First unit position

1, what is a number pair?

Number pair: consists of two numbers, separated by commas and enclosed in parentheses. The numbers in brackets are the number of columns and rows from left to right, that is, "columns first, then rows".

Function: Determine the position of a point. Longitude and latitude are the principles.

Example: In the grid diagram (plane rectangular coordinate system), it is represented by several pairs (3, 5) (third column, fifth row).

Note: (1) In the plane rectangular coordinate system, the coordinates on the X axis represent columns and the coordinates on the Y axis represent rows. For example, the number pair (3, 2) represents the third column and the second row.

(2) Logarithm (x, 5) remains unchanged, indicating a horizontal line, and the number of columns (5, y) remains unchanged, indicating a vertical line. (A number is uncertain and a point cannot be determined)

(column, row)

↓ ↓

Vertical columns are called rows and rows.

(looking from left to right) (looking from bottom to top)

(looking back from front to back)

2. The number of lines translated left and right remains unchanged; The number of columns in the chart that move up and down remains the same.

3. The distance between two points has nothing to do with the choice of reference point (0,0). Different reference points lead to different pairs, but the distance between two points remains the same.

Unit 2 Fractional Multiplication

Importance of (1) fractional multiplication:

1, the fractional multiplication of integers has the same meaning as integer multiplication, and it is a simple operation to find the sum of several identical addends.

Note: "Fraction times integer" means that the second factor must be an integer, not a fraction.

For example, ×7 means: What is the sum of seven? How much is seven times?

2. Multiplying a number by a fraction means finding the fraction of a number.

Note: "A number multiplied by a fraction" means that the second factor must be a fraction, not an integer. The first factor is anything. )

For example, x means: What do you want?

9 × means: What is the number of 9?

A × means: What is the number of A?

(2) Calculation rules of fractional multiplication:

1, the arithmetic of decimal times integer is: numerator times integer, denominator remains unchanged.

Note: (1) In order to simplify the calculation, the score can be reduced first and then calculated. (Integer and denominator divisor)

(2) The divisor is to subtract the greatest common factor from the following integer and denominator. (Integer cannot be multiplied by denominator, and the calculation result must be the simplest fraction).

2. The arithmetic of fractional multiplication is: use the product of molecular multiplication as numerator and the product of denominator multiplication as denominator. (numerator times numerator, denominator times denominator)

Note: (1) If the fractional multiplication formula contains a band score, the band score should be converted into a false score before calculation.

(2) The method of fractional simplification is to divide the numerator and denominator by their greatest common factor at the same time.

(3) In the process of multiplication, the divisor is to cross out two divisible numbers in the numerator and denominator, and then write the divisor above and below respectively. (numerator and denominator must not contain common factors after reduction, so the calculated result is the simplest fraction. )

(4) The basic nature of the fraction: the numerator and denominator are multiplied or divided by the same number (except 0) at the same time, and the size of the fraction remains unchanged.

(3) the relationship between products and elements:

A number (except 0) is multiplied by a number greater than 1, and the product is greater than this number. A×b=c, when b >; At 1, c>a.

A number (except 0) is multiplied by a number less than 1, and the product is less than this number. A×b=c, when b < 1

A number (except 0) is multiplied by a number equal to 1, and the product is equal to this number. A×b=c, and when b = 1, c = a.

Note: When comparing the size of factor and product, we should pay attention to the special situation when the factor is 0.

Attachment: the score of shape can be converted into () ×

(d) mixed operation of fractional multiplication

1, the mixed operation order of fractional multiplication is the same as that of integers. Multiply first, then divide, and then add and subtract. If there are parentheses, count them first and then count them outside.

2. The law of integer multiplication is also applicable to fractional multiplication; Algorithms can make some calculations simple.

Multiplicative commutative law: a×b=b×a

Law of multiplicative association: (a×b)×c=a×(b×c)

Multiplicative distribution law: a× (b c) = a× b a× c

(V) Meaning of reciprocal: Two numbers whose product is 1 are reciprocal.

