(100-96) - first action
Divide 320 by what happened in brackets - the second step
multiply by five - by the third action
plus 350 - by the fourth action
1 350+320=670:4=167.5=837.5
Similar tasks:
1. Fill in the blanks: 18t 4t = kg
6280g = kg g
48ts = kg
26302kg = t kg
7350kg = kg kg
35kg = g
2. Compare 18t 78kg 1t 878kg
22t 63kg 2t 263kg
380000g 38kg
5kg 320g 532g
3kg 490g 349g
3. Finish recording:
1/4 of a ton is kg
1/5 of a kilogram is g
1/10th of a quintal is kg
4. Express in smaller measures:
86ts =
3t =
25kg =
2t 3t =
5. Solve the problem.
Each of the three cars carried 28 quintals of grain, and the fourth - 16 quintals. All four vehicles were carrying tons of grain.
6. Solve the Problem.
The store brought 3 tons of watermelons. On the first day we sold 900 kg, on the second twice as much as on the first, and on the third day the rest. How many kilograms of watermelons were sold on the third day?
Solution:
7. Solve the problem. How many kilograms of flour are in two bags, if one contains 1/4 quintal and the other 1/4 quintal?
Answer:
8. Solve the Problem 1/2 kg of sweets cost 28 rubles. How much does 1 kg of sweets cost?
Answer:
9.* Solve the problem.
Gena has 900 rubles. And Valentin has 9 times less. How many rubles should Gena give to Valentin so that they have equal amounts of money?
Answer:
10. Solve the problem (orally):
72 kg of cucumbers were divided equally into 8 baskets. We sold three of these baskets. How many kilograms of cucumbers are left?
Answer:
1. Fill in the blanks:
3t 005 kg = kg
3t 5 c = kg
19kg = g
39ts = kg
5830kg = kg kg
46500kg = t kg
2. Compare
14t 260kg 14260kg
7670c 76t 7c
73000g 73kg
260000g 26kg
345t 34500ts
3. Finish recording:
1/4 part of a quintal is kg
1/5 of a ton is quintal
1/10th of a kilogram is g
4. Express in larger measures:
73ts =
640 kg =
2830g =
3200kg =
5. Solve the problem.
Each of the three buyers bought 18 kg of carrots, and the fourth - 46 kg. All four bought 1/2 of carrots
6. Solve the problem. 2 tons of carrots were collected from three participants. From the first plot, 500 kg were collected, from the second - 2 times more than from the first, and from the third - the rest of the carrots. How many kilograms of carrots were collected from the third plot?
Solution:
Answer:
7. Compare
1/4kg 1/2kg
1/2c 1/10c
1/10t 1/2t
8. Solve the problem.
A female blue whale loses 30 tons of weight while nursing a calf. This makes up 1/4 of its total mass. Determine the mass of the blue whale mother.
Answer:
9. Calculate and write down the answer:
816:6
x5
+490
:2
_________
100:2
x7
-250
:100
________
10.* Rearrange the digits in the number 810 so that it decreases by 630.
Answer.
To write a rational number m/n as a decimal fraction, you need to divide the numerator by the denominator. In this case, the quotient is written as finite or infinite decimal.
Write this number as a decimal fraction.
Solution. Divide the numerator of each fraction into a column by its denominator: A) divide 6 by 25; b) divide 2 by 3; V) divide 1 by 2, and then add the resulting fraction to one - the integer part of this mixed number.
Irreducible ordinary fractions whose denominators do not contain prime factors other than 2 And 5 , are written as a final decimal fraction.
IN example 1 when A) denominator 25=5·5; when V) the denominator is 2, so we get the final decimals 0.24 and 1.5. When b) the denominator is 3, so the result cannot be written as a finite decimal.
Is it possible to convert the following into a decimal fraction without long division? common fraction, the denominator of which does not contain any divisors other than 2 and 5? Let's figure it out! What fraction is called a decimal and is written without a fraction bar? Answer: fraction with denominator 10; 100; 1000, etc. And each of these numbers is a product equal number of twos and fives. In fact: 10=2 ·5 ; 100=2 ·5 ·2 ·5 ; 1000=2 ·5 ·2 ·5 ·2 ·5 etc.
Consequently, the denominator of an irreducible ordinary fraction will need to be represented as the product of “twos” and “fives”, and then multiplied by 2 and (or) 5 so that the “twos” and “fives” become equal. Then the denominator of the fraction will be equal to 10 or 100 or 1000, etc. To ensure that the value of the fraction does not change, we multiply the numerator of the fraction by the same number by which we multiplied the denominator.
Express the following common fractions as decimals:
Solution. Each of these fractions is irreducible. Let's expand the denominator of each fraction into prime factors.
20=2·2·5. Conclusion: one “A” is missing.
8=2·2·2. Conclusion: three “A”s are missing.
25=5·5. Conclusion: two “twos” are missing.
Comment. In practice, they often do not use factorization of the denominator, but simply ask the question: by how much should the denominator be multiplied so that the result is one with zeros (10 or 100 or 1000, etc.). And then the numerator is multiplied by the same number.
So, in case A)(example 2) from the number 20 you can get 100 by multiplying by 5, therefore, you need to multiply the numerator and denominator by 5.
When b)(example 2) from the number 8 the number 100 will not be obtained, but the number 1000 will be obtained by multiplying by 125. Both the numerator (3) and the denominator (8) of the fraction are multiplied by 125.
When V)(example 2) from 25 you get 100 if you multiply by 4. This means that the numerator 8 must be multiplied by 4.
periodic as a decimal. The set of repeating digits is called the period of this fraction. For brevity, the period of a fraction is written once, enclosed in parentheses.
When b)(example 1) there is only one repeating digit and is equal to 6. Therefore, our result 0.66... will be written like this: 0,(6) . They read: zero point, six in period.
If there are one or more non-repeating digits between the decimal point and the first period, then such a periodic fraction is called a mixed periodic fraction.
An irreducible common fraction whose denominator is together with others multiplier contains multiplier 2 or 5 , becomes mixed periodic fraction.
Write the numbers as a decimal fraction:
Any rational number can be written as an infinite periodic decimal fraction.
Write it as infinite periodic fraction numbers:
Solution.
Dear friends!
