Commonly used quantity relations

1. Number of copies × number of copies = total number of copies ÷ number of copies = total number of copies ÷ number of copies = number of copies

2. Multiples of 1 × multiples = multiples of multiples ÷ 1 multiples = multiples of multiples ÷ multiples = multiples of 1

3. Speed ??× time = distance distance ÷ speed = time distance ÷ time = speed

4. Unit price × quantity = total price total price ÷ unit price = quantity total price ÷ quantity = unit price

5. Work efficiency × working time = total amount of work total amount of work ÷ work efficiency = Total amount of work during working hours ÷ Working hours = Work efficiency

6. Addend + Addend = Sum and - One addend = Another addend

7. Minuend - Minuend = Minuend of difference - Difference = Minuend difference + Minuend = Minuend

8. Factor × factor = product ÷ one factor = another factor

9. Dividend ÷ Divisor = Quotient Divisor ÷ Quotient = Divisor Quotient × Divisor = Divisor

Primary school mathematics graph calculation formula

1. Square (C: Perimeter S: Area a: Side length)

Perimeter = side length × 4 C=4a

Area = side length × side length S=a×a

2. Cube (V : Volume a: Edge length)

Surface area = Edge length × Edge length × 6 S table = a × a × 6

Volume = Edge length × Edge length × Edge length V= a×a×a

3. Rectangle (C: perimeter S: area a: side length)

Perimeter = (length and width)×2 C=2(a b)

Area = length × width S = ab

4. Cuboid (V: volume s: area a: length b: width h: height)

( 1) Surface area (length × width × height width × height) × 2 S=2(ab ah bh)

(2) Volume = length × width × height V=abh

5. Triangle (s: area a: base h: height)

Area=base×height÷2 s=ah÷2

Triangle height=area×2÷base triangle Base = area × 2÷height

6. Parallelogram (s: area a: base h: height)

Area = base × height s=ah

7. Trapezoid (s: area a: upper base b: lower base h: height)

Area=(upper base and lower base)×height÷2 s=(a b)× h÷2

8. Circle (S: Area C: Perimeter л d=Diameter r=Radius)

(1) Perimeter = Diameter×л=2×л×Radius C=лd= 2лr

(2) Area = radius × radius × л

9. Cylinder (v: volume h: height s: base area r: base radius c: base perimeter)

(1) Side area = bottom perimeter × height = ch (2лr or лd) (2) Surface area = side area bottom area × 2

(3) Volume = bottom area × height (4) volume = side area ÷ 2 × radius

10. Cone (v: volume h: height s: base area r: base radius)

Volume = base Area × height ÷ 3

11. Total number ÷ total number of copies = average

12. Formula for sum and difference problem

(sum + difference) ÷2 = Large number (sum - difference) ÷ 2 = decimal

13. Sum and multiples problem

Sum ÷ (multiple - 1) = decimal decimal × multiple = large number (or sum - Decimal = large number)

14. Difference problem

Difference ÷ (multiple - 1) = decimal decimal × multiple = large number (or decimal + difference = large number)

15. Encounter problem

Meeting distance = speed sum × meeting time

Meeting time = meeting distance

÷Sum of speed

Sum of speed = distance to encounter ÷ time of encounter

16. Concentration problem

Weight of solute + weight of solvent = weight of solution

Weight of solute ÷ weight of solution × 100 = concentration

Weight of solution × concentration = weight of solute

Weight of solute ÷ concentration = weight of solution

17. Profit and discount issues

Profit = selling price - cost

Profit rate = profit ÷ cost × 100 = (sold price ÷ cost - 1) × 100

Amount of increase or decrease = principal × increase or decrease percentage

Interest = principal × interest rate × time

After-tax interest = principal × interest rate × time ×(1-20)

Common unit conversion

Length unit conversion

1 kilometer = 1000 meters 1 meter = 10 decimeters 1 decimeter = 10 Centimeter 1 meter = 100 centimeters 1 centimeter = 10 millimeters

Area unit conversion

1 square kilometer = 100 hectares 1 hectare = 10,000 square meters 1 square meter = 100 square decimeters

1 square decimeter = 100 square centimeters 1 square centimeter = 100 square millimeters

Volume unit conversion

1 cubic meter = 1000 cubic decimeters 1 cubic decimeter = 1000 cubic centimeters 1 cubic decimeter = 1 liter

