1 / 2 = X , which means x 1 / 2 = 5.

We must find such a number X , which, if multiplied by 1/2, would give 5. Since multiplying a certain number by 1/2 means finding 1/2 of this number, then, therefore, 1/2 of the unknown number X is equal to 5, and the whole number X twice as much, i.e. 5 2 = 10.

So 5: 1 / 2 = 5 2 = 10

Let's check:

Let's look at another example. Let's say you want to divide 6 by 2/3. Let's first try to find the desired result using the drawing (Fig. 19).

Fig.19

Let us draw a segment AB equal to 6 units, and divide each unit into 3 equal parts. In each unit, three thirds (3/3) of the entire segment AB is 6 times larger, i.e. e. 18/3. Using small brackets, we connect the 18 resulting segments of 2; There will be only 9 segments. This means that the fraction 2/3 is contained in 6 units 9 times, or, in other words, the fraction 2/3 is 9 times less than 6 whole units. Hence,

How to get this result without a drawing using calculations alone? Let's reason like this: we need to divide 6 by 2/3, i.e. we need to answer the question how many times 2/3 is contained in 6. Let's find out first: how many times 1/3 is contained in 6? In a whole unit there are 3 thirds, and in 6 units there are 6 times more, i.e. 18 thirds; to find this number we must multiply 6 by 3. This means that 1/3 is contained in b units 18 times, and 2/3 is contained in b units not 18 times, but half as many times, i.e. 18: 2 = 9. Therefore , when dividing 6 by 2/3 we did the following:

From here we get the rule for dividing a whole number by a fraction. To divide a whole number by a fraction, you need to multiply this whole number by the denominator of the given fraction and, making this product the numerator, divide it by the numerator of the given fraction.

Let's write the rule using letters:

To make this rule completely clear, it should be remembered that a fraction can be considered as a quotient. Therefore, it is useful to compare the rule found with the rule for dividing a number by a quotient, which was set out in § 38. Please note that the same formula was obtained there.

When dividing, abbreviations are possible, for example:

4. Dividing a fraction by a fraction.

Let's say we need to divide 3/4 by 3/8. What will the number that results from division mean? It will answer the question how many times the fraction 3/8 is contained in the fraction 3/4. To understand this issue, let's make a drawing (Fig. 20).

Let's take a segment AB, take it as one, divide it into 4 equal parts and mark 3 such parts. Segment AC will be equal to 3/4 of segment AB. Let us now divide each of the four original segments in half, then the segment AB will be divided into 8 equal parts and each such part will be equal to 1/8 of the segment AB. Let us connect 3 such segments with arcs, then each of the segments AD and DC will be equal to 3/8 of the segment AB. The drawing shows that a segment equal to 3/8 is contained in a segment equal to 3/4 exactly 2 times; This means that the result of division can be written as follows:

3 / 4: 3 / 8 = 2

Let's look at another example. Let's say we need to divide 15/16 by 3/32:

We can reason like this: we need to find a number that, after multiplying by 3/32, will give a product equal to 15/16. Let's write the calculations like this:

15 / 16: 3 / 32 = X

3 / 32 X = 15 / 16

3/32 unknown number X are 15/16

1/32 of an unknown number X is ,

32 / 32 numbers X make up .

Hence,

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

Let's write the rule using letters:

When dividing, abbreviations are possible, for example:

5. Division of mixed numbers.

When dividing mixed numbers, they must first be converted into improper fractions, and then the resulting fractions must be divided according to the rules for dividing fractions. Let's look at an example:

Let's convert mixed numbers to improper fractions:

Now let's divide:

Thus, to divide mixed numbers, you need to convert them into improper fractions and then divide using the rule for dividing fractions.

6. Finding a number from its given fraction.

Among the various fraction problems, sometimes there are those in which the value of some fraction of an unknown number is given and you need to find this number. This type of problem will be the inverse of the problem of finding the fraction of a given number; there a number was given and it was required to find some fraction of this number, here a fraction of a number was given and it was required to find this number itself. This idea will become even clearer if we turn to solving this type of problem.

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

Solution. The problem says that 50 glazed windows make up 1/3 of all the windows of the house, which means there are 3 times more windows in total, i.e.

The house had 150 windows.

Task 2. The store sold 1,500 kg of flour, which is 3/8 of the total flour stock the store had. What was the store's initial supply of flour?

Solution. From the conditions of the problem it is clear that 1,500 kg of flour sold constitute 3/8 of the total stock; This means that 1/8 of this reserve will be 3 times less, i.e. to calculate it you need to reduce 1500 by 3 times:

1,500: 3 = 500 (this is 1/8 of the reserve).

Obviously, the entire supply will be 8 times larger. Hence,

500 8 = 4,000 (kg).

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

From consideration of this problem, the following rule can be derived.

To find a number from a given value of its fraction, it is enough to divide this value by the numerator of the fraction and multiply the result by the denominator of the fraction.