Dear friends! You will soon be faced (or have already faced) with the need to decide percent problems. They start solving such problems in the 5th grade and finish... but they don’t finish solving problems involving percentages! These tasks are found both in tests and in exams: both transfer ones and the Unified State Exam and Unified State Exam. What to do? We need to learn to solve such problems. My book “How to Solve Percentage Problems” will help you with this.
Adding numbers.
- a+b=c, where a and b are terms, c is the sum.
- To find the unknown term, you need to subtract the known term from the sum.
Subtracting numbers.
- a-b=c, where a is the minuend, b is the subtrahend, c is the difference.
- To find the unknown minuend, you need to add the subtrahend to the difference.
- To find unknown subtrahend, you need to subtract the difference from the minuend.
Multiplying numbers.
- a·b=c, where a and b are factors, c is the product.
- To find unknown multiplier, you need to divide the product by a known factor.
Dividing numbers.
- a:b=c, where a is the dividend, b is the divisor, c is the quotient.
- To find the unknown dividend, you need to multiply the divisor by the quotient.
- To find unknown divisor, you need to divide the dividend by the quotient.
Laws of addition.
- a+b=b+a(commutative: rearranging the terms does not change the sum).
- (a+b)+c=a+(b+c)(combinative: in order to add a third number to the sum of two terms, you can add the sum of the second and third to the first number).
Addition table.
- 1+9=10; 2+8=10; 3+7=10; 4+6=10; 5+5=10; 6+4=10; 7+3=10; 8+2=10; 9+1=10.
- 1+19=20; 2+18=20; 3+17=20; 4+16=20; 5+15=20; 6+14=20; 7+13=20; 8+12=20; 9+11=20; 10+10=20; 11+9=20; 12+8=20; 13+7=20; 14+6=20; 15+5=20; 16+4=20; 17+3=20; 18+2=20; 19+1=20.
Laws of multiplication.
- a·b=b·a(commutative: rearranging the factors does not change the product).
- (a b) c=a (b c)(combinative: to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third).
- (a+b)c=ac+bc(distributive law of multiplication relative to addition: in order to multiply the sum of two numbers by a third number, you can multiply each term by this number and add the resulting results).
- (a-b) c=a c-b c(distributive law of multiplication relative to subtraction: in order to multiply the difference of two numbers by a third number, you can multiply the minuend and subtract by this number separately and subtract the second from the first result).
Multiplication table.
2·1=2; 3·1=3; 4·1=4; 5·1=5; 6·1=6; 7·1=7; 8·1=8; 9·1=9.
2·2=4; 3·2=6; 4·2=8; 5·2=10; 6·2=12; 7·2=14; 8·2=16; 9·2=18.
2·3=6; 3·3=9; 4·3=12; 5·3=15; 6·3=18; 7·3=21; 8·3=24; 9·3=27.
2·4=8; 3·4=12; 4·4=16; 5·4=20; 6·4=24; 7·4=28; 8·4=32; 9·4=36.
2·5=10; 3·5=15; 4·5=20; 5·5=25; 6·5=30; 7·5=35; 8·5=40; 9·5=45.
2·6=12; 3·6=18; 4·6=24; 5·6=30; 6·6=36; 7·6=42; 8·6=48; 9·6=54.
2·7=14; 3·7=21; 4·7=28; 5·7=35; 6·7=42; 7·7=49; 8·7=56; 9·7=63.
2·8=16; 3·8=24; 4·8=32; 5·8=40; 6·8=48; 7·8=56; 8·8=64; 9·8=72.
2·9=18; 3·9=27; 4·9=36; 5·9=45; 6·9=54; 7·9=63; 8·9=72; 9·9=81.
2·10=20; 3·10=30; 4·10=40; 5·10=50; 6·10=60; 7·10=70; 8·10=80; 9·10=90.
Divisors and multiples.
- Divider natural number A name the natural number to which A divided without remainder. (The numbers 1, 2, 3, 4, 6, 8, 12, 24 are divisors of the number 24, since 24 is divisible by each of them without a remainder) 1 is the divisor of any natural number. Greatest divisor any number is the number itself.
- Multiples natural number b is a natural number that is divisible by b. (The numbers 24, 48, 72,... are multiples of the number 24, since they are divisible by 24 without a remainder). The smallest multiple of any number is the number itself.
Signs of divisibility natural numbers.
- The numbers used when counting objects (1, 2, 3, 4,...) are called natural numbers. The set of natural numbers is denoted by the letter N.
- Numbers 0, 2, 4, 6, 8 called even in numbers. Numbers that end in even digits are called even numbers.
- Numbers 1, 3, 5, 7, 9 called odd in numbers. Numbers that end in odd digits are called odd numbers.
- Test for divisibility by number 2 . All natural numbers ending in an even digit are divisible by 2.
- Test for divisibility by number 5 . All natural numbers ending in 0 or 5 are divisible by 5.
- Divisibility test for the number 10 . All natural numbers ending in 0 are divisible by 10.
- Test for divisibility by number 3 . If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3.
- Divisibility test for the number 9 . If the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9.
- Test for divisibility by number 4 . If a number made up of the last two digits given number, is divisible by 4, then the given number itself is divisible by 4.
- Divisibility test for the number 11. If the difference between the sum of the digits in odd places and the sum of the digits in even places is divisible by 11, then the number itself is divisible by 11.
- A prime number is a number that has only two divisors: one and the number itself.
- A number that has more than two divisors is called composite.
- The number 1 is neither a prime number nor a composite number.
- Writing a composite number as a product only prime numbers is called factoring a composite number into prime factors. Any composite number can be uniquely represented as a product of prime factors.
- The greatest common divisor of given natural numbers is the largest natural number by which each of these numbers is divided.
- Largest common divisor given numbers equal to the product common prime factors in expansions of these numbers. Example. GCD(24, 42)=2·3=6, since 24=2·2·2·3, 42=2·3·7, their common prime factors are 2 and 3.
- If natural numbers have only one common divisor - one, then these numbers are called relatively prime.
- The least common multiple of given natural numbers is the smallest natural number that is a multiple of each of the given numbers. Example. LCM(24, 42)=168. Exactly this small number, which is divisible by both 24 and 42.