1 and reciprocal are two numbers, which are interdependent and cannot exist alone. A number cannot be called reciprocal. (It must be clear who is the reciprocal of who)

2. The only criterion to judge whether two numbers are reciprocal is whether the product of the multiplication of two numbers is "1".

For example: a×b= 1, then a and b are reciprocal.

3. Reciprocal method:

① Find the reciprocal of the fraction: exchange the positions of numerator and denominator.

② Find the reciprocal of an integer: 1 of an integer.

③ Find the reciprocal of the score: first turn it into a false score, and then find the reciprocal.

(4) Find the reciprocal of the decimal: first find the number of components, and then find the reciprocal.

4. The reciprocal of1is itself because 1× 1= 1.

0 has no reciprocal, because the product of any number multiplied by 0 is 0, and 0 cannot be used as the denominator.

5. Any number a(a≠0), whose reciprocal is; The reciprocal of non-zero integer a is; The reciprocal of the score is.

6. The reciprocal of the true score is a false score, and the reciprocal of the true score is greater than 1 and also greater than itself.

The reciprocal of the error score is less than or equal to 1.

The reciprocal of the score is less than 1.

(6) Fractional multiplication is applied to solve problems.

1, what is the score of a number? (by multiplication)

" 1"× =

How much is 25? Formula: 25× = 15

The number of A equals the number of B. Given that the number of A is 25, what is the number of B? Formula: 25× = 15

Note: Given the quantity of unit "1", find the fraction of the quantity of unit "1" and multiply it.

2. (what) is (what).

( )= ( " 1" ) ×

Example 1: It is known that the number A is the number B and the number B is 25. What is the number a?

A number = B number × that is, 25× = 15.

Note: (1) The quantity "b" between the word "yes" and the word "de" is the quantity of the unit "1", that is, the number b is regarded as the unit "1", and the number b is divided into five parts on average, and the number a is three of them.

(2) The words "Shi", "Zhan" and "Bi" are all equivalent to "=", while the word "De" is equivalent to "X".

(3) Number of units "1" × score = number corresponding to the score.

Example 2: The number of A is more (less) than that of B, and the number of B is 25. What's the number of a?

A number = B number B × that is, 25 25× = 25× (1) = 40 (or 10).

3. Find the quantity of the unit "1" skillfully: in sentences with scores, the quantity before the score is the corresponding quantity of the unit "1", or the quantity after the words "Zhan", "Yes" and "Bi" is the unit "1".

4. What is speed?

Speed is the distance traveled per unit time. Speed = distance/time/time = distance/speed/distance = speed × time

-Unit time refers to 1 hour, 1 minute, 1 second and other time units with the size of 1, such as minutes, hours and seconds.

5. How much is A more (less) than B?

Duo: (A-B) B.

Minus: (b-a) B.

Unit 3 Fractional Division

First, the significance of fractional division: fractional division is the inverse operation of fractional multiplication. Knowing the product of two numbers and one of the factors, we can find the other factor.

Second, the calculation rules of fractional division: dividing by a number (except 0) is equal to multiplying the reciprocal of this number.

1, dividend/divisor = dividend × the reciprocal of divisor. Example ÷3 =×3÷3×5

2. When division is converted into multiplication, the dividend must not be changed, and it becomes x, and the divisor becomes its reciprocal.

3. When there are decimals and fractions in the fractional division formula, the number of components and false fractions should be changed before calculation.

4. The change law of dividend and quotient:

① Divided by a number greater than 1, the quotient is less than the dividend: a÷b=c When b> is in 1, c

② Divided by a number less than 1, the quotient is greater than the dividend: a÷b=c when b.

③ Divided by a number equal to 1, the quotient equals the dividend: a÷b=c When b= 1, c = a..