We solved two problems on finding a number given its fraction. Such problems, as is especially clearly seen from the last one, are solved by two actions: division (when one part is found) and multiplication (when the whole number is found).

However, after we have learned the division of fractions, the above problems can be solved with one action, namely: division by a fraction.

For example, the last task can be solved in one action like this:

In the future, we will solve problems of finding a number from its fraction with one action - division.

7. Finding a number by its percentage.

In these problems you will need to find a number knowing a few percent of that number.

Task 1. At the beginning of this year I received 60 rubles from the savings bank. income from the amount I put into savings a year ago. How much money have I put in the savings bank? (The cash desks give depositors a 2% return per year.)

The point of the problem is that I put a certain amount of money in a savings bank and stayed there for a year. After a year, I received 60 rubles from her. income, which is 2/100 of the money I deposited. How much money did I put in?

Consequently, knowing part of this money, expressed in two ways (in rubles and fractions), we must find the entire, as yet unknown, amount. This is an ordinary problem of finding a number given its fraction. The following problems are solved by division:

This means that 3,000 rubles were deposited in the savings bank.

Task 2. Fishermen fulfilled the monthly plan by 64% in two weeks, harvesting 512 tons of fish. What was their plan?

From the conditions of the problem it is known that the fishermen completed part of the plan. This part is equal to 512 tons, which is 64% of the plan. We don’t know how many tons of fish need to be prepared according to the plan. Finding this number will be the solution to the problem.

Such problems are solved by division:

This means that according to the plan, 800 tons of fish need to be prepared.

Task 3. The train went from Riga to Moscow. When he passed the 276th kilometer, one of the passengers asked a passing conductor how much of the journey they had already covered. To this the conductor replied: “We have already covered 30% of the entire journey.” What is the distance from Riga to Moscow?

From the problem conditions it is clear that 30% of the route from Riga to Moscow is 276 km. We need to find the entire distance between these cities, i.e., for this part, find the whole:

§ 91. Reciprocal numbers. Replacing division with multiplication.

Let's take the fraction 2/3 and replace the numerator in place of the denominator, we get 3/2. We got the inverse of this fraction.

In order to obtain the inverse of a given fraction, you need to put its numerator in place of the denominator, and the denominator in place of the numerator. In this way we can get the reciprocal of any fraction. For example:

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

Two fractions that have the property that the numerator of the first is the denominator of the second, and the denominator of the first is the numerator of the second, are called mutually inverse.

Now let's think about what fraction will be the reciprocal of 1/2. Obviously, it will be 2 / 1, or just 2. By looking for the inverse fraction of the given one, we got an integer. And this case is not isolated; on the contrary, for all fractions with a numerator of 1 (one), the reciprocals will be integers, for example:

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

Since in finding reciprocal fractions we also encountered integers, in what follows we will talk not about reciprocal fractions, but about reciprocal numbers.

Let's figure out how to write the inverse of an integer. For fractions, this can be solved simply: you need to put the denominator in place of the numerator. In the same way, you can get the inverse of an integer, since any integer can have a denominator of 1. This means that the inverse of 7 will be 1/7, because 7 = 7/1; for the number 10 the inverse will be 1/10, since 10 = 10/1

This idea can be expressed differently: the reciprocal of a given number is obtained by dividing one by a given number. This statement is true not only for whole numbers, but also for fractions. In fact, if we need to write the inverse of the fraction 5/9, then we can take 1 and divide it by 5/9, i.e.

Now let's point out one thing property reciprocal numbers, which will be useful to us: the product of reciprocal numbers is equal to one. Indeed:

Using this property, we can find reciprocal numbers in the following way. Let's say we need to find the inverse of 8.

Let's denote it by the letter X , then 8 X = 1, hence X = 1/8. Let's find another number that is the inverse of 7/12 and denote it by the letter X , then 7/12 X = 1, hence X = 1: 7 / 12 or X = 12 / 7 .

We introduced here the concept of reciprocal numbers in order to slightly supplement the information about dividing fractions.

When we divide the number 6 by 3/5, we do the following:

Pay special attention to the expression and compare it with the given one: .

If we take the expression separately, without connection with the previous one, then it is impossible to solve the question of where it came from: from dividing 6 by 3/5 or from multiplying 6 by 5/3. In both cases the same thing happens. Therefore we can say that dividing one number by another can be replaced by multiplying the dividend by the inverse of the divisor.

The examples we give below fully confirm this conclusion.

GET OVER THESE RAKES ALREADY! 🙂

Multiplying and dividing fractions.

Attention!
There are additional
materials in Special Section 555.
For those who are very “not very. »
And for those who “very much so. ")

This operation is much more pleasant than addition and subtraction! Because it's easier. As a reminder, to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:

Everything is extremely simple. And please don't look for a common denominator! There is no need for him here...