- To find the LCM of several given natural numbers, you need to: 1) decompose each of the given numbers into prime factors; 2) write out the decomposition of the larger number and multiply it by the missing factors from the decomposition of other numbers.
- The least multiple of two relatively prime numbers is equal to the product of these numbers.
b-the denominator of a fraction shows how much equal parts divided;
a-the numerator of the fraction shows how many such parts were taken. The fraction bar means the division sign.
Sometimes instead of a horizontal fractional line they put an oblique line, and an ordinary fraction is written like this: a/b.
- U proper fraction the numerator is less than the denominator.
- U improper fraction the numerator is greater than the denominator or equal to the denominator.
If the numerator and denominator of a fraction are multiplied or divided by the same natural number, you get an equal fraction.
Dividing both the numerator and denominator of a fraction by their common divisor other than one is called reducing the fraction.
- A number consisting of an integer part and a fractional part is called a mixed number.
- To represent an improper fraction as a mixed number, you need to divide the numerator of the fraction by the denominator, then the partial quotient will be whole part mixed number, the remainder is the numerator of the fractional part, and the denominator remains the same.
- To represent a mixed number as an improper fraction, you need to multiply the integer part of the mixed number by the denominator, add the numerator of the fractional part to the resulting result and write it in the numerator of the improper fraction, leaving the denominator the same.
- Ray Oh with the starting point at the point ABOUT, on which are indicated single cut to and direction, called coordinate beam.
- Number, corresponding to the point coordinate ray, called coordinate this point. For example , A(3). Read: point A with coordinate 3.
- Lowest common denominator ( NCD) data irreducible fractions is the least common multiple ( NOC) denominators of these fractions.
- To reduce fractions to the smallest common denominator, you need to: 1) find the least common multiple of the denominators of these fractions, it will be the least common denominator. 2) find an additional factor for each fraction, why divide new denominator to the denominator of each fraction. 3) multiply the numerator and denominator of each fraction by its additional factor.
- From two fractions with same denominators the one with the larger numerator is greater, and the one with the smaller numerator is smaller.
- Of two fractions with the same numerators, the one with the smaller denominator is greater, and the one with the larger denominator is smaller.
- To compare fractions with different numerators and different denominators, you need to reduce fractions to the lowest common denominator, and then compare fractions with the same denominators.
Operations on ordinary fractions.
- To add fractions with the same denominators, you need to add their numerators and leave the denominator the same.
- If you need to add fractions with different denominators, first reduce the fractions to the lowest common denominator, and then add the fractions with the same denominators.
- To subtract fractions with like denominators, subtract the numerator of the second fraction from the numerator of the first fraction, leaving the denominator the same.
- If you need to subtract fractions with different denominators, then they are first brought to a common denominator, and then fractions with the same denominators are subtracted.
- When performing addition or subtraction operations mixed numbers these actions are performed separately for integer parts and for fractional parts, and then the result is written as a mixed number.
- The product of two ordinary fractions is equal to a fraction whose numerator is equal to the product of the numerators, and the denominator is equal to the product of the denominators of these fractions.
- To multiply a common fraction by a natural number, you need to multiply the numerator of the fraction by this number, but leave the denominator the same.
- Two numbers whose product is equal to one are called reciprocal numbers.
- When multiplying mixed numbers, they are first converted to improper fractions.
- To find a fraction of a number, you need to multiply the number by that fraction.
- To divide a common fraction by a common fraction, you need to multiply the dividend by the reciprocal of the divisor.
- When dividing mixed numbers, they are first converted into improper fractions.
- To divide a common fraction by a natural number, you need to multiply the denominator of the fraction by this natural number, and leave the numerator the same. ((2/7):5=2/(7·5)=2/35).
- To find a number by its fraction, you need to divide the number corresponding to it by this fraction.
- A decimal fraction is a number written in the decimal system and having digits less than one. (3.25; 0.1457, etc.)
- The places after the decimal point in a decimal fraction are called decimal places.
- The decimal will not change if you add or remove zeros at the end of the decimal.
To add decimal fractions, you need to: 1) equalize the number of decimal places in these fractions; 2) write them down one after the other so that the comma is written under the comma; 3) perform the addition, not paying attention to the comma, and put a comma in the sum under the commas in the added fractions.
To subtract decimal fractions, you need to: 1) equalize the number of decimal places in the minuend and the subtrahend; 2) sign the subtrahend under the minuend so that the comma is under the comma; 3) perform the subtraction, not paying attention to the comma, and in the resulting result place a comma under the commas of the minuend and the subtrahend.
- To multiply a decimal fraction by a natural number, you need to multiply it by this number, ignoring the comma, and in the resulting product, separate as many digits to the right with a comma as there were after the decimal point in this fraction.
- To multiply one decimal fraction by another, you need to perform the multiplication, not paying attention to the commas, and in the resulting result, separate as many digits on the right with a comma as there were after the decimal points in both factors together.
- To multiply a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point to the right by 1, 2, 3, etc. digits.
- To multiply a decimal by 0.1; 0.01; 0.001, etc. you need to move the decimal point to the left by 1, 2, 3, etc. digits.
- To divide a decimal fraction by a natural number, you need to divide the fraction by this number, as natural numbers are divided, and put a comma in the quotient when the division of the whole part is completed.
- To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point to the left by 1, 2, 3, etc. digits.
- To divide a number by a decimal fraction, you need to move the commas in the dividend and divisor as many digits to the right as there are after the decimal point in the divisor, and then divide by the natural number.
- To divide a decimal by 0.1; 0.01; 0.001, etc., you need to move the decimal point to the right by 1, 2, 3, etc. digits. (Dividing a decimal by 0.1, 0.01, 0.001, etc. is the same as multiplying that decimal by 10, 100, 1000, etc.)
To round a number to any digit, we underline the digit of this digit, and then we replace all the digits after the underlined one with zeros, and if they are after the decimal point, we discard them. If the first digit replaced by a zero or discarded is 0, 1, 2, 3 or 4, then the underlined digit is left unchanged. If the first digit replaced by a zero or discarded is 5, 6, 7, 8 or 9, then the underlined digit is increased by 1.
The arithmetic mean of several numbers.
The arithmetic mean of several numbers is the quotient of dividing the sum of these numbers by the number of terms.
The range of a number of numbers.