1 cubic centimeter = 1 milliliter 1 cubic meter = 1000 liters

Weight unit conversion

1 ton = 1000 kilograms 1 kilogram = 1000 grams 1 kilogram = 1 kilogram

RMB unit conversion

1 yuan = 10 jiao 1 jiao = 10 cents 1 yuan = 100 cents

Time unit conversion

1 century = 100 years 1 year = 12 months (31 days): 1\3\5\7\8\10\12 months ( 30 days) are: April\6\9\November

February 28 days in an ordinary year, February 29 in a leap year, 365 days in an ordinary year, 366 days in a leap year, 1 day = 24 hours

1 hour = 60 minutes 1 minute = 60 seconds 1 hour = 3600 seconds

Basic concepts

Chapter 1 Numbers and Number Operations

One concept

(1) Integers

1 The meaning of integers

Natural numbers and 0 are both integers.

2 Natural Numbers

When we count objects, the 1, 2, 3... used to express the number of objects are called natural numbers.

There is no object, represented by 0. 0 is also a natural number.

3 Counting units

One (one), ten, one hundred, one thousand, ten thousand, one hundred thousand, one million, ten million, billion... are all counting units.

The progress rate between each two adjacent counting units is 10. This counting method is called decimal notation.

4 digits

The counting units are arranged in a certain order, and the positions they occupy are called digits.

Divisibility of 5 numbers

If the integer a is divided by the integer b (b ≠ 0), the quotient of the division is an integer without a remainder, we say that a can be divided by b, or Say b can divide a.

If the number a can be divided by the number b (b ≠ 0), a is called a multiple of b, and b is called the divisor of a (or factor of a). Multiples and divisors are interdependent.

Because 35 is divisible by 7, 35 is a multiple of 7 and 7 is a divisor of 35.

The number of divisors of a number is limited, the smallest divisor is 1, and the largest divisor is itself. For example: the divisors of 10 are 1, 2, 5, and 10. The smallest divisor is 1 and the largest divisor is 10.

The number of multiples of a number is infinite, and the smallest multiple is itself. The multiples of 3 are: 3, 6, 9, 12... The smallest multiple is 3, and there is no largest multiple.

Numbers whose units digit is 0, 2, 4, 6, or 8 can all be divisible by 2. For example: 202, 480, 304 can all be divisible by 2. .

Numbers with 0 or 5 in the units digit can be divisible by 5. For example: 5, 30, and 405 can all be divisible by 5. .

If the sum of the digits of a number can be divisible by 3, the number can be divisible by 3. For example: 12, 108, and 204 can all be divisible by 3.

If the sum of the digits of a number is divisible by 9, the number will be divisible by 9.

A number that is divisible by 3 may not necessarily be divisible by 9, but a number that is divisible by 9 must be divisible by 3.

If the last two digits of a number are divisible by 4 (or 25), the number will be divisible by 4 (or 25). For example: 16, 404, and 1256 are all divisible by 4, and 50, 325, 500, and 1675 are all divisible by 25.

If the last three digits of a number are divisible by 8 (or 125), the number will be divisible by 8 (or 125). For example: 1168, 4600, 5000, and 12344 are all divisible by 8, and 1125, 13375, and 5000 are all divisible by 125.

A number that is divisible by 2 is called an even number.

A number that is not divisible by 2 is called an odd number.

0 is also an even number. Natural numbers can be divided into odd numbers and even numbers according to whether they are divisible by 2.

If a number has only two divisors, 1 and itself, such a number is called a prime number (or prime number). The prime numbers within 100 are: 2, 3, 5, 7, 11, 13, 17 , 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

If a number has other divisors besides 1 and itself, such a number is called a composite number. For example, 4, 6, 8, 9, and 12 are all composite numbers.

1 is neither a prime number nor a composite number. Except for 1, the natural numbers are either prime numbers or composite numbers. If natural numbers are classified according to the number of their divisors, they can be divided into prime numbers, composite numbers and 1.

Every composite number can be written as the multiplication of several prime numbers. Each prime number is a factor of this composite number and is called a prime factor of this composite number. For example, 15=3×5, 3 and 5 are called prime factors of 15.

Representing a composite number in the form of multiplying prime factors is called decomposing prime factors.

For example, decompose 28 into prime factors

The common divisor of several numbers is called the common divisor of these numbers. The largest one is called the greatest common divisor of these numbers. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12; the divisors of 18 are 1, 2, 3, 6, 9, and 18. Among them, 1, 2, 3, and 6 are the common divisors of 12 and 1 8, and 6 is their greatest common divisor.