Third, the mixed operation of fractional division

1, the mixed operation is calculated by trapezoidal equation, and the equal sign is written in the lower left corner of the first number.

2. Operation sequence:

① Division: it belongs to the same level operation and is calculated from left to right; Or convert all divisions into multiplication before calculation; Or follow the simple method of "dividing by several numbers is equal to multiplying the product of these numbers". Addition and subtraction are primary operations, and multiplication and division are secondary operations.

② Mixed operations: multiplication, division, addition and subtraction without brackets; Parentheses are counted in parentheses first, and then outside parentheses.

Note: (a b) ÷ c = a ÷ c b ÷ c

4. Ratio: The division of two numbers is also called the ratio of two numbers.

1. In the comparison formula, the number before the comparison symbol (:) is called the former item, the item after the comparison symbol is called the latter item, the comparison symbol is equivalent to the division symbol, and the quotient of the former item divided by the latter item is called the ratio.

Note: For example, 3: 4: 5 is pronounced as 3 to 4 to 5.

2. The ratio represents the relationship between two numbers, which can be expressed by a fraction, written in the form of a fraction, and read as several to several.

Example:12: 20 =12 ÷ 20 = = 0.612: 20.

Note: distinguish ratio from ratio: ratio is a number, usually expressed as a fraction, and can also be an integer or a decimal.

A ratio is a formula that represents the relationship between two numbers. It can be written as a ratio or a fraction.

3. The basic nature of the ratio: the first term and the second term of the ratio are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged.

3. Simplified proportion: the simplified result is still a proportion, not a number.

(1), the two terms before and after the ratio are divided by the greatest common divisor at the same time.

(2) The simplified method of the ratio of two fractions is to multiply the last term in the previous paragraph by the least common multiple of the denominator, and then simplify the integer ratio. You can also find the ratio and write it out in the form of ratio.

(3), the ratio of two decimal places, move the position of the decimal point to the right, and also convert it into an integer ratio first.

4. Find the ratio: write the ratio symbol as a division symbol and then calculate it. The result is a number (or fraction), which is equivalent to quotient, not ratio.

5, the difference between ratio and division, fraction:

Division divider division symbol () divider (non-0) quotient invariant division is an operation.

Fraction numerator Fraction line (-) Denominator (cannot be 0) The basic property of a fraction is a number.

The basic attribute ratio of the former item (:) and the latter item (not 0) indicates the relationship between two numbers.

Attachment: the quotient is unchanged: the dividend and divisor are multiplied or divided by the same number at the same time (except 0), and the quotient is unchanged.

The basic nature of a fraction: the numerator and denominator are multiplied or divided by the same number at the same time (except 0), and the size of the fraction remains unchanged.

Five, the application of fractional division and ratio

1, the known unit "1". Example: A is B, B is 25, what is A? Namely: A = B× (15× = 9)

2. The quantity of unknown unit "1" is divided. For example: A is B, A is 15, and what is B? Namely: A = B× (15 ÷ = 25) (it is recommended to solve the equation).

3. The basic quantitative relationship of fractional application problems (by fraction)

(1)A is a fraction of B?

A = B× fraction (Example: A is 15, what is it? 15× =9)

A fraction of B = (Example: 9 is b, what is b? 9÷ = 15)

What score = A-B (Example: 9 is what score of 15? 9÷ 15=) (the word "yes" is equivalent to "÷" and b is the unit "1")

(2) How much is A more (less) than B?

A difference b = (the quantity after the word "than" is the quantity of the unit "1") (for example, how much is 9 less than 15? ( 15-9)÷ 15= = = )

How much is B:–1(Example: 15 is less than 9? 15÷9= - 1= – 1= )

How many scores of C are: 1- (Example: how many scores are 9 less than 15? 1-9÷ 15= 1– = 1– = )

D A = B difference = B B X = B B X = B B( 1) (Example: A is less than 15, what is it? 15–15× =15× (1-) = 9 (mostly "+"and less "-")

E B = A ÷ (1) (Example: 9 is less than b, what is b? 9÷( 1- )=9 ÷ = 15) (mostly "+"and less "-")

(Example: 15 is greater than b, what is b? 15 ÷ (1+) =15 ÷ = 9) (mostly "+"and less "-")

4. Proportional distribution: the method of distributing a quantity according to a certain proportion is called proportional distribution.