The difference between the largest and lowest values of a series of data is called the range of a series of numbers.
Mode of number series.
The number that occurs with the highest frequency among the given numbers in a series is called the mode of the number series.
- One hundredth part is called a percentage.
- To express percentages as a fraction or a natural number, you need to divide the percentage by 100%. (4%=0.04; 32%=0.32).
- To express a number as a percentage, you need to multiply it by 100%. (0.65=0.65·100%=65%; 1.5=1.5·100%=150%).
- To find the percentage of a number, you need to express the percentage as a common or decimal fraction and multiply the resulting fraction by the given number.
- To find a number by its percentage, you need to express the percentage as an ordinary or decimal fraction and divide the given number by this fraction.
- To find what percentage the first number is from the second, you need to divide the first number by the second and multiply the result by 100%.
- The quotient of two numbers is called the ratio of these numbers. a:b or a/b– the ratio of numbers a and b, and a is the previous term, b is the next term.
- If the members of a given relation are rearranged, then the resulting relation is called the inverse of the given relation. The relationships b/a and a/b are mutually inverse.
- The ratio will not change if both terms of the ratio are multiplied or divided by the same number other than zero.
- The equality of two ratios is called proportion.
- a:b=c:d. This is a proportion. Read: A this applies to b, How c refers to d. The numbers a and d are called the extreme terms of the proportion, and the numbers b and c are called the middle terms of the proportion.
- The product of the extreme terms of a proportion is equal to the product of its middle terms. For proportion a:b=c:d or a/b=c/d the main property is written like this: a·d=b·c.
- To find the unknown extreme term of a proportion, you need to divide the product of the middle terms of the proportion by the known extreme term.
- To find the unknown average member proportions, you need to divide the product of the extreme terms of the proportion by the known middle term.
Let the value y depends on the size X. If when increasing X several times the size at increases by the same amount, then such values X And at are called directly proportional.
If two quantities are directly proportional, then the ratio of two arbitrarily taken values of the first quantity is equal to the ratio of two corresponding values of the second quantity.
The ratio of the length of a segment on a map to the length of the corresponding distance on the ground is called the map scale.
Let the value at depends on the size X. If when increasing X several times the size at decreases by the same amount, then such values X And at are called inversely proportional.
If two quantities are in reverse proportional dependence, then the ratio of two arbitrarily taken values of one quantity is equal to inverse relation corresponding values of another quantity.
- A set is a collection of some objects or numbers, compiled according to some general properties or laws (many letters on a page, many proper fractions with a denominator of 5, many stars in the sky, etc.).
- Sets consist of elements and can be finite or infinite. A set that does not contain a single element is called empty set and denote Ø.
- A bunch of IN called a subset of a set A, if all elements of the set IN are elements of the set A.
- Intersection of sets A And IN is a set whose elements belong to the set A and many IN.
- Union of sets A And IN is a set whose elements belong to at least one of these sets A And IN.
Lots of numbers.
- N– set of natural numbers: 1, 2, 3, 4,…
- Z– a set of integers: …, -4, -3, -2, -1, 0, 1, 2, 3, 4,…
- Q- a bunch of rational numbers, representable as a fraction m/n, Where m– whole, n– natural (-2; 3/5; √9; √25, etc.)
- A coordinate line is a straight line on which a positive direction, a reference point (point O) and a unit segment are given.
- Each point on the coordinate line corresponds to a certain number, which is called the coordinate of this point. For example, A(5). They read: point A with coordinate five. AT 3). They read: point B with coordinate minus three.
- Modulus of the number a (write |a|) call the distance from the origin to the point corresponding to a given number A. The modulus of any number is non-negative. |3|=3; |-3|=3, because the distance from the origin to the number -3 and to the number 3 is equal to three unit segments. |0|=0 .
- By definition of the modulus of a number: |a|=a, If a≥0 And |a|=-a, If A<0 .
Operations with rational numbers.
The sum of negative numbers is a negative number. The modulus of the sum is equal to the sum of the moduli of the terms (-3-5=-8).
The sum of two numbers with different signs has the sign of a term with a large absolute value. To find the modulus of the sum, you need to subtract the smaller from the larger modulus (-4+6=2; -7+3=-4).
The product of two negative numbers is a positive number. The modulus of the product is equal to the product of the moduli of these numbers (-5·(-6)=30).
The product of two numbers with different signs is a negative number. The modulus of the product is equal to the product of the moduli of these numbers (-3·7=-21; 4·(-7)=-28).
The quotient of two negative numbers is a positive number. The modulus of the quotient is equal to the quotient of the modulus of the dividend and divisor (-8:(-2)=4).
The quotient of two numbers with different signs is a negative number. The modulus of the quotient is equal to the quotient of the modulus of the dividend and divisor (-20:4=-5; 12:(-2)=-6).
- To write a rational number m/n as a decimal fraction, you need to divide the numerator by the denominator. In this case, the quotient is written either as a finite or infinite decimal fraction.
- Irreducible ordinary fractions, the denominators of which do not contain prime factors other than 2 and 5, are written as a final decimal fraction (3/2=1.5; 1/5=0.2).
- An infinite decimal fraction in which one or more digits invariably repeat in the same sequence is called periodic as a decimal. The set of repeating digits is called the period of this fraction. For brevity, the period of the fraction is written once, enclosing it in parentheses: 1/3=0,(3); 1/9=0,(1). If there is one or more non-repeating digits between the decimal point and the first period, then such a periodic fraction is called a mixed periodic fraction: 7/15 = 0.4 (6); 5/12=0.41 (6).
- An irreducible ordinary fraction, the denominator of which, together with other factors, contains a factor of 2 or 5, becomes a mixed periodic fraction.
- Any rational number can be written as an infinite periodic decimal fraction. Examples: 5=5,(0); 3/5=0.6 (0).
An infinite periodic decimal fraction is equal to an ordinary fraction, the numerator of which is the difference between the entire number after the decimal point and the number after the decimal point before the period, and the denominator consists of “nines” and “zeros”, and there are as many “nines” as there are digits in the period, and “ there are as many zeros as there are digits after the decimal point before the period. Examples:
1) 0,41 (6)=(416-41)/900=375/900=5/12
2) 0,10 (6)=(106-10)/900=96/900=8/75
3) 0,6 (54)=(654-6)/990=648/990=36/55
4) 0,(15)=(15-0)/99=15/99=5/33
5) 0,5 (3)=(53-5)/90=48/90=8/15.
