Lecture: Fractions, percentages, rational numbers

Rational numbers are those that can be expressed as a fraction.

So what are fractions anyway?

Fraction- a number that shows a certain number of parts of a whole, that is, units.

Fractions can be decimal and ordinary. As a mathematical operation, fraction- this is nothing but division. Every fraction is made up of numerator(divisible), which is at the top, denominator(divisor), which is at the bottom, and the line of a fraction, which directly performs the division function. The denominator of a fraction shows how many equal parts a whole is divided into. The numerator shows how many equal parts of the whole have been taken.

A fraction can be mixed, that is, it can have both a fractional and an integer part.

for example, 1; 5,03.

An ordinary fraction can have an arbitrary numerator and denominator.

for example, 1/5, 4/7, 7/11, etc.

The decimal in the denominator always has the numbers 10, 100, 1000, 10000, etc.

for example, 1/10 = 0.1; 6/100 = 0.06 etc.

You can perform the same mathematical operations on fractions as on integers:

1. Addition and subtraction of fractions

For these fractions, the smallest number that is divisible by one and the second denominator is the number 30.

To bring both fractions to a denominator of 30, you need to find an additional factor. To get the denominator 30 in the first fraction, it should be multiplied by 6. To get the denominator 30 in the second fraction, it should be multiplied by 5. So that the value of the fraction does not change, we multiply both the numerator and the denominator by these numbers. As a result of this we get:

To add or subtract numbers with same denominators, you should leave the denominator 30 as a result, and add the numerators:

2. Multiplication of fractions

When multiplying two fractions, multiply their numerators, then multiply the denominators, and write the result:

3. Division of fractions

When dividing two fractions, you need to flip the second fraction and perform the multiplication action:

4. Reducing fractions

If the numerator and denominator are a multiple of some identical number, then such a fraction can be reduced by dividing both the numerator and the denominator by the given number.

In the original fraction, both the numerator and denominator are divisible by 3, so the whole fraction can be reduced by that number.

5. Comparison of fractions

When comparing fractions, you need to use several rules:

- If there is a comparison of fractions that have the same denominator but a different numerator, then the fraction with the larger numerator will be larger. That is, this comparison is reduced to a comparison of numerators.

- If fractions have the same numerator but different denominators, then the denominators must be compared. That fraction will be greater, whose denominator is less.

- If fractions have different numerators and denominators, then they must be reduced to common denominator.

The common denominator is 42, therefore, the additional factor for the first fraction is 7, and the additional factor for the second fraction is 6. We get:

Now the comparison comes down to the first rule. The larger fraction is the one with the larger denominator:

Interest

Any number that is one hundredth of some integer is called one. percent.

1% = 1/100 = 0,01.

To convert a fraction to a percentage notation, it should be converted to a decimal fraction, and then multiplied by 100%.

For example,

Interest is used in three main cases:

1. If you need to find some percentage of a number. Imagine that you receive 10% of your parents' salary every month. However, if you don't know the math, you won't be able to calculate what your monthly income will be. So, this is easy enough to do.

Imagine that your parents receive 100,000 rubles a month. To find the amount that you should receive monthly, you need to divide the income of your parents by 100 and multiply by 10%, which you should receive:

100000: 100 * 10 = 10000 (rubles).

2. If you need to find out how much your parents receive monthly, if you know that they give you 6,000 rubles, and this, in turn, is 3%, then this action with interest is called finding a number by its percentage. To do this, you need to multiply the amount received by 100 and divide by your percentage:

6000 * 100: 3 = 200000 (rubles).

3. If you drink 1 liter of water during the day, and you, for example, need to drink 2 liters of water, then you can easily find the value of the percentage of water you drink. To do this, divide 1 liter by 2 liters and multiply by 100%.

1: 2 * 100% = 50%.

An ordinary fraction is a number of the form where the type is natural numbers, for example Number is called the numerator of the fraction, - the denominator. In particular, it may be that in this case the fraction has the form but more often they write simply. This means that any natural number can be represented as an ordinary fraction with a denominator of 1. Writing is another way of writing

Common fractions are divided into proper and improper fractions.

fractions. A fraction is called proper if its numerator is less than its denominator, and improper if its numerator is greater than or equal to its denominator.

Any improper fraction can be represented as a sum natural number and proper fraction(or in the form of a natural number, if the fraction is such that it is a multiple, for example

Example. Express an improper fraction as the sum of a natural number and a proper fraction: a)

Solution a)

It is customary to write the sum of a natural number and a proper fraction without an addition sign, that is, instead of writing instead of writing, a number written in this form is called a mixed number. It consists of two parts: integer and fractional. So, for the number 3 - the integer part is 3, and the fractional - Any improper fraction can be written as a mixed number (or as a natural number). The converse is also true: any mixed or natural number can be written as an improper fraction. For example, .