For example, it is known that the sum of Party A and Party B is 56 and the ratio of Party A and Party B is 3: 5. What are the numbers of Party A and Party B respectively?

Method 1: 56 ÷ (3+5) = 7a: 3× 7 = 21b: 5× 7 = 35.

Method 2: A: 56× =2 1 B: 56× =35.

For example, it is known that A is 2 1 and the ratio of A to B is 3: 5. How much is b?

Method 1: 2 1÷3=7 B: 5×7=35.

Method 2: The sum of Party A and Party B is 2 1÷ =56 B: 56× =35.

Method 2: A/B = B = A/= 2 1/= 35.

5. Draw a line graph:

(1) Find out the number of the unit "1", first draw the unit "1", and mark the known and unknown.

(2) Analyze the quantitative relationship.

(3) Find the equivalence relation.

(4) Column equation.

Note: Draw two line graphs for the relationship between two quantities, and draw one line graph for the relationship between part and whole.

Unit 4 circle

I. Characteristics of a circle

1, a circle is a plane figure surrounded by closed curves in a plane.

2. Features of the circle: beautiful appearance and easy rolling.

3. center o: the point of the center is called the center. The center of the circle is generally represented by the letter O. After the circle is folded in half for many times, the crease intersects with the center of the circle, that is, the center of the circle. The center of the circle determines the position of the circle.

Radius r: The line segment connecting the center of the circle and any point on the circle is called radius. In the same circle, there are countless radii, all of which are equal. The radius determines the size of the circle.

Diameter d: The line segment whose two ends pass through the center of the circle is called the diameter. The same circle has countless diameters, and all the diameters are equal. The diameter is the longest line segment in a circle.

The inner diameter of the same circle or equal circle is twice the radius: d=2r or r=d÷2= d=

4. Equal circles: circles with equal radii are called concentric circles, and equal circles can be completely overlapped by translation.

Concentric circles: Two circles with coincident centers and unequal radii are called concentric circles.

5. The circle is an axisymmetric figure: if a figure is folded in half along a straight line, the figures on both sides can completely overlap, and this figure is an axisymmetric figure. The straight line where the crease lies is called the symmetry axis.

Figures with symmetry axis: semicircle, sector, isosceles trapezoid, isosceles triangle and angle.

A figure with two axes of symmetry: a rectangle.

A figure with three axes of symmetry: an equilateral triangle

A figure with four axes of symmetry: a square

Figures with or without symmetry axis: circles and rings.

Step 6 draw a circle

(1) The distance between two feet of a compass is the radius of a circle.

(2) Draw a circle: fix the radius, center of the circle and make a circle.

Second, the circumference of the circle: the length of the curve around the circle is called the circumference of the circle, and the circumference is represented by the letter C.

1, the circumference of a circle is always more than three times the diameter.

2. Pi: The ratio of the circumference to the diameter of a circle is a fixed value, which is called Pi and is expressed by the letter π.

Namely: pi = = circumference ÷ diameter ≈3. 14.

Therefore, the circumference of a circle (c)= diameter (d)×π(pi)- circumference formula: c =πd, c = 2π r.

Note: Pi π is an infinite acyclic decimal, and 3. 14 is an approximation.

3. Circumference change law: how many times does the radius expand, how many times does the diameter expand, and the circumference expansion multiple is the same as the radius and diameter expansion multiple.

If r1:R2: R3 = d1:D2: D3 = c1:C2: C3.

4. Perimeter of semicircle = half of circumference+diameter = ×2πr=πr+d

Third, the area of the circle.

