How to add large fractions. Fractions

§ 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 link 6 / 8 - 5 / 8 is written here for clarity, but can be skipped from now on.

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 there are units 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 the number with terms several times) and the new action (finding the fraction of the 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 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 factor, 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 to 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 its products as defective, we will say this: over the past month the plant produced one percent defective. 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 this:

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, 2 each; 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 found rule 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 clear 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 meaning 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. In fact:

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.

Fractional expressions are difficult for a child to understand. Most people have difficulties with. When studying the topic “adding fractions with whole numbers,” the child falls into a stupor, finding it difficult to solve the problem. In many examples, before performing an action, a series of calculations must be performed. For example, convert fractions or convert an improper fraction to a proper fraction.

Let’s explain it clearly to the child. Let's take three apples, two of which will be whole, and cut the third into 4 parts. Separate one slice from the cut apple, and place the remaining three next to two whole fruits. We get ¼ of an apple on one side and 2 ¾ on the other. If we combine them, we get three apples. Let's try to reduce 2 ¾ apples by ¼, that is, remove another slice, we get 2 2/4 apples.

Let's take a closer look at operations with fractions that contain integers:

First, let's remember the calculation rule for fractional expressions with a common denominator:

At first glance, everything is easy and simple. But this only applies to expressions that do not require conversion.

How to find the value of an expression where the denominators are different

In some tasks you need to find the meaning of an expression where the denominators are different. Let's look at a specific case:
3 2/7+6 1/3

Let's find the value of this expression by finding a common denominator for two fractions.

For the numbers 7 and 3, this is 21. We leave the integer parts the same, and bring the fractional parts to 21, for this we multiply the first fraction by 3, the second by 7, we get:
6/21+7/21, do not forget that whole parts cannot be converted. As a result, we get two fractions with the same denominator and calculate their sum:
3 6/21+6 7/21=9 15/21
What if the result of addition is an improper fraction that already has an integer part:
2 1/3+3 2/3
In this case, we add up the integer parts and fractional parts, we get:
5 3/3, as you know, 3/3 is one, which means 2 1/3+3 2/3=5 3/3=5+1=6

Finding the sum is all clear, let’s look at the subtraction:

From all that has been said, the rule for operations with mixed numbers follows, which sounds like this:

• If you need to subtract an integer from a fractional expression, you do not need to represent the second number as a fraction; it is enough to perform the operation only on the integer parts.

Let's try to calculate the meaning of the expressions ourselves:

Let’s take a closer look at the example under the letter “m”:

4 5/11-2 8/11, the numerator of the first fraction is less than the second. To do this, we borrow one integer from the first fraction, we get,
3 5/11+11/11=3 whole 16/11, subtract the second from the first fraction:
3 16/11-2 8/11=1 whole 8/11

• Be careful when completing the task, do not forget to convert improper fractions into mixed fractions, highlighting the whole part. To do this, you need to divide the value of the numerator by the value of the denominator, then what happens takes the place of the whole part, the remainder will be the numerator, for example:

19/4=4 ¾, let’s check: 4*4+3=19, the denominator 4 remains unchanged.

Let's summarize:

Before starting a task related to fractions, it is necessary to analyze what kind of expression it is, what transformations need to be made on the fraction in order for the solution to be correct. Look for a more rational solution. Don't go the hard way. Plan all the actions, solve them first in draft form, then transfer them to your school notebook.

To avoid confusion when solving fractional expressions, you must follow the rule of consistency. Decide everything carefully, without rushing.

As we know from mathematics, a fractional number consists of a numerator and a denominator. The numerator is at the top and the denominator is at the bottom.

It is quite simple to perform mathematical operations of adding or subtracting fractional quantities with the same denominator. You just need to be able to add or subtract the numbers in the numerator (above), and the same bottom number remains unchanged.

For example, let's take the fractional number 7/9, here:

• the number “seven” on top is the numerator;
• the number “nine” below is the denominator.