Mathematics. Algebra. Geometry. Trigonometry

ALGEBRA: Numbers

2.2. Integers and rational numbers. Interest

Ordinary fractions.

Common fraction

is a number of the form , where m and n are natural numbers. The number m is called numerator,n- denominator. If n = 1, then the fraction has the form , but more often they simply write m, i.e. Any natural number can be represented as a common fraction with denominator 1.

The fraction is called correct, if its numerator is less than its denominator, and wrong, if its numerator is greater than or equal to its denominator. Any improper fraction can be represented as the sum of a natural number and a proper fraction (or as a natural number if m is a multiple of n).

It is customary to write the sum of a natural number and a proper fraction without an addition sign, that is, instead of writing. A number written in this form is called mixed number. It consists of an integer part and a fractional part.

Equality of fractions. Fraction reduction.

Two fractions and count equal, if ad = bc. It follows from the definition of equality that

= , because . The main property of a fraction:If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then a fraction equal to the given one will be obtained. Using the basic property of a fraction, it is sometimes possible to replace a given fraction with another one, the numerator and denominator of which are less than given. Such a replacement is called reduction fractions. If the numerator and denominator are coprime numbers, then reduction is not possible and such a fraction is called irreducible.

Arithmetic operations on ordinary fractions.

Let there be two fractions and

, . You can replace these fractions with others equal to them, so that the resulting fractions will have the same denominators. Such a transformation is called bringing fractions to a common denominator. Usually they try to reduce fractions to lowest common denominator, which is equal to N.O.K.().

1.Addition ordinary fractions is done like this:

a) If the denominators are the same, then add the numerators and leave the same denominator:;

2. Subtraction ordinary fractions is performed as follows:

a) if the denominators are the same, then

b) if the denominators of the fractions are different, then the fractions are first reduced to the lowest common denominator, and then rule a) is applied.

3. Multiplicationordinary fractions is performed as follows:

4. The division of ordinary fractions is performed as follows:

.

Decimals. Convert decimal to common fraction.

A decimal is another form of a fraction with a denominator. For example, . If the expansion of the denominator of a fraction into prime factors contains only 2 and 5, then this fraction can be written as a decimal; if the fraction is irreducible and the expansion of its denominator into prime factors includes other prime factors, then this fraction cannot be written as a decimal.

In a decimal fraction, you can attribute and discard zeros on the right - you get a fraction equal to it.

A fraction that has an infinite number of decimal places is called infinite decimal.

Theorem 10.

Any common fraction can be represented as an infinite decimal.

A consistently repeating group of digits (minimum) after a decimal point in a number is called a period, and an infinite decimal fraction that has a period is called periodic.

Let it be given by a periodic decimal fraction: , where is an m-digit number, then

, YU
YU - the formula for converting a periodic decimal fraction to an ordinary fraction.

Interest.

Among decimal fractions, the most commonly used fraction is 0.01, which is called percent and denoted 1

%. So 1% = 0.01; 25% = 0.25; 450% = 4.5 etc.

EXAMPLE The worker had to make 60 parts per shift. At the end of the working day, it turned out that he had completed 125

% tasks. How many parts did the worker make?

Solution: 1) 125

% = 1,25

2) 60H 1.25 = 75.

Answer: 75 parts.

coordinate line.

Let's take the line l, mark the point O on it, which we will take as the origin, set the direction and the unit segment . In this case, we say that given coordinate line. Each natural number or fraction corresponds to one point of the line l. If a point M of the line l corresponds to some number r, then this number is called coordinate point M and denoted by M(r). The numbers a and -a are called opposite. The numbers that correspond to points located on the coordinate line in a given direction are called positive; the numbers corresponding to the points located on the coordinate line in the direction opposite to the given one are called negative. The number 0 is considered neither positive nor negative. The point O, corresponding to the number 0, separates points with positive coordinates from points with negative coordinates on the coordinate line.

A given direction on a coordinate line is called positive(usually it goes to the right), and the direction opposite to the given one is negative

.

Integers and rational numbers.

The natural numbers 1, 2, 3, ... are also called positive integers. Numbers -1, -2, -3, ..., opposite to natural numbers, are called negative integers. The number 0 is also an integer. Whole numbers are natural numbers, their opposites and 0.

Whole numbers and fractions (positive and negative) make up the set rational numbers.

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In this article, we will begin to study rational numbers. Here we give definitions of rational numbers, give the necessary explanations and give examples of rational numbers. After that, we will focus on how to determine whether a given number is rational or not.