1, derivation of the formula of circular area

As shown in the figure, divide a circle into several parts along the diameter and cut it into a rectangle. The more copies, the closer the image is to a rectangle.

Radius of circle = width of rectangle

Half of the circumference = the length of the rectangle.

Rectangular area = length × width

So: area of circle = area of rectangle = length × width = half of circumference (πr)× radius of circle (r).

S circle = πr × r

S circle = πr×r = πr2

2. For several figures, the circumference of a circle is the shortest and that of a rectangle is the longest when the areas are equal; On the contrary, in the case of equal perimeters, the area of a circle is the largest, while the area of a rectangle is the smallest.

At the same time, the circular area is the largest. Taking advantage of this feature, baskets and plates are made into circles.

3. Change law of circular area: How many times does the radius expand and how many times does the circumference expand at the same time? The expansion multiple of circular area is the square of the expansion multiple of radius and diameter.

If: r1:R2: R3 = d1:D2: D3 = c1:C2: C3 = 2: 3: 4.

Then: s 1: S2: S3 = 4: 9: 16.

4. Annular area = great circle-small circle =πr big 2-πr small 2=π(r big 2-r small 2)

Sector area = πr2× (n stands for the degree of the central angle of the sector)

Runways: The perimeter of each runway is equal to the perimeter of the circle formed by two semi-circular runways plus the sum of two straight runways. Because the lengths of two straight runways are the same, the starting lines of two adjacent runways are different, and the distance between them is 2×π× runway width.

Note: For every one centimeter increase in the radius of a circle, the circumference increases by 2π one centimeter.

The diameter of the circle increases by b cm, and the circumference increases by πb cm.

6. The inscribed circle of any square, that is, the diameter of the largest circle is the side length of the square, and their area ratio is 4∶ 1

7. Public data

π=3. 14 2π=6.28 3π=9.42 4π= 12.56 5π= 15.7

Unit 5, Percentage

Meaning of 1. Percentage: indicates that one number is a percentage of another number.

Note: Percent is specially used to express a special ratio relation, which indicates the ratio of two numbers. So percentage is also called percentage or percentage, and percentage can't take units.

1, the difference and connection between percentage and score;

(1) connection: both can be used to express the proportional relationship between two quantities.

(2) Difference: the meaning is different: the percentage only indicates the proportional relationship, not the specific quantity, so it can't take the unit. Fractions not only indicate the proportional relationship, but also express the specific quantity in units.

The numerator of percentage can be decimal, and the numerator of fraction can only be integer.

Note: Percentages are widely used in life, and the problems involved are basically the same as fractions. Fractions with denominator of 100 are not percentages, and the denominator must be written as "%",so it is wrong to say that fractions with denominator of 100 are percentages. The two zeros of "%"should be lowercase, not to be confused with the number before the percentage. Generally speaking, attendance, survival rate, qualified rate and correct rate can reach 100%, rice yield and oil yield can not reach 100%, and the completion rate and percentage increase can exceed 100%. Generally, the powder yield is 70% and 80%, and the oil yield is 30% and 40%.

2. The relationship between decimals, fractions and percentages

(1) Percentalized Decimal: Move the decimal point to the left by two places and delete "%".

(2) Decimal percentage: move the decimal point two places to the right and add "%".

(3) Percentilization score: First, write the percentage as a score with the denominator of 100, and then simplify it to the simplest score.

(4) Fractional percentage: divide the numerator by the denominator to get a decimal, and then convert it into a percentage.

(5) Decimal fraction: Simplify fractions with decimal parts of 10, 100, 1000, etc.

(6) Fractional decimal: numerator divided by denominator.

Second, the percentage of application problems.

1. Find common percentages, such as compliance rate, pass rate, survival rate, germination rate and attendance rate. It means finding the percentage of one number to another.

2. Find out how much one number is more (or less) than another. In real life, people often use how much to increase, how much to decrease and how much to save to express increase or decrease.

