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. 1 multiple× multiple = multiple1multiple = multiple/multiple = 1 multiple

3. Speed × time = distance/speed = time/distance/time = speed.

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

5. Work efficiency × working hours = total workload ÷ work efficiency = working hours ÷ total workload ÷ working hours = work efficiency.

6. Appendix+Appendix = sum, and-one addend = another addend.

7. Minus-Minus = Minus-Minus = Minus+Minus = Minus

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

9. Dividend = quotient dividend = divisor quotient × divisor = dividend

Calculation formula of mathematical graphics in primary schools

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: side length)

Surface area = side length × side length× 6 s table =a×a×6 volume = side length× side length× side length v = a× a× a.

3. rectangle (c: perimeter s: area a: side length)

Circumference = (length+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+length× height+width× height )× 2s = 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 triangle base = area× 2 triangle height.

6. parallelogram (s: area a: bottom h: height)

Area = bottom × height s=ah

7. trapezoid (s: area a: upper bottom b: lower bottom h: height)

Area = (upper bottom+lower bottom) × 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: bottom area R: bottom radius C: bottom circumference)

(1) lateral area = bottom perimeter × height =ch(2лr or лd) (2) surface area = lateral area+bottom area ×2.

(3) Volume = bottom area × height (4) Volume = lateral area ÷2× radius.

10, cone (v: volume h: height s: bottom area r: bottom radius)

Volume = bottom area × height ÷3

1 1, total number of copies/total number of copies = average.

12, the formula of sum difference problem: (sum+difference) ÷ 2 = large number (sum-difference) ÷ 2 = decimal.

13, and multiple problem: sum ÷ (multiple-1) = decimal × multiple = large number (or sum- decimal = large number)

14. Difference multiple problem: difference ÷ (multiple-1) = decimal × multiple = large number (or decimal+difference = large number)

15, encountered a problem.

Meeting distance = speed and x meeting time; Meet time = meet distance ÷ speed and; Speed Sum = Meeting Distance/Meeting Time

16, concentration problem

Solute weight+solvent weight = solution weight/solution weight × 100% = concentration.

Solution weight × concentration = solute weight/solute concentration = solution weight.

17, profit and discount

Profit = selling price-cost; Profit rate = profit/cost × 100% = (selling price/cost-1) × 100%.

Up and down amount = principal × up and down percentage; Interest = principal × interest rate × time; After-tax interest = principal × interest rate × time × (1-20%)

Universal unit conversion

Length unit conversion

1 km =1000m1m =1decimeter/decimeter =10cm1m =10cm/kloc-.

Area unit conversion:

1 square kilometer = 100 hectare 1 hectare = 10000 square meter 1 square meter = 100 square decimeter.

1 dm2 = 100 cm2 1 cm2 = 100 mm2

Volume (volume) unit conversion:

1 m3 = 1000 cubic decimeter 1 cubic decimeter = 1000 cubic centimeter 1 cubic decimeter = 1 liter.

1 cm3 = 1 ml 1 m3 = 1000 liter

Weight unit conversion:1t =1000kg1kg =1000mg1kg =1kg.

RMB unit conversion: 1 yuan = 10 angle 1 angle = 10 minute 1 yuan = 100 minute.

Time unit conversion:

1 century = 100 1 year =1February (3 1 day) has:1\ 3 \ 5 \ 7 \ 8 \1.

February 28th in a normal year and February 29th in a leap year: 365 days in a normal year and 366 days in a leap year: 1 =24 hours.

1 hour = 60min 1 minute = 60s 1 hour = 3600s.

basic concept

Chapter I Number and Number Operation

A concept

(1) integer

Meaning of 1 integer: both natural numbers and 0 are integers.

2 natural number:

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

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

3 counting unit

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

The propulsion rate between every two adjacent counting units is 10. This counting method is called decimal counting method.

4 digits: Counting units are arranged in a certain order, and their positions are called digits.

Divisibility of 5 numbers

When the integer A is divided by the integer b(b ≠ 0), the quotient is an integer with no remainder, so we say that A is divisible by B, or that B is divisible by A. ..

If the number A is divisible by the number B (b ≠ 0), then A is called a multiple of B, and B is called a divisor of A (or a factor of A). Multiplication and divisor are interdependent.

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

The divisor of a number is finite, in which the smallest divisor is 1 and the largest divisor is itself. For example, the divisor of 10 is 1, 2,5, 10, where the smallest divisor is 1 0 and the largest divisor is 10.

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

Numbers in units of 0, 2, 4, 6 and 8 can be divisible by 2, for example, 202, 480 and 304 can be divisible by 2. .

Numbers in units of 0 or 5 can be divisible by 5, for example, 5,30,405 can be divisible by 5. .

The sum of the numbers in each bit of a number can be divisible by 3, so this number can be divisible by 3. For example, 12,108,204 can all be divisible by 3.

The sum of each digit of a number can be divisible by 9, and so can this number.

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

The last two digits of a number can be divisible by 4 (or 25), and this number can also be divisible by 4 (or 25). For example,16,404 and 1256 can all be divisible by 4, and 50,325,500 and 1675 can all be divisible by 25.

The last three digits of a number can be divisible by 8 (or 125), and this number can also be divisible by 8 (or 125). For example,1168,4600,5000, 12344 can all be divisible by 8, and 1 125,13375,5000 can all be/kloc-.

A number divisible by 2 is called an even number.

Numbers that are not divisible by 2 are called odd numbers.

0 is also an even number. Natural numbers can be divided into odd and even numbers according to their divisibility by 2.