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Definition and examples of rational numbers

In this subsection we give several definitions of rational numbers. Despite the differences in wording, all these definitions have the same meaning: rational numbers unite integers and fractional numbers, just as integers unite natural numbers, their opposite numbers, and the number zero. In other words, rational numbers generalize whole and fractional numbers.

Let's start with definitions of rational numbers which is perceived as the most natural.

From the sounded definition it follows that a rational number is:

• Any natural number n . Indeed, any natural number can be represented as an ordinary fraction, for example, 3=3/1.
• Any integer, in particular the number zero. Indeed, any integer can be written either as a positive common fraction, as a negative common fraction, or as zero. For example, 26=26/1 , .
• Any ordinary fraction (positive or negative). This is directly stated by the given definition of rational numbers.
• Any mixed number. Indeed, it is always possible to represent a mixed number as an improper common fraction. For example, and .
• Any finite decimal or infinite periodic fraction. This is so because the specified decimal fractions are converted to ordinary fractions. For example, , and 0,(3)=1/3 .

It is also clear that any infinite non-repeating decimal is NOT a rational number, since it cannot be represented as a common fraction.

Now we can easily bring examples of rational numbers. The numbers 4, 903, 100,321 are rational numbers, since they are natural numbers. The integers 58 , −72 , 0 , −833 333 333 are also examples of rational numbers. Ordinary fractions 4/9, 99/3, are also examples of rational numbers. Rational numbers are also numbers.

It can be seen from the above examples that there are both positive and negative rational numbers, and the rational number zero is neither positive nor negative.

The above definition of rational numbers can be formulated in a shorter form.

Definition.

Rational numbers call numbers that can be written as a fraction z/n, where z is an integer and n is a natural number.

Let us prove that this definition of rational numbers is equivalent to the previous definition. We know that we can consider the bar of a fraction as a sign of division, then from the properties of dividing integers and the rules for dividing integers, the following equalities follow and . Thus, which is the proof.

We give examples of rational numbers based on this definition. The numbers −5 , 0 , 3 , and are rational numbers, since they can be written as fractions with an integer numerator and a natural denominator of the form and respectively.

The definition of rational numbers can also be given in the following formulation.

Definition.

Rational numbers are numbers that can be written as a finite or infinite periodic decimal fraction.

This definition is also equivalent to the first definition, since any ordinary fraction corresponds to a finite or periodic decimal fraction and vice versa, and any integer can be associated with a decimal fraction with zeros after the decimal point.

For example, the numbers 5 , 0 , −13 , are examples of rational numbers because they can be written as the following decimals 5.0 , 0.0 , −13.0 , 0.8 and −7,(18) .

We finish the theory of this section with the following statements:

• integer and fractional numbers (positive and negative) make up the set of rational numbers;
• each rational number can be represented as a fraction with an integer numerator and a natural denominator, and each such fraction is some rational number;
• every rational number can be represented as a finite or infinite periodic decimal fraction, and each such fraction represents some rational number.

Is this number rational?

In the previous paragraph, we found out that any natural number, any integer, any ordinary fraction, any mixed number, any final decimal fraction, and also any periodic decimal fraction is a rational number. This knowledge allows us to "recognize" rational numbers from the set of written numbers.

But what if the number is given as some , or as , etc., how to answer the question, is the given number rational? In many cases, it is very difficult to answer it. Let us point out some directions for the course of thought.

If a number is specified as a numeric expression that contains only rational numbers and arithmetic signs (+, −, · and:), then the value of this expression is a rational number. This follows from how operations on rational numbers are defined. For example, after performing all the operations in the expression, we get a rational number 18 .

Sometimes, after simplifying expressions and more complex type, it becomes possible to determine whether a given number is rational.

Let's go further. The number 2 is a rational number, since any natural number is rational. What about number? Is it rational? It turns out that no, it is not a rational number, it is an irrational number (the proof of this fact by contradiction is given in the 8th grade algebra textbook listed below in the list of references). It is also proved that the square root of a natural number is a rational number only in those cases when under the root there is a number that is the perfect square of some natural number. For example, and are rational numbers, since 81=9 2 and 1024=32 2 , and the numbers and are not rational, since the numbers 7 and 199 are not full squares natural numbers.

Is the number rational or not? In this case, it is easy to see that, therefore, this number is rational. Is the number rational? It is proved that the kth root of an integer is a rational number only if the number under the root sign is the kth power of some integer. Therefore, it is not a rational number, since there is no integer whose fifth power is 121.

The method of contradiction allows us to prove that the logarithms of some numbers, for some reason, are not rational numbers. For example, let's prove that - is not a rational number.

Assume the opposite, that is, suppose that is a rational number and can be written as an ordinary fraction m/n. Then and give the following equalities: . The last equality is impossible, since on its left side there is not even number 5 n , and on the right side there is an even number 2 m . Therefore, our assumption is wrong, thus is not a rational number.