What percentage is A more than B (A-B)? B

What percentage is B less than A (A-B)?

3. Find the percentage of a number (unit "1") × percentage.

4. What percentage of a number is known? Find the percentage of this number = a number (unit "1").

5, discount discount, the meaning of discount: a few fold is a few tenths, that is, dozens of percent.

The discount percentage is universal.

20% off, 20% off, 20% off, 0.8%

15% off, 15% off, 15% off, 15% off

50%, 50%, 50%, half price.

6. The tax paid is called tax payable.

(Taxable amount) ÷ (Total income) = (tax rate)

(Taxable amount) = (total income) × (tax rate)

7. Interest rate

(1) The money in the bank is called the principal.

(2) When withdrawing money, the excess money paid by the bank is called interest.

(3) The ratio of interest to principal is called interest rate.

Interest = principal × interest rate× time

After-tax interest = interest-interest tax payable = interest-interest ×5%

Note: Interest on national debt and education savings is not taxed.

8. Classification of percentage application problems

(1) what percentage of b-(a-b) is A × 100% = × 100% = what percentage?

(2) What percentage of A is more (less) than B -× 100% = × 100%.

example

① A is 50, B is 40, and A is what percentage of B? How much is fifty percent of forty? )50÷40= 125%

② A is 50, B is 40, and B is what percentage of A? What percentage of 50 is 40? )40÷50=80%

③ B is 40, A is 125% of B. What is the number of A? What is 40 125%? )40× 125%=50

④ A is 50, B is 80% of A, what is the number of B? What is 80% of 50? )50×80%=40

⑤ B is 40, B is 80% of A, what is the number of A? Eighty percent of a number is 40. What's this number? )40÷80%=50

⑥ A is 50, A is 125% of B, what is the number of B? 125% of a number is 50. What's this number? )50÷ 125%=40

⑦ A is 50, B is 40. What percentage is A more than B? What percentage is over 50 and over 40? )(50-40)÷40× 100%=25%

8 A 50, B 40, what percentage is b less than a? What percentage is 40 less than 50? )(50-40)÷50× 100%=20%

Pet-name ruby A is 25% more than B, more than 10. How much is b? 10÷25%=40

Participating in A is 25% more than B, more than 10. How much is one? 10÷25%+ 10=50

? B is 20% less than A, less 10. How much is one? 10÷20%=50

? B is 20% less than A, less 10. How much is b? 10÷20%- 10=40

? B is 40, and A is 25% more than B. What's the number of A? How much is 25% over 40? )40×( 1+25%)=50

? A is 50, and B is 20% less than A. What's the number of B? How much is 25% over 50? )50×( 1-20%)=40

? B is 40% less than A. What's the number of 20% A? (40 is 20% less than what? )40÷( 1-20%)=50

? A is 50 more than B. What's the number of 25% B? (50 is 25% more than what? )40÷( 1+25%)=40

Unit 6, Statistics

1, the meaning of sector statistical chart: the total number is represented by the area of the whole circle, and the relationship between the number of parts and the total number is represented by the area of each sector in the circle, that is, the percentage of the number of parts to the total number, so it is also called percentage chart.

2. Advantages of commonly used statistical charts:

(1), and the bar chart directly shows the quantity of each quantity.

(2) The statistical chart of broken lines not only shows the increase and decrease of quantity intuitively, but also clearly shows the number of each quantity.

(3) The fan-shaped statistical chart intuitively shows the relationship between part and whole.

Unit 7, Mathematical Wide Angle

First, study the problem that chickens and rabbits were locked together in ancient China.

1, table solution is limited, and the number must be small, such as:

Count the legs of chickens (rabbits).

35 1 34

35 2 33

35 3 32

……

(One-to-one list method, fewer legs and smaller jumps; Multi-legged, big jump Jump one by one, holding the list)

2, using the hypothetical method to solve.