Two numbers whose common factor is only 1 are called coprime numbers. Two numbers that are in a mutually prime relationship have the following situations:

1 is coprime with any natural number.

Two adjacent natural numbers are relatively prime.

Two different prime numbers are relatively prime.

When the composite number is not a multiple of a prime number, the composite number and the prime number are relatively prime.

When the common divisor of two composite numbers is only 1, the two composite numbers are mutually prime. If any two of several numbers are mutually prime, it is said that these numbers are mutually prime.

If the smaller number is a divisor of the larger number, then the smaller number is the greatest common divisor of the two numbers.

If two numbers are coprime, their greatest common divisor is 1.

The common multiples of several numbers are called the common multiples of these numbers. The smallest one is called the least common multiple of these numbers. For example, the multiples of 2 are 2, 4, 6, 8, and 10. , 12, 14, 16, 18...

The multiples of 3 are 3, 6, 9, 12, 15, 18... Among them, 6, 12, 18... are common multiples of 2 and 3, 6 is their least common multiple. .

If the larger number is a multiple of the smaller number, then the larger number is the least common multiple of the two numbers.

If two numbers are relatively prime, then the product of the two numbers is their least common multiple.

The number of common divisors of several numbers is finite, while the number of common multiples of several numbers is infinite.

(2) Decimals

1 The meaning of decimals

Divide the integer 1 evenly into 10 parts, 100 parts, 1000 parts... What tenths are obtained? Hundreds, thousandths... can be expressed in decimals.

One decimal represents tenths, two decimals represent hundredths, and three decimals represent thousandths...

A decimal consists of an integer part, a decimal part and Made up of decimal points. The dot in a number is called the decimal point, the number to the left of the decimal point is called the integer part, the number to the left of the decimal point is called the integer part, and the number to the right of the decimal point is called the decimal part.

In decimals, the advance rate between each two adjacent counting units is 10. The progression rate between the highest fractional unit "tenth" of the decimal part and the lowest unit "one" of the whole number part is also 10.

2 Classification of decimals

Pure decimals: Decimals whose integer part is zero are called pure decimals. For example: 0.25 and 0.368 are both pure decimals.

With decimals: A decimal whose integer part is not zero is called a decimal. For example: 3.25 and 5.26 are both with decimals.

Finite decimal: A decimal whose digits in the decimal part is finite is called a finite decimal. For example: 41.7, 25.3, 0.23 are all finite decimals.

Infinite decimal: A decimal whose digits in the decimal part is infinite is called an infinite decimal. For example: 4.33… 3.1415926…

Infinite non-repeating decimal: The decimal part of a number with irregular arrangement of numbers and infinite digits. Such a decimal is called an infinite non-repeating decimal. For example: ∏

Recurring decimal: The decimal part of a number has a number or several numbers that repeatedly appear in sequence. This number is called a recurring decimal. For example: 3.555... 0.0333... 12.109109...

The decimal part of a recurring decimal, and the numbers that appear repeatedly in sequence are called the recurring sections of this recurring decimal. For example: the cyclic section of 3.99... is "9", and the cyclic section of 0.5454... is "54".

Pure repeating decimal: The repeating section starts from the first digit of the decimal part, which is called a pure repeating decimal. For example: 3.111… 0.5656…

Mixed cyclic decimal: The cyclic section does not start from the first digit of the decimal part, which is called a mixed cyclic decimal. 3.1222 …… 0.03333 ……

When writing recurring decimals, for simplicity, you only need to write a cyclic section for the cyclic part of the decimal, and dot a circle on the first and last digits of this cyclic section. point. If the loop section has only one number, just put a dot on it. For example: 3.777 ... abbreviated as 0.5302302 ... abbreviated as .

(3) Fractions

1 The meaning of fractions

Divide the unit "1" evenly into several parts, and the number representing such one or several parts is called Fraction.

In fractions, the horizontal line in the middle is called the fraction line; the number below the fraction line is called the denominator, which indicates how many parts the unit "1" can be divided into equally; the number below the fraction line is called the numerator, which indicates how many parts there are. .

Divide the unit "1" evenly into several parts and represent the number of one part, which is called a fractional unit.

2 Classification of Fractions

Proper fraction: The fraction whose numerator is smaller than the denominator is called a proper fraction. The true score is less than 1.

Improper fraction: A fraction in which the numerator is greater than the denominator or the numerator and denominator are equal is called an improper fraction. An improper fraction is greater than or equal to 1.