In conclusion, it is worth emphasizing that when clarifying the rationality or irrationality of numbers, one should refrain from sudden conclusions.

For example, one should not immediately assert that the product of irrational numbers π and e is an irrational number, this is “as if obvious”, but not proven. This raises the question: “Why would the product be a rational number”? And why not, because you can give an example of irrational numbers, the product of which gives a rational number:.

It is also unknown whether the numbers and many other numbers are rational or not. For example, there are irrational numbers whose irrational power is a rational number. To illustrate, we present the degree of the form , the base of this degree and the exponent are not rational numbers, but , and 3 is a rational number.

Bibliography.

• Mathematics. Grade 6: textbook. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
• Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

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2 MAIN WAVE 2013 CENTER URAL SIBERIA EAST: fractions percent rational numbers Theory: The set of rational numbers 1 1 ~ HOD ge N Z Main property 0 0. Proportion is the equality of two ratios. Property: Consequences Scheme Straight proportional dependence. Main properties 1. Order: 0 ; 0; Addition operation: ; HOK 3. Operation of multiplication and division: 4. Transitivity of the order relation: 5. Commutativity: 6. Associativity: 7. Distributivity: 8. Presence of zero: Presence of opposite numbers: Presence of one: Presence of reciprocal numbers: R R. 12. Relation of the order relation with the addition operation. The same rational number can be added to the left and right sides of a rational inequality. 2 B1

3 13. Connection of the order relation with the operation of multiplication. The left and right sides of a rational inequality can be multiplied by the same positive rational number Axiom of Archimedes. Whatever the rational number, you can take so many units that their sum will exceed a. Nk Rational inequalities one sign can be added term by term. Any rational fraction can be converted into a decimal equal to it by dividing the numerator by the denominator into a column. 1 remainder may be equal to zero and the quotient will be expressed as a finite decimal fraction, for example 3: 4 = zero in the remainder will never work out, since the remainder will repeat indefinitely and the quotient will be expressed as an infinite periodic decimal fraction. For example 2:3=0666 =06 7:13= = :15=21333 = ? Interest. The hundredth of a number is called its percentage. Three types of tasks for percentages A 100% 1. Finding percentages of a given number A p% x. x p% 100% To find p% of the number "A" you need to find 1% of "A" A: 100% and multiply by p%. 2. Finding a number by another number and its value as a percentage of the desired number. x 100% 100% x. p% p% To find a number by a given value "a" its p% you need to find 1% of the desired number by dividing the given value "a" by p% and multiply the result by 100% A 100% 3. Finding the percentage of numbers. 100% x% x% A We need to find the ratio of the number "a" to the number "A" and multiply by 100%. 3

4 CENTER Option 1;8. One tablet of the drug weighs 70 mg and contains 4% active substance. For a child under the age of 6 months, the doctor prescribes 105 mg of the active substance for each age of 5 months and weighing 8 kg during the day? Option 2. One tablet of the drug weighs 20 mg and contains 5% of the active substance. For a child under the age of 6 months, the doctor prescribes 04 mg of the active substance for each age of three months and weighing 5 kg during the day? Option 3. One tablet of the drug weighs 20 mg and contains 5% of the active substance. For a child under the age of 6 months, the doctor prescribes 1 mg of the active substance for each age of four months and weighing 7 kg during the day? Option 4;5. One tablet of the drug weighs 20 mg and contains 9% of the active substance. For a child under the age of 6 months, the doctor prescribes 135 mg of the active substance for each age of four months and weighing 8 kg during the day? Option 6. One tablet of the drug weighs 30 mg and contains 5% of the active substance. For a child under the age of 6 months, the doctor prescribes 075 mg of the active substance for each age of 5 months and weighing 8 kg during the day? Option 7. One tablet of the drug weighs 40 mg and contains 5% of the active substance. For a child under the age of 6 months, the doctor prescribes 125 mg of the active substance for each age of three months and weighing 8 kg during the day? Note that eight options are made up of six tasks with different numerical data but the same content. Necessary information for calculation, they wrote out in the table: Weight of one Percentage Variants Recipe mg Weight of a child kg tablets mg of active substance% 1 and and Solution of option 1. Idea: The percentage of the active substance in one tablet is known, which means you can find the corresponding amount of substance in mg. Knowing the weight of the child and the dosage of the active substance per 1 kg of weight, you can find the daily rate of the active substance. Then the number of tablets is the quotient of dividing the daily norm of the active substance by the amount of the active substance in one tablet. Actions: 1. Determine the amount of active substance in one tablet. We make up the proportion: we take the weight of one tablet of 70 mg as 100% and 4% of this weight will be x mg of the amount of active substance in one tablet. Let us write down this proportion schematically. From here we find the unknown term of the proportion. To do this, multiply x 4% of the known members of one diagonal and divide by the known member of the other diagonal: 70 4% x 28 mg. 100% 4