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

If you come across multiplication or division with integers and fractions, it’s okay. As with addition, we make a fraction from a whole number with one in the denominator - and go ahead! For example:

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

How can I make this fraction look decent? Yes, very simple! Use two-point division:

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

What determines the order of division? Either with brackets, or (as here) with the length of horizontal lines. Develop your eye. And if there are no brackets or dashes, like:

then divide and multiply in order, from left to right!

And another very simple and important technique. In actions with degrees, it will be so useful to you! Let's divide one by any fraction, for example, by 13/15:

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

That's it for operations with fractions. The thing is quite simple, but it gives more than enough errors. Take practical advice into account, and there will be fewer of them (mistakes)!

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not general words, not good wishes! This is a dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.

2. In examples with different types of fractions, we move on to ordinary fractions.

3. We reduce all fractions until they stop.

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

Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. The first time! Without a calculator! And draw the right conclusions.

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

So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. But only Then look at the answers.

We are looking for answers that match yours. I deliberately wrote them down in disarray, away from temptation, so to speak. Here they are, the answers, separated by semicolons.

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

Now we draw conclusions. If everything worked out, I’m happy for you! Basic calculations with fractions are not your problem! You can do more serious things. If not.

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

All these (and more!) examples are discussed in Special Section 555 “Fractions”. With detailed explanations of what, why and how. This analysis helps a lot with a lack of knowledge and skills!

Yes, and there is something about the second problem.) Quite practical advice, how to become more attentive. Yes Yes! Advice that can be applied every.

In addition to knowledge and attentiveness, success requires a certain automaticity. Where can I get it? I hear a heavy sigh... Yes, only in practice, nowhere else.

You can go to the website 321start.ru for training. There in the “Try” option there are 10 examples for everyone. With instant verification. For registered users - 34 examples from simple to severe. This is only in fractions.

If you like this site.

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

Here you can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

And here you can get acquainted with functions and derivatives.

Rule 1.

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

Rule 2.

To multiply a fraction by a fraction:

1. find the product of the numerators and the product of the denominators of these fractions

2. Write the first product as the numerator, and the second as the denominator.

Rule 3.

In order to multiply mixed numbers, you need to write them as improper fractions, and then use the rule for multiplying fractions.

Rule 4.

To divide one fraction by another, you must multiply the dividend by the reciprocal of the divisor.

Example 1.

Calculate

Example 2.

Calculate

Example 3.

Calculate

Example 4.

Calculate

Mathematics. Other materials

Raising a number to a rational power. (

Raising a number to a natural power. (

Generalized interval method for solving algebraic inequalities (Author A.V. Kolchanov)

Method for replacing factors when solving algebraic inequalities (Author Kolchanov A.V.)

Signs of divisibility (Lungu Alena)

Test yourself on the topic ‘Multiplication and division of ordinary fractions’

Multiplying fractions

We will consider the multiplication of ordinary fractions in several possible options.

Multiplying a common fraction by a fraction

This is the simplest case in which you need to use the following rules for multiplying fractions.

To multiply fraction by fraction, necessary:

multiply the numerator of the first fraction by the numerator of the second fraction and write their product into the numerator of the new fraction;

multiply the denominator of the first fraction by the denominator of the second fraction and write their product into the denominator of the new fraction;

Before multiplying numerators and denominators, check to see if the fractions can be reduced. Reducing fractions in calculations will make your calculations much easier.

Multiplying a fraction by a natural number

To make a fraction multiply by a natural number You need to multiply the numerator of the fraction by this number, and leave the denominator of the fraction unchanged.

If the result of multiplication is an improper fraction, do not forget to turn it into a mixed number, that is, highlight the whole part.

Multiplying mixed numbers

To multiply mixed numbers, you must first turn them into improper fractions and then multiply according to the rule for multiplying ordinary fractions.

Another way to multiply a fraction by a natural number

Sometimes when making calculations it is more convenient to use another method of multiplying a common fraction by a number.

To multiply a fraction by a natural number, you need to divide the denominator of the fraction by this number, and leave the numerator the same.

As can be seen from the example, this version of the rule is more convenient to use if the denominator of the fraction is divisible by a natural number without a remainder.

Dividing a fraction by a number

What is the fastest way to divide a fraction by a number? Let's analyze the theory, draw a conclusion, and use examples to see how dividing a fraction by a number can be done using a new short rule.

Typically, dividing a fraction by a number follows the rule for dividing fractions. We multiply the first number (fraction) by the inverse of the second. Since the second number is an integer, its inverse is a fraction, the numerator of which is equal to one, and the denominator is equal to the given number. Schematically, dividing a fraction by a natural number looks like this:

From this we conclude:

To divide a fraction by a number, you need to multiply the denominator by that number and leave the numerator the same. The rule can be formulated even more briefly:

When dividing a fraction by a number, the number goes into the denominator.

