Area of triangle = base × height ÷2. The formula S= a×h÷2.
Area of square = side length × side length formula S= a2
Area of rectangle = length× width Formula S= a×b
Area of parallelogram = base× height Formula S= a×h
Trapezoidal area = (upper bottom+lower bottom) × height ÷2 Formula S=(a+b)h÷2
Sum of internal angles: sum of internal angles of triangle = 180 degrees.
The surface area of a cuboid = (length× width+length× height+width× height )× 2 Formula: S=(a×b+a×c+b×c)×2.
Surface area of cube = side length × side length ×6 Formula: S=6a2.
Cuboid volume = length× width× height formula: V = abh
Volume of cuboid (or cube) = bottom area × height formula: V = abh.
Volume of cube = side length × side length × side length formula: V = a3.
Circumference = diameter × π formula: L = π d = 2π r
Area of circle = radius × radius× π formula: s = π R2.
Surface (side) area of cylinder: The surface (side) area of cylinder is equal to the perimeter of bottom multiplied by height. Formula: s = ch = π DH = 2π RH.
Surface area of cylinder: the surface area of cylinder is equal to the perimeter of the bottom multiplied by the height plus the area of the circles at both ends. Formula: S=ch+2s=ch+2πr2.
Volume of cylinder: the volume of cylinder is equal to the bottom area multiplied by the height. Formula: V=Sh
Volume of cone = 1/3 bottom× product height. Formula: V= 1/3Sh
arithmetic
1, additive commutative law: Two numbers are added to exchange the position of addend, and the sum is unchanged.
2. Additive associative law: A+B = B+A.
3. Multiplicative commutative law: a× b = b× a.
4. Multiplicative associative law: a × b × c = a ×(b × c)
5. Multiplicative distribution law: a× b+a× c = a× b+c.
6. The nature of division: a ÷ b ÷ c = a ÷(b × c)
7. Nature of division: In division, the dividend and divisor are expanded (or reduced) by the same multiple at the same time, and the quotient remains unchanged. O is divided by any number that is not O. Simple multiplication: the multiplicand and the end of the multiplier are multiplied by O. You can multiply 1 before o first, and zero does not participate in the operation, and add a few zeros at the end of the product.
8. Division with remainder: dividend = quotient × divisor+remainder
Equations, Algebras and Equality
Equation: An equation in which the value on the left of the equal sign equals the value on the right of the equal sign is called an equation. Basic properties of the equation: When both sides of the equation are multiplied (or divided) by the same number at the same time, the equation is still valid.
Equation: An equation with an unknown number is called an equation.
One-dimensional linear equation: An equation with an unknown number of degree 1 is called a one-dimensional linear equation. Example method and calculation of learning linear equation of one variable. That is, an example is given to illustrate that the formula is replaced by χ and calculated.
Algebra: Algebra means replacing numbers with letters.
Algebraic expression: Expressions expressed by letters are called algebraic expressions. For example 3x = AB+C.
mark
Fraction: divide the unit "1" into several parts on average, and the number representing such a part or points is called a fraction.
Comparison of fraction size: Compared with the fraction of denominator, the numerator is large and the numerator is small. Compare the scores of different denominators, divide them first and then compare them; If the numerator is the same, the denominator is big and small.
Addition and subtraction of fractions: add and subtract fractions with the same denominator, only add and subtract numerators, and the denominator remains the same. Fractions of different denominators are added and subtracted, first divided, then added and subtracted.
Fraction multiplied by integer, numerator is the product of fractional and integer multiplication, denominator remains unchanged.
Fractions are multiplied by fractions, the product of numerator multiplication is numerator, and the product of denominator multiplication is denominator.
Law of fractional addition and subtraction: Fractions with the same denominator are added and subtracted, only the numerator is added and subtracted, and the denominator remains the same. Fractions of different denominators are added and subtracted, first divided, then added and subtracted.
The concept of reciprocal: 1 If the product of two numbers is 1, we call one of them the reciprocal of the other. These two numbers are reciprocal. The reciprocal of 1 is 1, and 0 has no reciprocal.
A fraction divided by an integer (except 0) is equal to this fraction multiplied by the reciprocal of this integer.
The basic properties of a fraction: the numerator and denominator of a fraction are multiplied or divided by the same number (except 0), and the size of the fraction.
The law of division of fractions: dividing by a number (except 0) is equal to multiplying the reciprocal of this number.
True fraction: The fraction with numerator less than denominator is called true fraction.
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 a score: write a false score as an integer, and a true score is called with a score.
The basic nature of the fraction: the numerator and denominator of the fraction are multiplied or divided by the same number (except 0) at the same time, and the size of the fraction remains unchanged.