5 2. Determine the amount of the active substance prescribed by the doctor according to the prescription, taking into account the weight of the child. The dose of the substance must be multiplied by the weight of the child: mg. So the child needs to take 84 mg of the active substance per day Determine the number of tablets containing 84 mg of the active substance. 3 tab. 28 Answer 3. Other options are solved similarly. IN URAL Option 1;5. In the apartment where Anastasia lives, a flow meter is installed cold water counter. On September 1, the meter showed a consumption of 122 cubic meters of water, and on October 1, 142 cubic meters. What amount should Anastasia pay for cold water for September if the price of 1 cubic meter of cold water is 9 rubles 90 kopecks? Give your answer in rubles. Option 2. In the apartment where Maxim lives, a cold water meter is installed. On February 1, the meter showed a consumption of 129 cubic meters of water, and on March 1, 140 cubic meters. What amount should Maxim pay for cold water for February if the price of 1 cubic meter of cold water is 10 rubles 60 kopecks? Give your answer in rubles. Option 3. In the apartment where Alex lives, a cold water meter is installed. On June 1, the meter showed a consumption of 151 cubic meters of water, and on July 1, 165 cubic meters. What amount should Alexey pay for cold water for March if the price of 1 cubic meter of cold water is 20 rubles 80 kopecks? Give your answer in rubles. Option 4. In the apartment where Asya lives, a flow meter is installed hot water counter. On May 1, the meter showed a consumption of 84 cubic meters of water, and on June 1, 965 cubic meters. What amount should Anastasia pay for hot water in January if the price of 1 cubic meter of hot water is 72 rubles 60 kopecks? Give your answer in rubles. Option 6;8. In the apartment where Anfisa lives, a hot water meter is installed. On September 1, the meter showed a consumption of 239 cubic meters of water, and on October 1, 349 cubic meters. What amount should Anfisa pay for hot water for September if the price of 1 cubic meter of hot water is 78 rubles 60 kopecks? Give your answer in rubles. Option 7. In the apartment where Alla lives, a hot water meter is installed. On July 1, the meter showed a consumption of 772 cubic meters of water, and on August 1, 797 cubic meters. What amount should Alla pay for hot water in July if the price of 1 cubic meter of hot water is 144 rubles 80 kopecks? Give your answer in rubles. The URAL region solved the problem of paying for water consumption according to the meter. Numerical data for calculation by options were entered in the table: Vari Counter readings at the beginning Counter readings at the beginning Price of 1 cubic meter of the calendar month cubic meters of the next calendar month cubic meters 1 and ruble 90 kopecks ruble 60 kopecks ruble 80 kopecks ruble 60 kopecks 6 and ruble 60 kopecks ruble 80 kopecks Solution 1. Idea: The meter readings are known at the beginning of the calendar month of cubic meters and at the beginning of the next calendar month of cubic meters. So you can find out the water consumption for the month payable. Knowing the number of cubic meters of water used and the price of one cubic meter of water, you can find the amount that must be paid for this water. 5

6 Actions: Determine the water consumption for the month Determine the amount payable for the consumed water for the month p Answer 198. Other options are solved similarly. TO SIBERIA Option 1. 1 kilowatt-hour of electricity costs 1 ruble 40 kopecks. The electricity meter on June 1 showed kilowatt-hours and on July 1 it showed kilowatt-hours. How much do you need to pay for electricity in June? Give your answer in rubles. Option 2. 1 kilowatt-hour of electricity costs 1 ruble 20 kopecks. The electricity meter on November 1 showed 669 kilowatt-hours and on December 1 it showed 846 kilowatt-hours. How much do you need to pay for electricity in November? Give your answer in rubles. Option 3. 1 kilowatt-hour of electricity costs 2 rubles 40 kopecks. The electricity meter on October 1 showed kilowatt-hours and on November 1 it showed kilowatt-hours. How much do you need to pay for electricity in October? Give your answer in rubles. Option 4;5. 1 kilowatt-hour of electricity costs 2 rubles 50 kopecks. The electricity meter on January 1 showed kilowatt-hours and on February 1 it showed kilowatt-hours. How much do you need to pay for electricity in January? Give your answer in rubles. Option 6. 1 kilowatt-hour of electricity costs 1 ruble 30 kopecks. The electricity meter on September 1 showed kilowatt-hours and on October 1 it showed kilowatt-hours. How much do you need to pay for electricity in September? Give your answer in rubles. Option 7;8. 1 kilowatt-hour of electricity costs 1 ruble 70 kopecks. The electricity meter on April 1 showed kilowatt-hours and on May 1 it showed kilowatt-hours. How much do you need to pay for electricity in April? Give your answer in rubles. The SIBERIA region solved the problem of paying for electricity consumption by the meter. Numerical data for calculation by options were entered in the table: Options Meter readings at the beginning of the calendar month kWh Meter readings at the beginning of the next calendar month kWh 7 kopecks and 70 kopecks ruble Solution 1. Idea: Known meter readings at the beginning of the kilowatt-hour calendar month and at the beginning of the next kilowatt-hour calendar month. So you can find out the electricity consumption for the month payable. Knowing the number of kilowatt-hours of electricity consumed and the price of one kilowatt-hour, you can find the amount that must be paid for this electricity. Actions: Determine the electricity consumption for the month Determine the amount payable for the electricity consumed for the month. 6