The set of real numbers.
- Any infinite non-periodic decimal fraction called irrational number. Examples: π ; √2 ; e etc.
- All rational and irrational numbers form the set of real numbers. The set of real numbers is denoted by the letter R.
The median of a given series of numbers.
To find the median of a given series, you need to arrange these numbers in ascending or descending order. The number that is in the middle of the resulting series will be the median of this series of numbers. If the number of given numbers is even, then the median of the series is equal to the arithmetic mean of the two numbers in the middle of the series ordered in ascending or descending order.
- Expressions in which numbers, arithmetic symbols and parentheses can be used along with letters are called algebraic expressions.
- The letter values for which the algebraic expression makes sense are called valid letter values.
- If in an algebraic expression you replace the letters with their values and perform the indicated actions, then the resulting number is called the value of the algebraic expression.
- Two expressions are said to be identically equal if, for any admissible values of the variables, the corresponding values of these expressions are equal.
- A formula is an algebraic expression written as an equality and expressing the relationship between two or more variables. Example: path formula s=v t(s - distance traveled, v - speed, t - time).
- If there is a “+” sign before the brackets or there is no sign at all, then when the brackets are opened, the signs of the algebraic terms are preserved.
- If the parentheses are preceded by the sign " — ", then when opening the brackets, the signs of the algebraic terms change to opposite signs.
Terms that have the same letter part are called similar terms. Finding the algebraic sum of like terms is called reducing like terms. To bring similar terms, you need to add their coefficients and multiply the resulting result by the common letter part.
- An equality with a variable is called an equation.
- Solving an equation means finding its many roots. An equation may have one, two, several, many roots, or none at all.
- Each value of a variable at which a given equation turns into a true equality is called a root of the equation.
- Equations that have the same roots are called equivalent equations.
- Any term of the equation can be transferred from one part of the equality to another, while changing the sign of the term to the opposite.
- If both sides of an equation are multiplied or divided by the same non-zero number, you get an equation equivalent to the given equation.
- a-b – positive number, That a>b.
- If, when comparing numbers a and b, the difference a-b is a negative number, then a
- If inequalities are written by signs< или >, then they are called strict inequalities.
- If inequalities are written with the signs ≤ or ≥, then they are called non-strict inequalities.
Properties of numerical inequalities.
- If the right side of the inequality is swapped with its left side, then the sign of the inequality will change to the opposite. (If a>b, That b ).
- If the number a more number b, and the number b more number c, then the number a more number c. (If a> b, b> c, That a> c).
- If you add the same number to both sides of the inequality, the sign of the inequality will not change. (If a>b, That a+c>b+c, where c is any number).
- Any term can be moved from one part of the inequality to another by changing the sign of this term to the opposite.
- If both sides of an inequality are multiplied or divided by the same positive number, then the sign of the inequality does not change. (If a>b; c is a positive number, then ac>bc; a/c>b/c).
- If both sides of an inequality are multiplied or divided by the same negative number, then the sign of the inequality is reversed. (If a>b; c is a negative number, then ac
- If the number is positive a greater than a positive number b, then the number 1/ a less number 1/ b. (If a> b; a>0; b>0 , That 1/ a<1/ b).
Numerical intervals.
The interval between the points corresponding to the numbers a and b specified on the coordinate line represents the numerical interval between the numbers a and b. Types of numerical intervals: interval, line segment, half-interval, Ray, open Ray. Solutions to numerical inequalities can be depicted on numerical intervals.
A) b) Inequality of the form x≤a. Answer: (-∞; a]. V) d) Inequality of the form x≥a. Answer: . G) Straight on a plane. Central symmetry. Function. Inverse function. The rule for finding a function inverse to a given one: 1) from this equality they express x through y; 2) in the resulting equality, instead of x write y, and instead y write x. The graphs of mutually inverse functions are symmetrical to each other with respect to the straight line y=x (bisectors of the I and III coordinate angles). Linear function. Direct proportionality. Direct proportionality is a function defined by a formula of the form y=kx, where x is an independent variable, k- coefficient straight proportionality. The graph of direct proportionality is a straight line passing through the origin. Inverse proportionality. Inverse proportionality is a function defined by a formula of the form y=k/x, where x is an independent variable different from zero, k- coefficient reverse proportionality. The inverse proportionality graph is a hyperbola consisting of two branches. For k>0, the branches of the hyperbola are located in I and III, and for k<0 – во II и IV координатных четвертях. Linear equation in two variables and its graph. Systems of linear equations with two variables. Solving systems of linear inequalities with one variable. Absolute and relative errors. Page 1 of 1 1 Option No. 3329663 When completing tasks 1-23, the answer is one number, which corresponds to the number of the correct answer, or a number, a sequence of letters or numbers. The answer should be written without spaces or any additional characters. If the option is given by the teacher, you can enter the answers to the assignments in Part C or upload them to the system in one of the graphic formats. The teacher will see the results of completing assignments in Part B and will be able to evaluate the uploaded answers to Part C. The scores assigned by the teacher will appear in your statistics. Version for printing and copying in MS Word 1. square it, 2. add 1. The first of them squares the number on the screen, the second increases it by 1. Write down the order of commands in a program that converts the number 2 to the number 36 and contains no more than 4 commands. Enter only command numbers. (For example, the program 2122
- This program add 1 square it add 1 add 1. This program converts the number 1 to the number 6. Answer: 1. add 1, 2. multiply by 5. The first of them increases the number on the screen by 1, the second multiplies it. For example, program 121 specifies the following sequence of commands: add 1 multiply by 5 add 1 This program converts, for example, the number 7 to the number 41. Write down in your answer a program that contains no more than five commands and converts the number 2 to the number 280. Answer: The input of the algorithm is a natural number N. The algorithm constructs a new number from it R in the following way. 1. Constructing a binary notation for a number N. 2. Two more digits are added to this entry on the right according to the following rule: a) all the digits of the binary notation are added, and the remainder of dividing the sum by 2 is added to the end of the number (on the right). For example, record 10000 is converted to record 100001; b) the same actions are performed on this entry - the remainder of dividing the sum of digits by 2 is added to the right. The record obtained in this way (it contains two digits more than in the record of the original number N) is the binary representation of the desired number R. Enter the smallest number N, for which the result of the algorithm is greater than 97. In the answer, write this number in the decimal number system. Answer: The machine receives a five-digit number as input. Based on this number, a new number is constructed according to the following rules. 