Calculation formula of quantitative relationship
Unit price × quantity = total price 2, single output × quantity = total output
Speed × time = distance 4, work efficiency × time = total workload.
Appendix+Appendix = and one addend = and+another addend.
Negative-negative = differential negative = negative-differential negative = negative+difference.
Factor × factor = product One factor = product ÷ another factor.
Frequency divider/frequency divider = frequency divider = frequency divider/frequency divider = quotient × frequency divider
Length unit:
1 km = 1 km 1 km = 1000 m
1 m = 10 decimeter 1 decimeter =10 cm1cm =10 mm.
Area unit:
1 km2 = 1 00ha1hectare =10000m2
1 m2 = 100 square decimeter 1 square decimeter = 100 square centimeter 1 square centimeter = 100 square millimeter
1 mu = 666.666 square meters.
volume unit
1 m3 = 1000 cubic decimeter
1 cm3 = 1000 cm3
1 liter = 1 cubic decimeter = 1000 ml 1 ml = 1 cubic centimeter.
Unit right
1 ton = 1 000kg1kg = 1 000g = 1 kg =1kg.
compare
What is the ratio? When two numbers are divided, it is called the ratio of two numbers. For example, the first and second terms of the ratio of 2÷5 or 3:6 or 1/3 are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged.
What is proportion? Two formulas with equal ratios are called proportions. For example, 3: 6 = 9: 18
The basic property of proportion: in proportion, the product of two external terms is equal to the product of two internal terms.
Solution ratio: the unknown term in the proportion is called solution ratio. Such as 3: χ = 9: 18.
Proportion: two related quantities, one of which changes and the other changes. If the ratio (i.e. quotient k) corresponding to these two quantities is constant, these two quantities are called proportional quantities, and the relationship between them is called proportional relationship. For example: y/x=k( k must be) or kx = y.
Inverse proportion: two related quantities, one of which changes and the other changes accordingly. If the product of the corresponding two numbers in these two quantities is certain, these two quantities are called inverse proportional quantities, and their relationship is called inverse proportional relationship. For example: x×y = k( k must be) or k/x = y.
per cent
Percentage: a number that indicates that one number is a percentage of another number, which is called percentage. Percentages are also called percentages or percentages.
To convert decimals into percentages, just move the decimal point two places to the right and add hundreds of semicolons at the end. In fact, to convert a decimal into a percentage, just multiply this decimal by 100%. To convert percentages to decimals, simply remove the percent sign and move the decimal point two places to the left.
When a fraction is converted into a percentage, the fraction is generally converted into a decimal (three decimal places are generally reserved when it is not used up), and then the decimal is converted into a percentage. In fact, to turn a fraction into a percentage, you must first turn the fraction into a decimal and then multiply it by 100%.
Divide the percentage into components, and rewrite the percentage into components first, so that the quotation that can be lowered can be made into the simplest score.
We should learn to decompose fractions into components and fractions into decimals.
Multiplication and divisor
Maximum common divisor: The common divisor of several numbers is called the common divisor of these numbers. There is a finite common factor. The largest one is called the greatest common divisor of these numbers.
Least common multiple: The common multiple of several numbers is called the common multiple of these numbers. There are infinite common multiples. The smallest one is called the least common multiple of these numbers.
Prime number: the common divisor has only 1 two numbers, which is called prime number. Two adjacent numbers must be prime numbers. Two consecutive odd numbers must be coprime. 1 and any number coprime.
Comprehensive score: the difference between scores of different denominators is changed into the same denominator score equal to the original score, which is called comprehensive score. (Common divisor is the least common multiple)
Decrement: divide the numerator and denominator of a fraction by the common divisor at the same time, and the fraction value remains unchanged. This process is called dropping points.
Simplest fraction: The numerator and denominator are fractions of prime numbers, which are called simplest fraction. At the end of the score calculation, the score must be converted into the simplest score.
Prime number (prime number): If a number only has 1 and its two divisors, it is called a prime number (or prime number).
Composite number: a number. If there are other divisors besides 1 and itself, such numbers are called composite numbers. 1 is neither prime nor composite.
Prime factor: If a prime number is a factor of a certain number, then this prime number is the prime factor of this number.
Prime factor decomposition: A composite number is represented by the complementary way of prime factors, which is called prime factor decomposition.
Multiple characteristics:
Characteristics of multiples of 2: You are 0, 2, 4, 6, 8.
Characteristics of multiples of 3 (or 9): The sum of the numbers on each digit is multiples of 3 (or 9).
Characteristics of multiples of 5: You are 0, 5.
Characteristics of multiples of 4 (or 25): The last two digits are multiples of 4 (or 25).
Characteristics of multiples of 8 (or 125): the last three digits are multiples of 8 (or 125).