Divide a fraction by a number:

To divide a fraction by a number, we rewrite the numerator unchanged, and multiply the denominator by this number. We reduce 6 and 3 by 3.

When dividing a fraction by a number, we rewrite the numerator and multiply the denominator by that number. We reduce 16 and 24 by 8.

When dividing a fraction by a number, the number goes into the denominator, so we leave the numerator the same and multiply the denominator by the divisor. We reduce 21 and 35 by 7.

Multiplying and dividing fractions

Last time we learned how to add and subtract fractions (see lesson “Adding and Subtracting Fractions”). The most difficult part of those actions was bringing fractions to a common denominator.

Now it's time to deal with multiplication and division. The good news is that these operations are even simpler than addition and subtraction. First, let's consider the simplest case, when there are two positive fractions without a separated integer part.

To multiply two fractions, you must multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.

To divide two fractions, you need to multiply the first fraction by the “inverted” second fraction.

From the definition it follows that dividing fractions reduces to multiplication. To “flip” a fraction, just swap the numerator and denominator. Therefore, throughout the lesson we will mainly consider multiplication.

As a result of multiplication, a reducible fraction can arise (and often does arise) - it, of course, must be reduced. If after all the reductions the fraction turns out to be incorrect, the whole part should be highlighted. But what definitely won't happen with multiplication is reduction to a common denominator: no criss-cross methods, greatest factors and least common multiples.

Task. Find the meaning of the expression:

By definition we have:

Multiplying fractions with whole parts and negative fractions

If fractions contain an integer part, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.

If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the multiplication or removed altogether according to the following rules:

1. Plus by minus gives minus;
2. Two negatives make an affirmative.

Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was necessary to get rid of the whole part. For a work, they can be generalized in order to “burn” several disadvantages at once:

3. We cross out the negatives in pairs until they completely disappear. In extreme cases, one minus can survive - the one for which there was no mate;
4. If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out because there was no pair for it, we take it outside the limits of multiplication. The result is a negative fraction.

We convert all fractions to improper ones, and then take the minuses out of the multiplication. We multiply what is left according to the usual rules. We get:

Let me remind you once again that the minus that appears in front of a fraction with a highlighted whole part refers specifically to the entire fraction, and not just to its whole part (this applies to the last two examples).

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

Reducing fractions on the fly

Multiplication is a very labor-intensive operation. The numbers here turn out to be quite large, and to simplify the problem, you can try to reduce the fraction further before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:

In all examples, the numbers that have been reduced and what remains of them are marked in red.

Please note: in the first case, the multipliers were reduced completely. In their place there remain units that, generally speaking, need not be written. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.

However, never use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:

You can't do that!

The error occurs because when adding, the numerator of a fraction produces a sum, not a product of numbers. Consequently, it is impossible to apply the basic property of a fraction, since this property deals specifically with the multiplication of numbers.

There are simply no other reasons for reducing fractions, so the correct solution to the previous problem looks like this:

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

Dividing fractions.

Dividing a fraction by a natural number.

Examples of dividing a fraction by a natural number

Dividing a natural number by a fraction.

Examples of dividing a natural number by a fraction

Division of ordinary fractions.

Examples of dividing ordinary fractions

Dividing mixed numbers.

To divide one mixed number by another, you need to:
• convert mixed fractions to improper fractions;
• multiply the first fraction by the reciprocal of the second;
• reduce the resulting fraction;
• If you get an improper fraction, convert the improper fraction into a mixed fraction.

Examples of dividing mixed numbers

1 1 2: 2 2 3 = 1 2 + 1 2: 2 3 + 2 3 = 3 2: 8 3 = 3 2 3 8 = 3 3 2 8 = 9 16

2 1 7: 3 5 = 2 7 + 1 7: 3 5 = 15 7: 3 5 = 15 7 5 3 = 15 5 7 3 = 5 5 7 = 25 7 = 7 3 + 4 7 = 3 4 7

Any obscene comments will be deleted and their authors will be blacklisted!

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

Fractions. Multiplying and dividing fractions.

Multiplying a common fraction by a fraction.

To multiply ordinary fractions, you need to multiply the numerator by the numerator (we get the numerator of the product) and the denominator by the denominator (we get the denominator of the product).

Formula for multiplying fractions:





Before you begin multiplying numerators and denominators, you need to check whether the fraction can be reduced. If you can reduce the fraction, it will be easier for you to make further calculations.

Note! There is no need to look for a common denominator here!!

Dividing a common fraction by a fraction.

Dividing an ordinary fraction by a fraction occurs like this: you turn the second fraction over (i.e., change the numerator and denominator) and after that the fractions are multiplied.

Formula for dividing ordinary fractions:





Multiplying a fraction by a natural number.

Note! When multiplying a fraction by a natural number, the numerator of the fraction is multiplied by our natural number, and the denominator of the fraction is left the same. If the result of the product is an improper fraction, then be sure to highlight the whole part, turning the improper fraction into a mixed fraction.