Example 1. Addition:

5/49 + 4/49 = (5+4) / 49 =9/49.

Example 2. Subtraction:

6/35−3/35 = (6−3) / 35 = 3/35.

Subtracting simple fractional values ​​that have different denominators

To perform the mathematical operation of subtracting quantities that have different denominators, you must first bring them to a single denominator. When performing this task, it is necessary to adhere to the rule that this common denominator must be the smallest of all possible options.

Example 3

Given two simple quantities with different denominators (lower numbers): 7/8 and 2/9.

It is necessary to subtract the second from the first value.

The solution consists of several steps:

1. Find the common lower number, i.e. something that is divisible by both the lower value of the first fraction and the second. This will be the number 72, since it is a multiple of the numbers eight and nine.

2. The bottom digit of each fraction has increased:

• the number “eight” in the fraction 7/8 has increased ninefold - 8*9=72;
• the number “nine” in the fraction 2/9 has increased eightfold - 9*8=72.

3. If the denominator (lower digit) has changed, then the numerator (upper digit) must also change. According to the existing mathematical rule, the top number must be increased by exactly the same amount as the bottom one. That is:

• the numerator “seven” in the first fraction (7/8) is multiplied by the number “nine” - 7*9=63;
• We multiply the numerator “two” in the second fraction (2/9) by the number “eight” - 2*8=16.

4. As a result of our actions, we got two new quantities, which, however, are identical to the original ones.

• first: 7/8 = 7*9 / 8*9 = 63/72;
• second: 2/9 = 2*8 / 9*8 = 16/72.

5. Now it is possible to subtract one fractional number from another:

7/8−2/9 = 63/72−16/72 =?

6. Carrying out this action, we return to the topic of subtracting fractions with the same lower digits (denominators). This means that the subtraction action will be carried out on top, in the numerator, and the bottom digit will be transferred without changes.

63/72−16/72 = (63−16) / 72 = 47/72.

7/8−2/9 = 47/72.

Example 4

Let's complicate the problem by taking several fractions with different but multiple numbers at the bottom to solve.

The values ​​given are: 5/6; 1/3; 1/12; 7/24.

They must be taken away from each other in this sequence.

1. We bring the fractions using the above method to a common denominator, which will be the number “24”:

• 5/6 = 5*4 / 6*4 = 20/24;
• 1/3 = 1*8 / 3*8 = 8/24;
• 1/12 = 1*2 / 12*2 = 2/24.

7/24 - we leave this last value unchanged, since the denominator is the total number “24”.

2. We subtract all quantities:

20/24−8/2−2/24−7/24 = (20−8−2−7)/24 = 3/24.

3. Since the numerator and denominator of the resulting fraction are divisible by one number, they can be reduced by dividing by the number “three”:

3:3 / 24:3 = 1/8.

4. We write the answer like this:

5/6−1/3−1/12−7/24 = 1/8.

Example 5

Three fractions with non-multiple denominators are given: 3/4; 2/7; 1/13.

You need to find the difference.

1. We bring the first two numbers to a common denominator, it will be the number “28”:

• ¾ = 3*7 / 4*7 = 21/28;
• 2/7 = 2*4 / 7*4 = 8/28.

2. Subtract the first two fractions from each other:

¾−2/7 = 21/28−8/28 = (21−8) / 28 = 13/28.

3. Subtract the third given fraction from the resulting value:

4. We bring the numbers to a common denominator. If it is not possible to select the same denominator in an easier way, then you just need to perform the steps by multiplying all the denominators in sequence by each other, not forgetting to increase the value of the numerator by the same figure. In this example we do this:

• 13/28 = 13*13 / 28*13 = 169/364, where 13 is the lower digit of 5/13;
• 5/13 = 5*28 / 13*28 = 140/364, where 28 is the lower number from 13/28.

5. Subtract the resulting fractions:

13/28−5/13 = 169/364−140/364 = (169−140) / 364 = 29/364.