Characteristics of multiples of 7 (1 1 or 13): the difference (big-small) between the last three digits and other digits is a multiple of 7 (1 1 3).
Characteristics of multiples of 17 (or 59): the difference (big-small) between the last three digits and the rest digits is a multiple of 17 (or 59).
Characteristics of multiples of 19 (or 53): the difference (big-small) between the last three digits and other seven digits is a multiple of 19 (or 53).
Characteristics of multiples of 23 (or 29): the difference (big-small) between the last four digits and the other five digits is multiples of 23 (or 29).
Of the two numbers in the multiple relation, the greatest common divisor is smaller and the smallest common multiple is larger.
The coprime relation between two numbers, the greatest common divisor is 1, and the least common multiple is the product.
When two numbers are divided by their greatest common divisor, the quotient is coprime.
The product of two numbers and the least common multiple is equal to the product of these two numbers.
The common divisor of two numbers must be the greatest common divisor of these two numbers.
1 is neither prime nor composite.
A prime number greater than 3 divided by 6 must get 1 or 5.
Odd and even numbers
Even numbers: Numbers are numbers of 0, 2, 4, 6 and 8.
Odd number: The number is not 0, 2, 4, 6 or 8.
Even even = even Qiqi = Qiqi.
Even numbers add up to even numbers, and odd numbers add up to odd numbers.
Even × even = even × odd = odd × even = even.
The sum of two adjacent natural numbers is odd, and the product of adjacent natural numbers is even.
If one number in the multiplication is even, then the product must be even.
Odd ≠ even number
separable
If c | a, c | b, then c | (a b)
If, then b | a, c | a
If b | a, c | a and (b, c)= 1, then BC | a.
If c | b, b | a, then c | a
decimal
Natural number: an integer used to represent the number of objects, called natural number. 0 is also a natural number.
Pure Decimal: Decimal in units of 0.
With Decimal: Decimal with more than 0 digits.
Cyclic decimal: a decimal, starting from a certain bit of the decimal part, and one or several numbers are repeated in turn. Such decimals are called cyclic decimals. Like 3. 14 14 14.
Acyclic decimal: a decimal, starting from the decimal part, without one number or several numbers appearing repeatedly. Such a decimal is called acyclic decimal. Like 3. 14 1592654.
Infinite cycle decimal: a decimal, from the decimal part to the infinite digits, and one or several numbers are repeated in turn. Such decimals are called infinite cyclic decimals. For example, 3. 14 14 14 ...
Infinite acyclic decimal: a decimal, from decimal part to infinite digits, is called infinite acyclic decimal without one number or several numbers appearing repeatedly. Such as 3. 14 1592654. ...
profit
Interest = principal × interest rate × time (time is usually in years or months, which should correspond to the unit of interest rate).
Interest rate: The ratio of interest to principal is called interest rate. The ratio of interest to principal for one year is called annual interest rate. The ratio of interest to principal in January is called monthly interest rate.
Distance = speed × time; Distance ÷ time = speed; Distance/speed = time
Edit the key questions in this paragraph.
Determine the location, distance, meeting distance, speed sum = meeting time, meeting time = speed and meeting time × speed sum = meeting distance.
Meeting questions (straight line)
Distance of a+distance of b = total distance.
Encounter a problem (ring)
Distance of a+distance of b = circumference of the ring.
Edit this paragraph to keep up with the question.
Catch-up time = distance difference/speed difference/speed difference = distance difference/catch-up time × speed difference = distance difference
Catch up with the problem (straight line)
Distance difference = chaser distance-chased distance = speed difference x chasing time
Follow-up question (ring)
Fast Distance-Slow Distance = perimeter of the curve
Edit this paragraph of running water problem
Downstream range = (ship speed+current speed) × downstream time = (ship speed-current speed )× downstream time = ship speed+current speed = still water speed = (downstream speed+current speed) ÷2 Current speed: (downstream speed-current speed) ÷2 Ship.
Edit this paragraph to solve the problem.
When a ship sails in a river, it is pushed or propelled by running water besides its own speed. In this case, calculating the speed, time and distance of the ship is called the problem of sailing in running water. Running water problem is one of the travel problems, so the relationship between the three quantities (speed, time and distance) in the travel problem will be used repeatedly here. In addition, there are two basic formulas for running water: downstream speed = ship speed+water speed, and (1) countercurrent speed = ship speed-water speed. (2) The ship speed here is based on the reciprocal operation relationship of addition and subtraction, which can be obtained from the formula (1): current speed = downstream speed-ship speed, and ship speed = downstream speed-current speed. Formula (2) shows that current speed = ship speed-upstream speed, and ship speed = upstream speed+current speed. That is to say, as long as we know any two of the three quantities: the speed of the ship in still water, the actual speed of the ship and the current speed, we can find the third quantity. In addition, given the current speed and the current speed of the ship, according to formula (1) and formula (2), we can add and subtract them: ship speed = (current speed+current speed) ÷2, water speed = (current speed-current speed) ÷2.