Dividing fractions involving natural numbers.

It's not as scary as it seems. As with addition, we convert the whole number into a fraction with one in the denominator. For example:



Multiplying mixed fractions.

Rules for multiplying fractions (mixed):

• convert mixed fractions to improper fractions;
• multiplying the numerators and denominators of fractions;
• reduce the fraction;
• If you get an improper fraction, then we convert the improper fraction into a mixed fraction.

Note! To multiply a mixed fraction by another mixed fraction, you first need to convert them to the form of improper fractions, and then multiply according to the rule for multiplying ordinary fractions.



The second way to multiply a fraction by a natural number.

It may be more convenient to use the second method of multiplying a common fraction by a number.

Note! To multiply a fraction by a natural number, you must divide the denominator of the fraction by this number, and leave the numerator unchanged.



From the example given above, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.

Multistory fractions.

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

To bring such a fraction to its usual form, use division through 2 points:



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

Note, For example:

When dividing one by any fraction, the result will be the same fraction, only inverted:

Practical tips for multiplying and dividing fractions:

1. The most important thing when working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It's better to write a few extra lines in your draft than to get lost in mental calculations.

2. In tasks with different types of fractions, go to the type of ordinary fractions.

3. We reduce all fractions until it is no longer possible to reduce.

4. We transform multi-level fractional expressions into ordinary ones using division through 2 points.

• Under- and under- Reworked song "Spring Tango" (The time comes - birds fly from the south) - music. Valery Milyaev I didn’t hear enough, I didn’t understand, I didn’t get it, in the sense that I didn’t guess, I wrote all the verbs with inseparably, I didn’t know about the prefix nedo. It happens, […]
• Page not found In the third final reading, a package of Government documents providing for the creation of special administrative regions (SAR) was adopted. As a result of leaving the European Union, the UK will not be included in the European VAT area and […]
• The Joint Investigative Committee will appear in the fall The Joint Investigative Committee will appear in the fall The investigation of all law enforcement agencies will be brought under one roof on the fourth attempt Already in the fall of 2014, according to Izvestia, President Vladimir Putin […]
• A patent for an algorithm What a patent for an algorithm looks like How a patent for an algorithm is prepared Preparing technical descriptions of methods for storing, processing, and transmitting signals and/or data specifically for patenting purposes usually does not present any special difficulties, and […]
• WHAT IS IMPORTANT TO KNOW ABOUT THE NEW BILL ON PENSIONS December 12, 1993 CONSTITUTION OF THE RUSSIAN FEDERATION (taking into account amendments made by the Laws of the Russian Federation on amendments to the Constitution of the Russian Federation dated December 30, 2008 N 6-FKZ, dated December 30, 2008 N 7-FKZ, […]
• Funny ditties about a woman's pension for the hero of the day, men for the hero of the day, men - in chorus for the hero of the day, women - dedication to pensioners, women, funny. Competitions for pensioners will be interesting. Presenter: Dear friends! Just a moment! Sensation! Only […]

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:

Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.

From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.

If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells about a flying arrow:

A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.

Wednesday, July 4, 2018

The differences between set and multiset are described very well on Wikipedia. Let's see.

As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.

Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.

First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...

And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.

Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

Sunday, March 18, 2018

The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.

Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.

1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.

2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic symbols into numbers. This is not a mathematical operation.

4. Add the resulting numbers. Now this is mathematics.

The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.

From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With the large number 12345, I don’t want to fool my head, let’s consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.

As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of ​​a rectangle in meters and centimeters, you would get completely different results.

Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.

The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.

What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.

Sign on the door He opens the door and says:

Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?

Female... The halo on top and the arrow down are male.

If such a work of design art flashes before your eyes several times a day,

Then it’s not surprising that you suddenly find a strange icon in your car:

Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: minus sign, number four, designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.

1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.

We will consider the multiplication of ordinary fractions in several possible options.

Multiplying a common fraction by a fraction

This is the simplest case in which you need to use the following rules for multiplying fractions.

To multiply fraction by fraction, necessary:

• multiply the numerator of the first fraction by the numerator of the second fraction and write their product into the numerator of the new fraction;
• multiply the denominator of the first fraction by the denominator of the second fraction and write their product into the denominator of the new fraction;

Before multiplying numerators and denominators, check to see if the fractions can be reduced. Reducing fractions in calculations will make your calculations much easier.



Multiplying a fraction by a natural number

To make a fraction multiply by a natural number You need to multiply the numerator of the fraction by this number, and leave the denominator of the fraction unchanged.

If the result of multiplication is an improper fraction, do not forget to turn it into a mixed number, that is, highlight the whole part.



Multiplying mixed numbers

To multiply mixed numbers, you must first turn them into improper fractions and then multiply according to the rule for multiplying ordinary fractions.