Mixed numbers: Improper fractions can be written as numbers composed of integers and proper fractions, usually called mixed numbers.

3 Reduction and Common Fraction

Converting a fraction into a fraction that is equal to it but has a smaller numerator and denominator is called a reduction.

A fraction whose numerator and denominator are coprime numbers is called the simplest fraction.

Converting fractions with different denominators into fractions with the same denominator that are equal to the original fractions is called a common fraction.

(4) Percent

1 A number that expresses what percent of another number is called a percent, also called a percentage or a percentage. Percentages are usually represented by "". The percent sign is the symbol that represents a percentage.

Two methods

(1) How to read and write numbers

1. How to read integers: from high to low, read level by level . When reading "100 million" or "10,000", first read it according to the pronunciation of "one", and then add the word "billion" or "ten thousand" at the end. The 0 at the end of each level is not read out, and if there are several consecutive 0s in other digits, only one zero is read.

2. How to write integers: from high to low, write level by level. If there is no unit on any digit, write 0 on that digit.

3. How to read decimals: When reading decimals, the integer part is read as an integer, the decimal point is read as "dot", and the decimal part is read sequentially from left to right on each digit. number.

4. How to write decimals: When writing decimals, write the integer part as an integer. The decimal point is written in the lower right corner of the ones place, and the decimal part writes the numbers on each digit in sequence.

5. How to read fractions: When reading fractions, read the denominator first, then "divided" and then the numerator. The numerator and denominator should be read as integers.

6. How to write fractions: first write the fraction line, then the denominator, and finally the numerator. Write it as an integer.

7. How to read percentages: When reading percentages, read percent first, then read the number before the percent sign. When reading, read it as an integer.

8. How to write percentages: Percents are usually not written as fractions, but are represented by adding a percent sign "" after the original numerator.

(2) Rewriting of numbers

For the convenience of reading and writing, a large multi-digit number is often rewritten into a number using "ten thousand" or "hundred million" as the unit. . Sometimes you can omit the number after a certain digit of the number and write it as an approximate number as needed.

1. Accurate number: In real life, for the convenience of counting, a larger number can be rewritten into a number in units of tens of thousands or billions. The rewritten number is the exact number of the original number. For example, if 1254300000 is rewritten as a number in tens of thousands, it will be 1254.3 million; if it is rewritten as a number in hundreds of millions, it will be 1.2543 billion.

2. Approximate number: According to actual needs, we can also omit the mantissa after a certain digit of a larger number and use an approximate number to represent it. For example: 1302490015 omitting the last digit after billion is 1.3 billion.

3. Rounding method: If the number in the highest digit of the mantissa to be omitted is 4 or smaller than 4, remove the mantissa; if the number in the highest digit of the mantissa to be omitted is 5 or larger than 5, just Round off the integer and add 1 to the previous digit. For example: omitting the last digit after 3459 million is approximately 350,000. Omitting the last digit after 472509742 billion is approximately 4.7 billion.

4. Size comparison

1. Compare the size of integers: Compare the size of integers. The number with more digits is larger. If the digits are the same, look at the highest bit. If the number on the top is larger, that number will be larger; if the number on the highest digit is the same, look at the next digit. Whichever digit has the larger number will be the larger number.

2. Compare the size of decimals: first look at their integer parts, the number with a larger integer part is larger; if the integer parts are the same, the number with a larger number in the tenth place is larger; the number in the tenth place is larger If the numbers on the percentile are the same, the number with a larger percentile will be larger...

3. Compare the sizes of fractions: If the denominator is the same, the fraction with the larger numerator will be larger; if the numerator is the same Numbers, the smaller the denominator, the larger the fraction.

If the denominator and numerator of a fraction are different, first make the common fraction and then compare the two numbers.

(3) Mutual conversion of numbers

1. Convert decimals into fractions: How many decimals there are originally, just write a few zeros after 1 as the denominator, and remove the original decimals The decimal point is used as the numerator to reduce the points.

2. Convert fractions to decimals: Use the denominator to remove the numerator. Those that can be divided into finite decimals are converted into finite decimals. Some cannot be divided into finite decimals, and those that cannot be converted into finite decimals are generally kept to three decimal places.

3. A simplest fraction, if the denominator contains no other prime factors except 2 and 5, this fraction can be converted into a finite decimal; if the denominator contains prime factors other than 2 and 5, This fraction cannot be converted into a finite decimal.

4. Convert decimals into percentages: Just move the decimal point two places to the right and add a percent sign at the end.