Engineering problem formula
(1) general formula:
Efficiency × working hours = total workload; Total workload ÷ working time = working efficiency; Total amount of work ÷ efficiency = working hours.
Work efficiency × working hours = total workload ÷ work efficiency = working hours.
Total workload ÷ working time = working efficiency
(2) Assuming that the total workload is "1", the formula for solving engineering problems is:
1÷ working time = the fraction of the total amount of work completed in unit time;
1What is the score that can be completed per unit time = working time.
(Note: If the hypothetical method is used to solve the engineering problem, you can arbitrarily assume that the total workload is 2, 3, 4, 5 ... Especially if the total workload is the least common multiple of several working hours, the fractional engineering problem can be transformed into a relatively simple integer engineering problem, and the calculation will become simpler. )
1, number of copies × number of copies = total number of copies/number of copies = total number of copies/number of copies = number of copies.
Total number ÷ Total number of copies = average value
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. Appendix+Appendix = sum, and-one addend = another addend.
6. Minus-Minus = Minus-Minus = Minus+Minus = Minus
7, factor × factor = product product ÷ one factor = another factor
Dividend = quotient dividend = divisor quotient × divisor = dividend
Mathematical graphic 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=a2
2. Cube: side length of volume A.
Surface area = side length × side length× 6s Table =a×a×6=6a2
Volume = side length × side length× side length V=a×a×a=a3
3. Rectangle: C circumference, S area and A side length.
Circumference = (length+width) ×2 C=2(a+b)
Area = length × width S=ab
Five, the main points of problem solving
(A) the interpretation of nouns
1. statistical table: statistical data and its indicators are listed in tabular form, which is called statistical table. A narrow statistical table only represents statistical indicators.
2. Statistical chart: A statistical chart is a geometric representation of statistical indicators, that is, it visually represents the quantitative relationship between things in the form of the position of points, the rise and fall of line segments, the length of straight lines or the size of areas.
(2) Short answer questions
1. statistical table can replace the lengthy text description, which is convenient for the calculation, analysis and comparison of indicators. Whether its formulation is reasonable or not has an important influence on the quality of statistical analysis.
Statistical charts can intuitively reflect and analyze the quantitative relationship between things through the position of points, the rise and fall of line segments, the length of straight lines and the size of areas. Because statistics is a rough representation of quantity, it is best to attach corresponding statistical tables.
Generally speaking, a statistical table consists of four parts: title, subtitle, line and number (sometimes with remarks).
Matters needing attention in compiling statistical tables:
(1) The title summarizes the contents of the table and is written at the top of the table, usually indicating the time and place.
(2) Headings use horizontal headings and vertical headings to represent the subject and predicate respectively, with concise words and distinct levels.
(3) There should not be too many lines, which are generally represented by three and a half lines, namely, the top line, the bottom line, the horizontal line under the vertical line sign and the half line in the total.
(4) All tables use Arabic numerals. The decimal places of the same indicator should be consistent and aligned many times. Don't leave spaces in the table.
(5) Notes should not be listed in the table. If necessary, you can mark "*" in the table and explain it outside the table.
3. Statistical chart usually consists of four parts: title, course, scale and legend.
Matters needing attention in drawing statistical charts:
(1) According to the nature of the data and the purpose of analysis, select the appropriate graph.
(2) The title should briefly explain the content, place and time of the map, which is located at the bottom of the map and generally needs to indicate the time and place.
(3) The statistical chart has a vertical axis and a horizontal axis, and the two axes should have titles, and the titles should indicate the units. The vertical axis is from bottom to top, and the horizontal axis is from left to right. Numbers are always from small to large, and some numbers require the vertical axis scale to start from 0.
(4) The aspect ratio of graphics (except circular graphics) is generally around 7:5 or 5:7, which is more beautiful.
(5) When comparing different things, you can use different lines or colors to represent them, but you need to use legends to explain them. Legends are generally placed in the upper right corner of the graph or in an appropriate position below the graph.
Draw a semi-logarithmic line chart with the horizontal axis as the arithmetic scale and the vertical axis as the logarithmic scale. Indicate the dynamic change trend of the ratio between quantities, such as the ratio A/B, let X=A/B, and use the logarithmic algorithm, LGX = LGA-LGB, that is, the ratio of scales on the vertical axis is expressed by the difference of logarithmic values, so it reflects the change of the development speed of A and B phenomena.