Another way to multiply a fraction by a natural number

Sometimes when making calculations it is more convenient to use another method of multiplying a common fraction by a number.

To multiply a fraction by a natural number, you need to divide the denominator of the fraction by this number, and leave the numerator the same.



As can be seen from the example, this version of the rule is more convenient to use if the denominator of the fraction is divisible by a natural number without a remainder.

Operations with fractions

Adding fractions with like denominators

There are two types of addition of fractions:

• Adding fractions with like denominators
• Adding fractions with different denominators

First, let's learn the addition of fractions with like denominators. Everything is simple here. To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged. For example, let's add the fractions and . Add the numerators and leave the denominator unchanged:



This example can be easily understood if we remember the pizza, which is divided into four parts. If you add pizza to pizza, you get pizza:

Example 2. Add fractions and .

Again, we add up the numerators and leave the denominator unchanged:



The answer turned out to be an improper fraction. When the end of the task comes, it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part of it. In our case, the whole part is easily isolated - two divided by two equals one:



This example can be easily understood if we remember about a pizza that is divided into two parts. If you add more pizza to the pizza, you get one whole pizza:

Example 3. Add fractions and .



This example can be easily understood if we remember the pizza, which is divided into three parts. If you add more pizza to the pizza, you get pizza:

Example 4. Find the value of an expression

This example is solved in exactly the same way as the previous ones. The numerators must be added and the denominator left unchanged:

Let's try to depict our solution using a drawing. If you add pizzas to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.

As you can see, there is nothing complicated about adding fractions with the same denominators. It is enough to understand the following rules:

1. To add fractions with the same denominator, you need to add their numerators and leave the denominator the same;
2. If the answer turns out to be an improper fraction, then you need to highlight the whole part of it.

Adding fractions with different denominators

Now let's learn how to add fractions with different denominators. When adding fractions, the denominators of the fractions must be the same. But they are not always the same.

For example, fractions can be added because they have the same denominators.

But fractions cannot be added right away, since these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

There are several ways to reduce fractions to the same denominator. Today we will look at only one of them, since the other methods may seem complicated for a beginner.

The essence of this method is that first we look for the least common multiple (LCM) of the denominators of both fractions. The LCM is then divided by the denominator of the first fraction to obtain the first additional factor. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and a second additional factor is obtained.

The numerators and denominators of the fractions are then multiplied by their additional factors. As a result of these actions, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to add such fractions.

Example 1. Let's add the fractions and

These fractions have different denominators, so you need to reduce them to the same (common) denominator.

First of all, we find the least common multiple of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 2. The least common multiple of these numbers is 6

LCM (2 and 3) = 6

Now let's return to fractions and . First, divide the LCM by the denominator of the first fraction and get the first additional factor. LCM is the number 6, and the denominator of the first fraction is the number 3. Divide 6 by 3, we get 2.

The resulting number 2 is the first additional multiplier. We write it down to the first fraction. To do this, make a small oblique line over the fraction and write down the additional factor found above it:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction and get the second additional factor. LCM is the number 6, and the denominator of the second fraction is the number 2. Divide 6 by 2, we get 3.

The resulting number 3 is the second additional multiplier. We write it down to the second fraction. Again, we make a small oblique line over the second fraction and write down the additional factor found above it:

Now we have everything ready for addition. It remains to multiply the numerators and denominators of the fractions by their additional factors:



Look carefully at what we have come to. We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to add such fractions. Let's take this example to the end:



This completes the example. It turns out to add .

Let's try to depict our solution using a drawing. If you add pizza to a pizza, you get one whole pizza and another sixth of a pizza:

Reducing fractions to the same (common) denominator can also be depicted using a picture. Reducing the fractions and to a common denominator, we got the fractions and . These two fractions will be represented by the same pieces of pizza. The only difference will be that this time they will be divided into equal shares (reduced to the same denominator).



The first drawing represents a fraction (four pieces out of six), and the second drawing represents a fraction (three pieces out of six). Adding these pieces we get (seven pieces out of six). This fraction is improper, so we highlighted the whole part of it. As a result, we got (one whole pizza and another sixth pizza).

Please note that we have described this example in too much detail. In educational institutions it is not customary to write in such detail. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the found additional factors by your numerators and denominators. If we were at school, we would have to write this example as follows:



But there is also another side to the coin. If you do not take detailed notes in the first stages of studying mathematics, then questions of the sort begin to appear. “Where does that number come from?”, “Why do fractions suddenly turn into completely different fractions? «.

To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:

3. Find the LCM of the denominators of fractions;
4. Divide the LCM by the denominator of each fraction and obtain an additional factor for each fraction;
5. Multiply the numerators and denominators of fractions by their additional factors;
6. Add fractions that have the same denominators;
7. If the answer turns out to be an improper fraction, then select its whole part;

Example 2. Find the value of an expression .

Let's use the diagram we provided above.