7 p Answer The rest of the options are solved in a similar way. TO THE EAST Option 1; 5; 8. In the apartment where Ekaterina lives, a cold water meter is installed. On September 1, the meter showed a consumption of 189 cubic meters of water, and on October 1, 204 cubic meters. What amount should Catherine pay for cold water for September if the price of 1 cubic meter of cold water is 16 rubles 90 kopecks? Give your answer in rubles. Option 2. In the apartment where Valery lives, a cold water meter is installed. On March 1, the meter showed a consumption of 182 cubic meters of water, and on April 1, 192 cubic meters. What amount should Valery pay for cold water for March if the price of 1 cubic meter of cold water is 23 rubles 10 kopecks? Give your answer in rubles. Option 3. In the apartment where Marina lives, a cold water meter is installed. On July 1, the meter showed a consumption of 120 cubic meters of water, and on August 1, 131 cubic meters. How much should Marina pay for cold water in July if the price of 1 cubic meter of cold water is 20 rubles 60 kopecks? Give your answer in rubles. Option 4. In the apartment where Yegor lives, a hot water meter is installed. On November 1, the meter showed a consumption of 879 cubic meters of water, and on December 1, 969 cubic meters. What amount should Yegor pay for hot water in November if the price of 1 cubic meter of hot water is 108 rubles 20 kopecks? Give your answer in rubles. Option 6. In the apartment where Mikhail lives, a hot water meter is installed. On March 1, the meter showed a consumption of 708 cubic meters of water, and on April 1, 828 cubic meters. What amount should Mikhail pay for hot water for March if the price of 1 cubic meter of hot water is 72 rubles 20 kopecks? Give your answer in rubles. Option 7. In the apartment where Anastasia lives, a hot water meter is installed. On January 1, the meter showed a consumption of 894 cubic meters of water, and on February 1, 919 cubic meters. What amount should Anastasia pay for hot water in January if the price of 1 cubic meter of hot water is 103 rubles 60 kopecks? Give your answer in rubles. The tasks of the "VOSTOK" region coincided with the tasks of the "URAL" region with a difference in numerical data. Options Meter readings at the beginning of the calendar month cubic meters Meter readings at the beginning of the next calendar month cubic meters Price of 1 cubic meters 1 and 5 and ruble 90 kopecks ruble 10 kopecks ruble 60 kopecks ruble 20 kopecks ruble 20 kopecks ruble 60 kopecks Therefore, the idea of ​​a solution and actions will be similar to those considered earlier for the URAL region. AT

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MATHEMATICS. EVERYTHING FOR THE TEACHER! DECIMAL FRACTIONS AND ACTIONS ON THEM DIDACTIC AND LIBRARY BLIO IOTE didactic materials on the topic "Decimal fractions": cards for individual

Algorithm for finding the range of admissible values ​​of an algebraic fraction. Example. Find the range of acceptable values: x 25 (x 5) (2x + 4). 1. Write out the denominator of an algebraic fraction; 2. Equate the issued

Topic 3. “Relationships. Proportions. Percentage" The ratio of two numbers is the quotient of dividing one of them by the other. The ratio shows how many times the first number is greater than the second or what part of the first number

Finding numbers Example 1. The numerators of three fractions are proportional to the numbers 1, 2, 5, and the denominators, respectively, to the numbers 1, 3, 7. The arithmetic mean of the fractions is equal. Find these fractions. Decision. By condition

Quarter 1 What numbers are natural? How to read a number? How to write a number in numbers? Relations between units How to draw a coordinate ray and mark points on this ray? Formulas for Numbers

Lesson number Lesson topic CALENDAR - THEMATIC PLANNING Grade 6 Number of hours Chapter 1. Ordinary fractions. 1. Divisibility of numbers 24 hours 1-3 Divisors and multiples 3 Divisor, multiple, least multiple of natural