1. The first, third and fifth digits, as well as the second and fourth digits, are added separately. 2. The resulting two numbers are written one after another in non-decreasing order without separators. Example. Original number: 63,179. Sums: 6 + 1 + 9 = 16; 3 + 7 = 10. Result: 1016. Specify the smallest number when processed by the machine to produce the result 621. Answer: 1. The first and second digits, as well as the second and third digits, are multiplied separately. 2. The resulting two numbers are written one after another in non-increasing order without separators. Example. Original number: 179. Products: 1*7 = 7; 7*9 = 63. Result: 637. Specify the smallest number, when processed, the machine produces the result 205. Answer: The machine receives a four-digit number as input. Based on this number, a new number is constructed according to the following rules: 1. The first and second, as well as the third and fourth digits of the original number are multiplied. Example. Original number: 2466. Products: 2 × 4 = 8; 6 × 6 = 36. Result: 368. Specify the smallest number, as a result of which the machine will produce the number 124. Answer: A word is formed from the letters of the Russian alphabet. It is known that the word is formed according to the following rules: a) there are no repeating letters in the word; b) all letters of the word are in direct or reverse alphabetical order, possibly excluding the first. Which of the following words satisfies all of the conditions listed? Answer: The Accord-4 performer has two teams, which are assigned numbers: 1. subtract 1 2. multiply by 4 By executing the first of them, Accord-4 subtracts 1 from the number on the screen, and by executing the second, it multiplies this number by 4. Write down the order of commands in a program that contains no more than five commands and converts the number 5 to the number 62. If there is more than one such program, then write down any of them. In your answer, indicate only the command numbers. Yes, for the program multiply by 4 you need to write: 211. This program converts, for example, the number 7 to the number 26. Answer: The Calculator performer has two teams, which are assigned numbers: 1. subtract 1 2. divide by 3 When performing the first of them, the Calculator subtracts 1 from the number on the screen, and when performing the second, it divides it by 3 (if division is impossible, the Calculator turns off). Write down the order of commands in the program for obtaining number 1 from number 37, containing no more than 5 commands, indicating only the command numbers. (For example, program 21121 is a program divide by 3 divide by 3 This program, for example, converts the number 60 to the number 5.) Answer: Masha forgot the password to start the computer, but remembered the algorithm for obtaining it from the hint string “KBMAM9KBK”: if all sequences of characters “MAM” are replaced with “RP”, “KBK” with “1212”, and then the last three characters are removed from the resulting string, then the resulting sequence will be the password. Define a password: Answer: Anya invited her friend Natasha to visit, but did not tell her the code for the digital lock of her entrance, but sent the following message: “In the sequence 4, 1, 9, 3, 7, 5, from all numbers that are greater than 4, subtract 3, and then remove all odd numbers from the resulting sequence.” After completing the steps indicated in the message, Natasha received the following code for the digital lock: 4) 4, 1, 6, 3, 4, 2 Answer: Lyuba forgot the password to start the computer, but remembered the algorithm for obtaining it from the characters “QWER3QWER1” in the hint line. If all sequences of “QWER” characters are replaced with “QQ”, and the combinations of “3Q” characters are removed from the resulting string, then the resulting sequence will be the password: Answer: Performer ThreeFive has two teams, which are assigned numbers: 1. add 3, 2. multiply by 5. By completing the first of them, ThreeFive adds 3 to the number on the screen, and by completing the second, it multiplies this number by 5. Write down the order of commands in a program that contains no more than 5 commands and converts the number 1 to the number 515. In your answer, indicate only the command numbers, do not put spaces between the numbers. Yes, for the program multiply by 5 add 3 add 3 you need to write: 211. This program converts, for example, the number 4 to the number 26. Answer: The performer Kvadrator has two teams, which are assigned numbers: 1. add 1, 2. square it. The first of these commands increases the number on the screen by 1, the second - squares it. The program for the performer Quadrator is a sequence of command numbers. For example, 21211 is a program square it add 1 square it add 1 add 1 This program converts the number 2 to the number 27. Write a program that converts the number 2 to the number 102 and contains no more than 6 commands. If there is more than one such program, then write down any of them. Answer: The machine receives a three-digit number as input. Based on this number, a new number is constructed according to the following rules. 1. The first and second, as well as the second and third digits of the original number are added. 2. The resulting two numbers are written one after another in descending order (without separators). Example. Original number: 348. Sums: 3 + 4 = 7; 4 + 8 = 12. Result: 127. Specify the smallest number, as a result of which the machine will produce the number 1412. Answer: The machine receives a four-digit octal number as input. Based on this number, a new number is constructed according to the following rules. 1. The first and second, as well as the third and fourth digits are added. 2. The resulting two numbers in the octal number system are written one after another in ascending order (without separators). Example. Original number: 4531. Sums: 4+5 = 9; 3+1 = 4. Result: 49. Determine which of the following numbers can be the result of the machine. Answer: In some information system, information is encoded in binary six-bit words. When transmitting data, their distortion is possible, therefore, a seventh (control) digit is added to the end of each word so that the sum of the digits of the new word, including the control one, is even. For example, 0 will be added to the right of the word 110011, and 1 will be added to the right of the word 101100. After receiving the word, it is processed. In this case, the sum of its digits, including the control one, is checked. If it is odd, it means that there was a failure when transmitting this word, and it is automatically replaced by the reserved word 0000000. If it is even, it means that there was no failure or there was more than one failure. In this case, the accepted word is not changed. Original message 1100101 0001001 0011000 was adopted as 1100111 0001100 0011000 What will the received message look like after processing? 