A number with only two divisors of 1 is called a prime number (or prime number), and the prime numbers within 100 are: 2, 3, 5, 7,1,13, 17.

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

1 is not a prime number or a composite number, and natural numbers are either prime numbers or composite numbers except 1. 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 product of several prime numbers. Every prime number is a factor of this composite number, which is called the prime factor of this composite number. For example, 15=3×5, and 3 and 5 are called prime factors of 15.

Multiplying a composite number by a prime factor is called prime factor decomposition.

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 divisor of 12 is 1, 2, 3, 4, 6,12; The divisors of 18 are 1, 2,3,6,9 and 18. Where 1, 2,3,6 are the common divisors of 12 and 1 8, and 6 is their greatest common divisor.

The common divisor is only 1, which is called prime number. There are the following situations:

1 is coprime with any natural number.

Two adjacent natural numbers are coprime.

Two different prime numbers are coprime.

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

When the common divisor of two composite numbers is only 1, these two composite numbers are coprime. If any two numbers are coprime, they are said to be coprime.

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

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

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

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

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

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

The common divisor of several numbers is finite, while the common multiple of several numbers is infinite.

(2) Decimals

The meaning of 1 decimal

Divide the integer 1 into 10, 100, 1000 ... a tenth, a percentage, a thousandth ... can be expressed in decimals.

One decimal place indicates a few tenths, two decimal places indicate a few percent, and three decimal places indicate a few thousandths. ...

Decimal system consists of integer part, decimal part and decimal part. The point in the number is called the decimal point, 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 series between every two adjacent counting units is 10. The propulsion rate between the highest decimal unit "one tenth" of the decimal part and the lowest unit "one" of the integer part is also 10.

2 Classification of decimals

Pure decimals: Decimals with zero integer parts are called pure decimals. For example, 0.25 and 0.368 are pure decimals.

With decimals: decimals whose integer part is not zero are called with decimals. For example, 3.25 and 5.26 are all decimals.

Finite decimals: The digits in the decimal part are finite decimals, which are called finite decimals. For example, 4 1.7, 25.3 and 0.23 are all finite decimals.

Infinite decimal: The digits in the decimal part are infinite decimal, which is called infinite decimal. For example: 4.33...3. 145438+05926 ...

Infinite acyclic decimal: the decimal part of a number with irregular arrangement and unlimited digits. Such decimals are called infinite cyclic decimals. For example: ∈

Cyclic decimal: the decimal part of a number, in which one or several numbers appear repeatedly in turn, is called cyclic decimal. For example: 3.555 … 0.0333 …12.15438+009 …

The decimal part of cyclic decimal is called the cyclic part of cyclic decimal. For example, the period of 3.99 ... is "9", and the period of 0.5454 ... is "54".

Pure cyclic decimal: the cyclic segment starts from the first digit of the decimal part, which is called pure cyclic decimal. For example: 3.111.5656 ...

Mixed cycle decimal: the cycle section does not start from the first digit of the decimal part. This is called mixed cyclic decimal. 3. 1222 …… 0.03333 ……

When writing a cyclic decimal, for simplicity, the cyclic part of the decimal only needs one cyclic segment, and a dot is added to the first and last digits of this cyclic segment. If there is only one number in the circle, just click a point on it. For example: 3.777 ... Jane writing 0.5302302 ... Jane writing.

(3) scores

1 significance of the score

Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction.

In the score, the middle horizontal line is called the dividing line; The number below the fractional line is called the denominator, indicating how many copies the unit "1" is divided into on average; The number below the fractional line is called the numerator, indicating how many copies there are.

Divide the unit "1" into several parts on average, and the number representing one part is called fractional unit.

2 Classification of scores

True fraction: The fraction with numerator less than denominator is called true fraction. The true score is less than 1.

False fraction: Fractions with numerator greater than denominator or numerator equal to denominator are called false fractions. False score is greater than or equal to 1.

With fraction: False fraction can be written as a number consisting of integer and true fraction, which is usually called with fraction.

3 Reduction and comprehensive score

Changing a fraction into a fraction equal to it, but with smaller numerator and denominator, is called divisor.

The denominator of a molecule is a fraction of a prime number, which is called simplest fraction.

Dividing the scores of different denominators by the scores of the same denominator equals the original score, which is called the total score.

4) Percentage

1 indicates that one number is the percentage of another number, which is called percentage, also called percentage or percentage. Percentages are usually expressed as "%". The percent sign is a symbol indicating percentage.

Operating rule

1. additive commutative law;

When two numbers are added, the positions of addends are exchanged, and their sum is unchanged, that is, A+B = B+A.

2. Additive associative law:

Add three numbers, first add the first two numbers, then add the third number; Or add the last two numbers first, and then add the first number, and their sum remains the same, that is, (a+b)+c=a+(b+c).

3. Multiplicative commutative law;

When two numbers are multiplied, the position of the exchange factor remains unchanged, that is, a× b = b× a.

4. Multiplicative associative law:

Multiply three numbers, first multiply the first two numbers and then multiply the third number; Or multiply the last two numbers first, and then multiply them with the first number, and their products are unchanged, that is, (a×b)×c=a×(b×c).

5. Multiplicative distribution law:

When the sum of two numbers is multiplied by a number, you can multiply the two addends by this number, and then add the two products, that is, (a+b) × c = a× c+b× c.

6. The essence of subtraction:

If you subtract several numbers from a number continuously, you can subtract the sum of all subtractions from this number, and the difference remains unchanged, that is, a-b-c=a-(b+c).