Subject. Development of the concept of number Tutorial developed in accordance with work program general educational academic discipline ODP.0 Mathematics. The study guide contains: theoretical

“Agreed” “Approve” Deputy Director for OWR School principal Year 6 Calendar-thematic planning in mathematics (correspondence form of education) 2018-2019 academic year Textbook: Vilenkin N.Ya., Zhokhov

Fractional-rational expressions Expressions containing division by an expression with variables are called fractional (fractional-rational) expressions Fractional Expressions for some values ​​of the variables do not have

Topic 1 “Numeric expressions. Procedure. Comparison of numbers. A numeric expression is one or more numerical values(numbers) interconnected by signs of arithmetic operations: addition,

Calendar-thematic planning mathematics Grade 6 (5 hours per week, total 170 hours) of the lesson Lesson topic 1-3 Addition and subtraction of fractions with the same denominators, addition and subtraction of decimals

Chapter 1 Fundamentals of Algebra Number Sets Consider the basic number sets. The set of natural numbers N includes numbers of the form 1, 2, 3, etc., which are used to count objects. A bunch of

RATIONAL NUMBERS Ordinary fractions Definition Fractions of the form called ordinary fractions Ordinary fractions, regular and improper Definition A fraction, proper if< при, где Z, N Z, N Z,

1 IRRATIONAL AND REAL NUMBERS Irrational numbers The simplest example on measuring the length of the diagonal of a unit square shows that the operation of taking the square root of a rational number

26. Tasks for integers Find the greatest common divisor of numbers (1 8): 1. 247 and 221. 2. 437 and 323. 3. 357 and 391. 4. 253 and 319. 5. 42 4 and 54 3. 6. 78 4 and 65 2. 7. 77 3 and 242 2. 8. 51 3 and 119 2. 9. Sum

Contents: 1. Addition and subtraction of natural numbers. Comparison of natural numbers. 2. Numeric and alphabetic expressions. The equation. 3. Multiplication of natural numbers. 4. Division of natural numbers. Ordinary

LECTURE 6 LINEAR COMBINATIONS AND LINEAR DEPENDENCE MAIN LEMMA ON LINEAR DEPENDENCE BASIS AND DIMENSION OF A LINEAR SPACE RANK OF A SYSTEM OF VECTORS 1 LINEAR COMBINATIONS AND LINEAR DEPENDENCE

The main property of a fraction RULES AND LO SAMPLES OF TASKS AND I Bring the fraction to a new denominator: 1) Multiply (or divide) the denominator of the fraction by the number. 2) Multiply (or divide) the numerator of the fraction by the same number.

I option 8B class, October 4, 007 1 Insert the missing words: Definition 1 Arithmetic square root from the number of which a is equal from the number a (a 0) is denoted as follows: by the expression The action of finding

Question What are natural numbers? Answer Natural numbers are called numbers that are used in counting. What are classes and digits in writing numbers? What are numbers called when added? Formulate an associative

For foreign listeners preparatory department AUTHOR: Starovoitova Natalya Alexandrovna Department of pre-university training and career guidance 1 2 3 8 4 Numbers; ; ; ; 2 3 7 5 4 - ordinary fractions.

ARITHMETIC Actions with natural numbers and ordinary fractions. Procedure) If there are no brackets, then the actions of the th degree are performed first (raising to natural degree), then the th power (multiplication

CONTENT Mathematical symbols... 3 Comparison of numbers... 4 Addition... 5 Connection between the components of addition... 5 Commutative law of addition... 6 Associative law of addition... 6 Procedure...

REFERENCE MATERIAL FOR PREPARING FOR THE ANSWER TO THE THEORETICAL QUESTION OF THE MATHEMATICS TRANSFER EXAM IN THE 6TH CLASS (in the reference material, hyperlinks to Internet resources are highlighted in blue) TICKET

Standard variant"Complex Numbers Polynomials and rational fractions» Task Given two complex numbers and cos sn Find and write the result in algebraic form write the result in trigonometric

Chapter INTRODUCTION TO ALGEBRA .. SQUARE THREE-MEMBER ... The Babylonian problem of finding two numbers by their sum and product. One of the oldest problems in algebra was proposed in Babylon, where

Topic 1. Direction of reference Analysis of problem solving by topic Chapter 1 " Negative numbers» The tasks for this topic are practical, important for understanding the use of + signs and, for developing skills

ADDITION To add 1 to a number means to get the number following the given one: 4+1=5, 1+1=14, etc. Adding the numbers 5 means adding one to 5 three times: 5+1+1+1=5+=8. SUBTRACT Subtract 1 from a number means