Answer: ¾−2/7−5/13 = 29/364.

Mixed fractions

In the examples discussed above, only proper fractions were used.

As an example:

• 8/9 is a proper fraction;
• 9/8 is incorrect.

It is impossible to turn an improper fraction into a proper fraction, but it is possible to turn it into mixed. Why do you divide the top number (numerator) by the bottom (denominator) to get a number with a remainder? The integer resulting from division is written down like this, the remainder is written in the numerator at the top, and the denominator at the bottom remains the same. To make it clearer, let's look at a specific example:

Example 6

Convert the improper fraction 9/8 to the correct one.

To do this, divide the number “nine” by “eight”, resulting in a mixed fraction with an integer and a remainder:

9: 8 = 1 and 1/8 (this can be written differently as 1+1/8), where:

• number 1 is the integer resulting from division;
• another number 1 is the remainder;
• the number 8 is the denominator, which remains unchanged.

An integer is also called a natural number.

The remainder and denominator are a new, but proper fraction.

When writing the number 1, it is written before the proper fraction 1/8.

Subtracting mixed numbers with different denominators

From the above, we give the definition of a mixed fractional number: "Mixed number - this is a quantity that is equal to the sum of a whole number and a proper ordinary fraction. In this case, the whole part is called natural number, and the number that is left is his fractional part».

Example 7

Given: two mixed fractional quantities consisting of a whole number and a proper fraction:

• the first value is 9 and 4/7, that is (9+4/7);
• the second value is 3 and 5/21, that is (3+5/21).

It is required to find the difference between these quantities.

1. To subtract 3+5/21 from 9+4/7, you must first subtract integer values ​​from each other:

4/7−5/21 = 4*3 / 7*3−5/21 =12/21−5/21 = (12−5) / 21 = 7/21.

3. The resulting result of the difference between two mixed numbers will consist of the natural (integer) number 6 and the proper fraction 7/21 = 1/3:

(9 + 4/7) - (3 + 5/21) = 6 + 1/3.

Mathematicians from all countries have agreed that the “+” sign when writing mixed quantities can be omitted and only the whole number left before the fraction without any sign.

Has your child brought homework from school and you don't know how to solve it? Then this mini lesson is for you!

How to add decimals

It is more convenient to add decimal fractions in a column. To add decimals, you need to follow one simple rule:

• The place must be under the place, the comma under the comma.

As you can see in the example, the whole units are located under each other, the tenths and hundredths digits are located under each other. Now we add the numbers, ignoring the comma. What to do with the comma? The comma is moved to the place where it stood in the integer category.

Adding fractions with equal denominators

To perform addition with a common denominator, you need to keep the denominator unchanged, find the sum of the numerators and get a fraction that will be the total sum.

Adding fractions with different denominators using the common multiple method

The first thing you need to pay attention to is the denominators. The denominators are different, whether one is divisible by the other, or whether they are prime numbers. First you need to bring it to one common denominator; there are several ways to do this:

• 1/3 + 3/4 = 13/12, to solve this example we need to find the least common multiple (LCM) that will be divisible by 2 denominators. To denote the smallest multiple of a and b – LCM (a;b). In this example LCM (3;4)=12. We check: 12:3=4; 12:4=3.
• We multiply the factors and add the resulting numbers, we get 13/12 - an improper fraction.

• In order to convert an improper fraction into a proper one, divide the numerator by the denominator, we get the integer 1, the remainder 1 is the numerator and 12 is the denominator.

Adding fractions using the cross-cross multiplication method

To add fractions with different denominators, there is another method using the “cross to cross” formula. This is a guaranteed way to equalize the denominators; to do this, you need to multiply the numerators with the denominator of one fraction and vice versa. If you are just at the initial stage of learning fractions, then this method is the simplest and most accurate way to get the correct result when adding fractions with different denominators.

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” from shamans that mathematicians use. But that's 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.

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, degree designation). 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.