1) 0000000 0001100 0011000 2) 0000000 0000000 0011000 3) 1100111 0000000 0011000 4) 1100111 0001100 0000000 Answer: The performer Calculator1 has two teams, which are assigned numbers: 1. add 1, 2. multiply by 5. By performing the first of them, Calculator1 adds 1 to the number on the screen, and by performing the second, it multiplies it by 5. The program for this executor is a sequence of command numbers. For example, program 121 specifies the following sequence of commands: add 1, multiply 5, add 1, This program converts, for example, the number 7 to the number 41. Write in your answer a program that contains no more than six commands and converts the number 1 to the number 77. Answer: The CALCULATOR executor has only two commands, which are assigned numbers: 2. multiply by 2 By executing command number 1, the CALCULATOR subtracts from the number on the screen 1, and by executing command number 2, multiplies the number on the screen by 2. Write a program containing more than 4 teams, which from the number 3 gets the number 16. Indicate only the team numbers. For example, program 21211 is a program: multiply by 2 multiply by 2 which converts the number 1 to 0. Answer: Vasya forgot the password for Windows XP, but remembered the algorithm for obtaining it from the hint string “B265C42GC4”: if all sequences of characters “C4” are replaced with “F16”, and then all three-digit numbers are removed from the resulting string, then the resulting sequence will be the password. Define a password: Answer: Performer TwoFive has two teams, which are assigned numbers: 1. subtract 2 2. divide by 5 By performing the first of them, TwoFive subtracts 2 from the number on the screen, and by performing the second, it divides this number by 5 (if division is completely impossible, TwoFive is turned off). Write down the order of commands in a program that contains no more than 5 commands and converts the number 152 to the number 2. In your answer, indicate only the command numbers, do not put spaces between the numbers. Yes, for the program divide by 5 you need to write 211. This program converts, for example, the number 55 to the number 7. Answer: In some information system, information is encoded in binary six-bit words. When transmitting data, their distortion is possible, therefore, a seventh (control) digit is added to the end of each word so that the sum of the digits of the new word, including the control one, is even. For example, 0 will be added to the right of the word 110011, and 1 will be added to the word 101100. After receiving the word, it is processed. In this case, the sum of its digits, including the control one, is checked. If it is odd, it means that there was a failure when transmitting this word, and it is automatically replaced by the reserved word 0000000. If it is even, it means that there was no failure or there was more than one failure. In this case, the accepted word is not changed. The original message 1100101 0001001 1111000 was received as 1100111 0001100 1111000. What will the received message look like after processing? 1) 0000000 0001100 1111000 2) 0000000 0000000 1111000 3) 1100101 0000000 1111000 4) 1100111 0001100 0000000 Answer: Mitya invited his friend Vasya to visit, but did not tell him the code for the digital lock of his entrance, but sent the following message: “In the sequence 4, 1, 8, 2, 6, divide all numbers greater than 3 by 2, and then remove them from the resulting sequence all even numbers." After completing the steps indicated in the message, Vasya received the following code for the digital lock: Answer: The cashier forgot the password to the safe, but remembered the algorithm for obtaining it from the string “AYY1YABC55”: if you sequentially remove the string of characters “YY” and “ABC” from the string, and then swap the characters A and Y, then the resulting sequence will be the password. Define a password.
Secrets fast multiplication and divisions 1. Multiplication and division by 5, 50, 500, etc. Multiplication by 5, 50, 500, etc. is replaced by multiplication by 10, 100, 1000, etc., followed by division by 2 of the resulting product (or division by 2 and multiplication by 10, 100, 1000, etc. = 100:2, etc.) 54*5=(54*10):2=540:2=*5 = (54:2)*10= 270). To divide a number by 5.50, 500, etc., you need to divide this number by 10,100,1000, etc. and multiply by 2. 10800: 50 = 10800:100*2 =216 10800: 50 = 10800*2:100 =216 2. Multiplication and division by 25, 250, 2500, etc. Multiplication by 25, 250, 2500, etc. is replaced by multiplication by 100, 1000, 10000, etc. and the result is divided by = 100: 4) 542*25=(542*100):4=13*25=248: 4*100 = 6200) (if the number is divisible by 4, then multiplication does not take time; any student can do it). To divide a number by 25, 25,250,2500, etc., this number must be divided by 100,1000,10000, etc. and multiplied by 4 31200: 25 = 31200:100*4 = 1248. 3. Multiplication and division by 125, 1250, 12500, etc. Multiplication by 125, 1250, etc. is replaced by multiplication by 1000, 10000, etc. and the resulting product must be divided by = 1000: 8) 72*125=72*1000:8=9000 If the number is divisible by 8, then first divide by 8, and then multiply by 1000, 10000, etc. 48*125 = 48:8*1000 = 6000 To divide a number by 125, 1250, etc., you need to divide this number by 1000, 10000, etc. and multiply by 8. 7000: 125 = 7000:1000*8 = 56. 4. Multiplication and division by 75, 750, etc. To multiply a number by 75, 750, etc., you need to divide this number by 4 and multiply by 300, 3000, etc. (75 = 300: 4) 48* 75 = 48:4*300 = 3600 To divide a number by 75,750, etc., you need to divide this number by 300, 3000, etc. and multiply by 4 7200: 75 = 7200: 300*4 = 96. 5.Multiply by 15, 150. When multiplying by 15, if the number is odd, multiply it by 10 and add half of the resulting product: 23x15=23x(10+5)=230+115=345; if the number is even, then we proceed even simpler - we add half of it to the number and multiply the result by 10: 18x15=(18+9)x10=27x10=270. When multiplying a number by 150, we use the same technique and multiply the result by 10, since 150 = 15x10: 24x150=((24+12)x10)x10=(36x10)x10=3600. In the same way, quickly multiply a two-digit number (especially an even one) by a two-digit number ending in 5: 24*35 = 24*(30 +5) = 24*30+24:2*10 = 720+120=840. 6. Multiplying two-digit numbers less than 20. To one of the numbers you need to add the number of units of the other, multiply this amount by 10 and add to it the product of the units of these numbers: 18x16=(18+6)x10+8x6= 240+48=288. Using the described method, you can multiply two-digit numbers less than 20, as well as numbers that have the same number of tens: 23x24 = (23+4)x20+4x6=27x20+12=540+12=562. Explanation:
(10+a)*(10+b) = 100 + 10a + 10b + a*b = 10*(10+a+b) + a*b = 10*((10+a)+b) + a* b. 7.Multiplying a two-digit number by 101. Perhaps the simplest rule: assign your number to yourself. Multiplication is complete. 57 * 101 = 5> 5757 Explanation: (10a+b)*101 = 1010a + 101b = 1000a + 100b + 10a + b 8. Multiplying a number by 11. You should “spread apart” the digits of the number being multiplied by 11, and enter the sum of these digits into the resulting gap, and if this sum is more than 9, then, as with normal addition, the unit should be moved to the highest digit. Example: Explanation: We have: We compose the product: 5 units, 5+2=7 tens, 2+6=8 hundreds, 6+3=9 thousand, 3+4=7 tens of thousands, 4 hundreds of thousands. 43625*11=479875. When the multiplicand is between 1000 and 10000 (for example, 7543), then you can use the following method of multiplying by 11. First, divide the multiplicand 7543 into two-digit faces, then find the product of the first face (75) on the left by 11, as indicated in the multiplication a two-digit number by 11. The resulting number (75*11=725) will give hundreds of the product, since hundreds of the multiplicand were multiplied. Then you need to multiply the second side (43) by 11, we get the units of the product: 43*11=473. Finally, we add up the resulting products: 825 hundred. +473=82739. Therefore, 7543*11=82739. Let's look at another example: 8324*11. 83`24; 83 hundred *11=913 cells. 24*11=264; 913 cells +264=91564. Therefore, 8324*11=91564. 9. Multiplication by 22, 33, ..., 99. To multiply a two-digit number 22.33, ...,99, you need to represent this factor as the product of a single-digit number by 11. First multiply by single digit number, and then at 11: 15 *33= 15*3*11=45*11=495. 10.