(1) If they are all rabbits.

(2) If they are all chickens

(3) If they each lift one leg.

(4) If the rabbit lifts its two front legs.

3. Solve (general law) by algebraic method.

Note: This question is one of the famous interesting questions in ancient China. About 1500 years ago, this interesting question was recorded in Sun Tzu's calculation. The book describes it like this: "There are chickens and rabbits in the same cage today, with 35 heads on the top and 94 feet on the bottom. The geometry of chicken and rabbit? These four sentences mean: there are several chickens and rabbits in a cage, counting from the top, there are 35 heads; It's 94 feet from the bottom. How many chickens and rabbits are there in each cage?

Second, monks divide steamed bread.

100 monks eat 100 buns, one big monk eats three, and three little monks eat one. How many monks are there?

There is a famous arithmetic problem in Cheng Dawei's masterpiece "Directing at Arithmetic Unity" in Ming Dynasty:

A hundred steamed buns and a hundred monks,

The three monks are even less controversial.

One of the three little monks,

How many monks are there? "

If translated into vernacular, it means: there are 100 monks sharing 100 steamed buns, which is exactly the ending. If the big monk is divided into three parts and the young monk is divided into three parts, how many people are there in each part?

Method one, solve by equation:

Solution: Let the big monk have X people and the little monk have (100-x) people. According to the meaning of the question, the equation is listed:

3x + ( 100-x)= 100

x=25

100-25=75 people

Method two, chickens and rabbits in the same cage:

(1) Suppose 100 people are big monks, how many steamed buns should they eat?

3× 100=300 (pieces).

How much did you eat?

300- 100=200 (pieces).

(3) Why did you eat 200 more? This is because the young monk is regarded as a big monk. So how many steamed buns does each young monk count when he is regarded as a big monk?

3- = (piece)

(4) Each young monk bought 8/3 steamed buns, and one * * * bought 200 steamed buns, so the young monk has:

Little monk: 200÷75 (person)

Big monk: 100-75=25 (person)

Method 3, grouping method:

Because the big monk has three steamed buns and the little monk has three steamed buns. We can group three little monks and 1 big monks, so that four monks in each group are exactly divided into four steamed buns, so that the total number of 100 monks is divided into 100÷(3+ 1)=25 groups, because each group has 1. Because there are three young monks in each group, there are 25×3=75 young monks.

This is the solution in "Command Algorithm Unifies Clans". The original words are: "Take one hundred monks as the truth, divide by three to get four, and get twenty-five great monks." The so-called "real" is "dividend" and "law" is "divisor". The formula is:

100(3+ 1)= 25 (group)

Big monk: 25× 1=25 (person)

Little monk: 100-25=75 (person) or 25×3=75 (person)

This shows the wisdom of the ancient working people in China.

Three, integer, fraction, percentage application problem structure type

(1) How many times (or fractions or percentages) is A greater than B?

Answer: a number divided by b number

There are 40 poplars and 50 willows on campus. What percentage of willows are poplars? (Still a fraction? )

(2) What is the number of A (or fraction or percentage)?

To solve the problem of score application, we must first determine the unit "1". After the unit "1" is determined, a specific quantity always corresponds to a specific fraction (fraction). This relationship is called "quantity-rate correspondence", which is the key to solve the problem of score application.

Find the multiple (fraction or percentage) of a number by multiplication, and the unit is "1"× fraction = corresponding quantity.

Example: There are 180 students in grade six, and the number of students in grade five is 56 students in grade six. How many students are there in Grade Five?

180×56 = 150

(3) How many times a number is known (or fraction or percentage), how to find the application problem of a number (that is, how to find the standard quantity or unit "1").

Solution: corresponding quantity ÷ corresponding score = unit "1"

Example: Boys in Grade 6 of Yuhong Primary School 120, accounting for 35% of the students who participated in the interest activity group. How many students are there in the sixth grade?

120÷35 =200 (person)