5. Convert percentages to decimals: To convert percentages to decimals, just remove the percent sign and move the decimal point two places to the left.

6. Convert fractions into percentages: Usually, the fractions are converted into decimals first (when division cannot be completed, three decimal places are usually retained), and then the decimals are converted into percentages.

7. Convert percentages into decimals: First rewrite the percentage into fractions, and reduce those that can be reduced into the simplest fraction.

(4) Divisibility of Numbers

1. Decompose a composite number into prime factors, usually using short division. First divide by prime numbers that can divide the composite number, keep dividing until the quotient is a prime number, and then write the divisor and quotient in the form of continuous multiplication.

2. The method to find the greatest common divisor of several numbers is: first divide by the common divisors of these numbers continuously until the obtained quotient only has a common divisor of 1, and then divide all the divisors Multiply and find the product. This product is the greatest common divisor of these numbers.

3. The method to find the least common multiple of several numbers is: first divide by the common divisor of these numbers (or part of them) until they are mutually prime (or two by two). So far, then multiply all the divisors and quotients to find the product. This product is the least common multiple of these numbers.

4. Two numbers that are in a mutually prime relationship: 1 is mutually prime with any natural number; two adjacent natural numbers are mutually prime; when the composite number is not a multiple of a prime number, the composite number and the prime number are mutually prime. Prime; Two composite numbers are relatively prime when their common divisor is only 1.

(5) Reduction and common division

Reduction method: use the common divisor of the numerator and the denominator (except 1) to remove the numerator and denominator; usually divide until the final Up to simple fractions.

The method of common fractions: first find the lowest common multiple of the denominators of the original fractions, and then convert each fraction into a fraction using this lowest common multiple as the denominator.

Three properties and laws

(1) The law of constant quotient

The law of constant quotient: In division, the dividend and divisor expand at the same time or at the same time If you reduce it by the same amount, the quotient remains the same.

(2) Properties of decimals

Properties of decimals: Adding zero or removing zero from the end of a decimal will keep the size of the decimal unchanged.

(3) The movement of the decimal point position causes changes in the size of the decimal

1. If the decimal point is moved one place to the right, the original number will be expanded by 10 times; if the decimal point is moved two places to the right, the original number will be expanded by 10 times. The original number will be expanded 100 times; if the decimal point is moved three places to the right, the original number will be expanded 1000 times...

2. If the decimal point is moved one place to the left, the original number will be reduced 10 times; Move two places to the left, and the original number will be reduced by 100 times; move the decimal point three places to the left, and the original number will be reduced by 1,000 times...

3. When the decimal point is not moved left or right enough. , the bits should be padded with "0".

(4) Basic properties of fractions

Basic properties of fractions: The numerator and denominator of a fraction are multiplied or divided by the same number (except zero), and the size of the fraction remains unchanged. .

(5) The relationship between fractions and division

1. Divisor ÷ divisor = dividend/divisor

2. Because zero cannot be used as a divisor, the denominator of the fraction is Cannot be zero.

3. The dividend is equivalent to the numerator, and the divisor is equivalent to the denominator.

The meaning of the four operations

(1) Four arithmetic operations on integers

1 Integer addition:

Combine two numbers into one number The operation is called addition.

In addition, the numbers added are called addends, and the resulting numbers are called sums. The addend is the part number and the sum is the total number.

Addends Addends = sum and one addend = sum - another addend

2 Integer subtraction:

The sum and of two addends are known The operation of finding one of the addends and the other addend is called subtraction.

In subtraction, the known sum is called the minuend, the known addend is called the subtrahend, and the unknown addend is called the difference. The minuand is the total number, and the subtrahend and difference are the partial numbers respectively.

Addition and subtraction are inverse operations of each other.

3 Integer multiplication:

The simple operation of finding the sum of several identical addends is called multiplication.

In multiplication, the same addends and the number of the same addends are called factors. The sum of the same addends is called the product.

In multiplication, 0 multiplied by any number yields 0. Multiplying 1 with any number yields any number.

A factor Factors, the operation of finding another factor is called division.

In division, the known product is called the dividend, a known factor is called the divisor, and the factor required is called the quotient.

Multiplication and division are the inverse operations of each other.

In division, 0 cannot be used as a divisor. Since 0 multiplied by any number is 0, any number divided by 0 cannot get a definite quotient.

Divisor ÷ Divisor = Quotient Divisor = Divisor ÷ Quotient Divisor = Quotient × Divisor

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