Step 1. Find the LCM for the denominators of the fractions

Find the LCM for the denominators of both fractions. The denominators of the fractions are the numbers 2, 3 and 4. You need to find the LCM for these numbers:

Step 2. Divide the LCM by the denominator of each fraction and get an additional factor for each fraction

Divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it above the first fraction:

Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. Divide 12 by 3, we get 4. We get the second additional factor 4. We write it above the second fraction:

Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We get the third additional factor 3. We write it above the third fraction:

Step 3. Multiply the numerators and denominators of the fractions by their additional factors

We multiply the numerators and denominators by their additional factors:

Step 4. Add fractions with the same denominators

We came to the conclusion that fractions that had different denominators turned into fractions that had the same (common) denominators. All that remains is to add these fractions. Add it up:



The addition didn't fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is moved to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning of the new line. The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.

Step 5. If the answer turns out to be an improper fraction, then highlight its whole part

Our answer turned out to be an improper fraction. We have to highlight a whole part of it. We highlight:



We received an answer

Subtracting fractions with like denominators

There are two types of subtraction of fractions:

8. Subtracting fractions with like denominators
9. Subtracting fractions with different denominators

First, let's learn how to subtract fractions with like denominators. Everything is simple here. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, but leave the denominator the same.

For example, let's find the value of the expression . To solve this example, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same. Let's do this:

This example can be easily understood if we remember the pizza, which is divided into four parts. If you cut pizzas from a pizza, you get pizzas:

Example 2. Find the value of the expression.

Again, from the numerator of the first fraction, subtract the numerator of the second fraction, and leave the denominator the same:

This example can be easily understood if we remember the pizza, which is divided into three parts. If you cut pizzas from a pizza, you get pizzas:

Example 3. Find the value of an expression

This example is solved in exactly the same way as the previous ones. From the numerator of the first fraction you need to subtract the numerators of the remaining fractions:

The answer was an improper fraction. If the example is completed, then it is customary to get rid of the improper fraction. Let's get rid of the improper fraction in the answer. To do this, let’s select its entire part:

As you can see, there is nothing complicated about subtracting fractions with the same denominators. It is enough to understand the following rules:

To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same;

If the answer turns out to be an improper fraction, then you need to highlight its whole part.

Subtracting fractions with different denominators

For example, you can subtract a fraction from a fraction because the fractions have the same denominators. But you cannot subtract a fraction from a fraction, since these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

The common denominator is found using the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written above the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written above the second fraction.

The fractions are then multiplied by their additional factors. As a result of these operations, fractions that had different denominators are converted into fractions that have the same denominators. And we already know how to subtract such fractions.

Example 1. Find the meaning of the expression:

First we find the LCM of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 4. The least common multiple of these numbers is 12

LCM (3 and 4) = 12

Now let's return to fractions and

Let's find an additional factor for the first fraction. To do this, divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. Write a four above the first fraction:

We do the same with the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. Write a three over the second fraction:

Now we are ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to subtract such fractions. Let's take this example to the end:

We received an answer

Let's try to depict our solution using a drawing. If you cut pizza from a pizza, you get pizza

This is the detailed version of the solution. If we were at school, we would have to solve this example shorter. Such a solution would look like this:

Reducing fractions to a common denominator can also be depicted using a picture. Reducing these fractions to a common denominator, we got the fractions and . These fractions will be represented by the same pizza slices, but this time they will be divided into equal shares (reduced to the same denominator):

The first picture shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting three pieces from eight pieces, we get five pieces out of twelve. The fraction describes these five pieces.

Example 2. Find the value of an expression

These fractions have different denominators, so first you need to reduce them to the same (common) denominator.

Let's find the LCM of the denominators of these fractions.

The denominators of the fractions are the numbers 10, 3 and 5. The least common multiple of these numbers is 30

LCM(10, 3, 5) = 30

Now we find additional factors for each fraction. To do this, divide the LCM by the denominator of each fraction.

Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is the number 10. Divide 30 by 10, we get the first additional factor 3. We write it above the first fraction:

Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it above the second fraction:

Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is the number 5. Divide 30 by 5, we get the third additional factor 6. We write it above the third fraction:

Now everything is ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same (common) denominators. And we already know how to subtract such fractions. Let's finish this example.

The continuation of the example will not fit on one line, so we move the continuation to the next line. Don't forget about the equal sign (=) on the new line:

The answer turned out to be a regular fraction, and everything seems to suit us, but it is too cumbersome and ugly. It would be necessary to make it simpler and more aesthetically pleasing. What can be done? You can shorten this fraction. Recall that reducing a fraction is the division of the numerator and denominator by the greatest common divisor of the numerator and denominator.

To correctly reduce a fraction, you need to divide its numerator and denominator by the greatest common divisor (GCD) of the numbers 20 and 30.

GCD should not be confused with NOC. The most common mistake of many beginners. GCD is the greatest common divisor. We find it to reduce a fraction.