Multiplying two-digit numbers by 111. First, let’s take as a multiplicand a two-digit number whose sum of digits is less than 10. Let’s explain with numerical examples: Since 111=100+10+1, then 45*111=45*(100+10+1). When multiplying a two-digit number, the sum of the digits of which is less than 10, by 111, it is necessary to insert twice the sum of the digits (i.e., the numbers represented by them) of its tens and units 4+5=9 into the middle between the digits. 4500+450+45=4995. Therefore, 45*111=4995. When the sum of the digits of a two-digit multiplicand is greater than or equal to 10, for example 68*11, you need to add the digits of the multiplicand (6+8) and insert 2 units of the resulting sum into the middle between the numbers 6 and 8. Finally, add 1100 to the composed number 6448. Therefore, 68*111=7548. 11. Multiply by 37. When multiplying a number by 37, if the given number is a multiple of 3, it is divided by 3 and multiplied by 111. 27*37=(27:3)*(37*3)=9*111=999 If the given number is not a multiple of 3, then 37 is subtracted from the product or 37 is added to the product. 23*37=(24-1)*37=(24:3)*(37*3)-37=888-37=851. 12. Square any two-digit number. If you memorize the squares of all numbers from 1 to 25, then it is easy to find the square of any two-digit number greater than 25. In order to find the square of any two-digit number, you need to multiply the difference between this number and 25 by 100 and to the resulting product add the square of the complement of the given number to 50 or the square of its excess over 50. Let's look at an example: 372=12*100+132=1200+169=1369 (M–25)*100+ (50-M) 2=100M-2500+2500–100M+M2=M2 . 13. Multiplication of numbers close to 100. When increasing (decreasing) one of the factors by several units, multiply the resulting integer and the added (subtracted) units by another factor and subtract the second product from the first product (add the resulting products) 98∙8=(100-2) ∙8=100∙8-2∙8=800-16=784. This technique of representing one of the factors as a difference allows you to easily multiply by 9, 99, 999. To do this, just multiply the number by 1000) and subtract the number that was multiplied from the resulting integer: 154x9=154x10-154==1386. But it’s even easier to familiarize children with the rule - “to multiply a number by 9 (99, 999), it is enough to subtract from this number the number of its tens (hundreds, thousands), increased by one, and to the resulting difference add the addition of its units digit to 10 (complement to the number formed by the last two (three) digits of this number): 154x9=(154-16)x10+(10-4)=138x10+6=1380+6=1386 14. Multiplication of two-digit numbers whose units add up to 10. Let two be given double digit numbers, whose sum is 10: M=10m + n, K=10a + 10 – n. Let's compose their work. M * K= (10m+n) * (10a + 10 – n) =100am + 100m – 10mn + 10an + +10n – n2 = m * (a + 1) * 100 + n * (10a + 10 – n) – 10mn = (10m) * * (10 * (a + 1)) + n * (K – 10m). Let's look at a few examples: 17 * 23= 10 * 30 + 7 * 13= 300 + 91= 391; 33 * 67= 30 * 70 + 3 * 37= 2100 + 111= 2211. 15 .
Multiplying by a number written in nines only. In order to find the product of a number written only in nines by a number that has the same number of digits, you need to subtract one from the factor and add another number to the resulting number, all the digits of which complement the digits of the specified resulting number to 9. 137 * 999= 136 863; The presence of such a method is seen from the following method of solving the given examples: 8 * 9= 8 * (10 – 1)= 80 – 8= 72, 46 * 99= 46 * (100 – 1)= 4600 – 54= 4554. 16. Squaring a number ending in 5. Multiply the number of tens by next number tens and add 25. 15*15 = 225 = 10*20+ 25 (or 1*2 and add 25 to the right) 35*35 =30*40 +25= 1225 (3*4 and add 25 to the right) 65*65 = 60*70+25=4225 (6*7 and add 25 to the right)
Inequality of the form x 
Inequality of the form x>a. Answer: (a; +∞).
Double inequality of the form a≤x≤b. Answer: .
Example:
Multiplication is done in the same way three-digit numbers by 1001, four-digit ones by 10001, etc.
34 * 11 = 374, since 3 + 4 = 7, we place the seven between the three and the four
68 * 11 = 748, since 6 + 8 = 14, we place the four between the seven (six plus the transferred one) and eight
10a+b - arbitrary number, where a is the number of tens, b is the number of units.
(10a+b)*11 = 10a*11 + b*11 = 110a + 11b = 100a + 10a + 10b + b = 100a + 10*(a+b) + b,
where do we have a hundreds, a+b tens and b units. i.e. the result contains a*(a+1) hundreds, two tens and five units.