2. General linear and Euclidean spaces A set X is said to be a linear space over the field of real numbers, or simply a real linear space, if for any elements

LECTURE The concept of a matrix and its properties Actions on matrices The concept of a matrix An order (dimension) matrix is ​​a rectangular table of numbers or literal expressions containing columns: () i rows

Arithmetic - class ANSWERS: Topic Multiplication and division of decimal fractions)) 00.0 Topic Addition and subtraction of fractions with different denominators)) Topic Division of ordinary fractions))) and Topic Proportions) Topic

3 Dear reader! In your hands is a modern reference book that will support you in your studies in grades 5-11, help you prepare for exams, and make it easy to enter a university. In the directory

Lesson Topic Note Divisibility of numbers 16 h.

Topic 1. Sets. Numerical Sets N, Z, Q, R 1. Sets. Operations on sets. 2. The set of natural numbers N. 3. The set of integers Z. Divisibility of integers. divisibility signs. 4. Rational

Moscow: AST Publishing House: Astrel, 2016. 284, p. (Academy of Primary Education). 978-5-17-098011-6 978-5-271-47746-1 978-5-17-098011-6 978-5-271-47746-1 Contents Dear Adults!... 6 Numbers

Website of elementary mathematics by Dmitry Gushchin wwwthetspru Gushchin D D REFERENCE MATERIALS FOR PREPARING FOR THE USE IN MATHEMATICS ASSIGNMENT B7: CALCULATIONS AND TRANSFORMATIONS Content elements and types to be checked

Contents Equation................................... Integer expressions.. ................................... Expressions with powers............. ................. 3 Monomial ............................... ..............

VV Rasin REAL NUMBERS Yekaterinburg 2005 Federal Agency for Education Ural State University them. A. M. Gorky V. V. Rasin REAL NUMBERS Yekaterinburg 2005 UDC 517.13(075.3)

Equations In algebra, two types of equalities are considered - identities and equations. Identity is an equality that holds for all admissible) values ​​of the letters included in it. For identities, signs are used

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PREPARATION FOR THE OGE Reference materials for students in grade 9 Algebra Natural numbers and operations on them The concept of a natural number refers to the simplest, initial concepts of mathematics and is not defined

Consider the first way to solve the SLE according to Cramer's rule for a system of three equations with three unknowns: The answer is calculated using Cramer's formulas: D, D1, D2, D3 are determinants

Systems of Equations Let two equations with two unknowns f(x, y)=0 and g(x, y)=0 be given, where f(x, y), g(x, y) are some expressions with variables x and y. If the task is to find all general solutions data

Math class. Teacher Demidova Elena Nikolaevna quarter .. divisibility of NUMBERS Divisors and multiples. Tests for divisibility by 0, and. Signs of divisibility by and by 9. Prime and composite numbers. Decomposition into simple

Grade 6 (FSES LLC) of the lesson Main view Content (section, topics) learning activities Repetition of the course of mathematics of the 5th grade (hours) Number of hours Material of the textbook Correction Repetition of the course of mathematics.

Class. Degree with arbitrary real exponent, its properties. Power function, its properties, graphs .. Recall the properties of a degree with rational indicator. a a a a a for natural times

Lecture 2 Solution of systems linear equations. 1. Solution of systems of 3 linear equations by Cramer's method. Definition. A system of 3 linear equations is a system of the form In this system, the required quantities,

Lesson 16 Relations. Proportions. Percentage Quotient 12: 6 = 2 is the ratio of numbers 12 and 6. The ratio of numbers 12 and 6 is equal to the number 2. number 2. Quotient 2: = 2 is the ratio of numbers 2 and. The ratio of numbers 2 and equal

Problem 1 Unified State Examination -2015 (basic) If you only need an answer, the first example is 2.65 - the second example is 3.2 - the third example is -1.1 This is a task for actions with ordinary fractions. Here is a little theory for those who are slightly

Chapter I. Elements of linear algebra Linear algebra is the part of algebra that studies linear spaces and subspaces, linear operators, linear, bilinear and quadratic functions on linear spaces.

Progressions A sequence is a function of a natural argument. Specifying a sequence by a general term formula: a n = f(n), n N, for example, a n = n + n + 4, a = 43, a = 47, a 3 = 3,. Sequencing

Topic 1.4. Solution of systems of two (three) linear equations of Cramer's formula Gabriel Cramer (1704 1752) Swiss mathematician. This method is applicable only in the case of systems of linear equations, where the number of variables

Mathematics Grade 6 LEARNING CONTENT Arithmetic Natural numbers. Divisibility of natural numbers. Signs of divisibility by, 5, 9, 0. Prime and composite numbers. Decomposition of a natural number into prime factors.