And LCM is the least common multiple. We find it in order to bring fractions to the same (common) denominator.

Now we will find the greatest common divisor (GCD) of the numbers 20 and 30.

So, we find GCD for numbers 20 and 30:

GCD (20 and 30) = 10

Now we return to our example and divide the numerator and denominator of the fraction by 10:

We received a beautiful answer

Multiplying a fraction by a number

To multiply a fraction by a number, you need to multiply the numerator of the given fraction by that number and leave the denominator the same.

Example 1. Multiply a fraction by the number 1.

Multiply the numerator of the fraction by the number 1

The recording can be understood as taking half 1 time. For example, if you take pizza once, you get pizza

From the laws of multiplication we know that if the multiplicand and the factor are swapped, the product will not change. If the expression is written as , then the product will still be equal to . Again, the rule for multiplying a whole number and a fraction works:

This notation can be understood as taking half of one. For example, if there is 1 whole pizza and we take half of it, then we will have pizza:

Example 2. Find the value of an expression

Multiply the numerator of the fraction by 4

The expression can be understood as taking two quarters 4 times. For example, if you take 4 pizzas, you will get two whole pizzas

And if we swap the multiplicand and the multiplier, we get the expression . It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:

Multiplying fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer turns out to be an improper fraction, you need to highlight the whole part of it.

Example 1. Find the value of the expression.

We received an answer. It is advisable to reduce this fraction. The fraction can be reduced by 2. Then the final solution will take the following form:

The expression can be understood as taking a pizza from half a pizza. Let's say we have half a pizza:

How to take two thirds from this half? First you need to divide this half into three equal parts:

And take two from these three pieces:

We'll make pizza. Remember what pizza looks like when divided into three parts:

One piece of this pizza and the two pieces we took will have the same dimensions:

In other words, we are talking about the same size pizza. Therefore the value of the expression is

Example 2. Find the value of an expression

Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer was an improper fraction. Let's highlight the whole part of it:

Example 3. Find the value of an expression

The answer turned out to be a regular fraction, but it would be good if it was shortened. To reduce this fraction, it must be divided by the gcd of the numerator and denominator. So, let’s find the gcd of numbers 105 and 450:

GCD for (105 and 150) is 15

Now we divide the numerator and denominator of our answer by gcd:

Representing a whole number as a fraction

Any whole number can be represented as a fraction. For example, the number 5 can be represented as . This will not change the meaning of five, since the expression means “the number five divided by one,” and this, as we know, is equal to five:

Reciprocal numbers

Now we will get acquainted with a very interesting topic in mathematics. It's called "reverse numbers".

Definition. Reverse to number a is a number that, when multiplied by a gives one.

Let's substitute in this definition instead of the variable a number 5 and try to read the definition:

Reverse to number 5 is a number that, when multiplied by 5 gives one.

Is it possible to find a number that, when multiplied by 5, gives one? It turns out it is possible. Let's imagine five as a fraction:

Then multiply this fraction by itself, just swap the numerator and denominator. In other words, multiply a fraction by itself, only upside down:

What will happen as a result of this? If we continue to solve this example, we get one:

This means that the inverse of the number 5 is the number , since when you multiply 5 by you get one.

The reciprocal of a number can also be found for any other integer.

• the reciprocal of 3 is a fraction
• the reciprocal of 4 is a fraction

You can also find the reciprocal of any other fraction. To do this, just turn it over.

Multiplying and dividing fractions.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

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

For example:

Everything is extremely simple. And please don't look for a common denominator! There is no need for him here...

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

For example:

If you come across multiplication or division with integers and fractions, it’s okay. As with addition, we make a fraction from a whole number with one in the denominator - and go ahead! For example:

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

How can I make this fraction look decent? Yes, very simple! Use two-point division:

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

What determines the order of division? Either with brackets, or (as here) with the length of horizontal lines. Develop your eye. And if there are no brackets or dashes, like:

then divide and multiply in order, from left to right!

And another very simple and important technique. In actions with degrees, it will be so useful to you! Let's divide one by any fraction, for example, by 13/15:

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

That's it for operations with fractions. The thing is quite simple, but it gives more than enough errors. Take practical advice into account, and there will be fewer of them (mistakes)!

Practical tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not general words, not good wishes! This is a dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.

2. In examples with different types of fractions, we move on to ordinary fractions.

3. We reduce all fractions until they stop.

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

5. Divide a unit by a fraction in your head, simply turning the fraction over.

Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. The first time! Without a calculator! And draw the right conclusions...

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

So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. But only Then look at the answers.

Calculate:

Have you decided?

We are looking for answers that match yours. I deliberately wrote them down in disarray, away from temptation, so to speak... Here they are, the answers, written with semicolons.

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

Now we draw conclusions. If everything worked out, I’m happy for you! Basic calculations with fractions are not your problem! You can do more serious things. If not...

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

If you like this site...

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

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.