1. 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.

2. The calculation rule of fractional multiplication by integer: fractional multiplication by integer, the product of fractional numerator multiplied by integer is numerator, and the denominator remains unchanged.

3. A number multiplied by a fraction can be regarded as finding a fraction of this number.

4. Calculation rules of fractional multiplication: fractional multiplication, the product of molecular multiplication is numerator, and the product of denominator multiplication is denominator.

5. The commutative law, associative law and distributive law of integer multiplication are also applicable to fractional multiplication.

6. Two numbers whose product is 1 are reciprocal.

7. To find the reciprocal of a number (except 0), just switch the numerator and denominator of this number. The reciprocal of 1 is 1. 0 has no reciprocal.

The reciprocal of the true score is greater than1; The reciprocal of the false score is less than or equal to1; The reciprocal of the score is less than 1.

Note: the reciprocal must be a pair of two numbers, and a single number cannot be called reciprocal.

8. When a number (except 0) is multiplied by a true fraction, the product is less than itself.

9. Multiply a number (except 0) by a false fraction, and the product is equal to or greater than itself.

10. A number (except 0) times a fraction, and the product is greater than itself.

1 1. General steps to solve fractional application problems.

(1) Find out the key sentences with scores.

(2) Find out the quantity of the unit "1" (hereinafter referred to as "standard quantity")

(3) Draw a line graph. Standard quantity and comparison quantity are the whole and part relationship. Just draw a line segment. Standard quantity and comparison quantity are not the whole and part relationship. Just draw two lines.

(4) Write the equivalence relation according to the line segment diagram: standard quantity × corresponding score = comparison quantity.

(5) According to the known conditions and problems.

12. The concept of attention in multiplication application problems.

(1) Ideas for solving multiplication application problems: Given a number, what is the score of this number? Unit "1"× corresponding score = corresponding quantity.

(2) The method of finding the unit "1": find it from the key sentences with scores, and pay attention to the rules before "de" and after "yes, ratio, equivalence, occupation and equality".

(3) A score greater than B means A score greater than B, and B score less than A means B score less than A.

decimal

1. Meaning of fractional division: The meaning of fractional division is the same as that of integer division, and it is an operation to find the other factor by knowing the product of two factors and one of them.

2. Fraction divided by integer (except 0) is equal to fraction multiplied by the reciprocal of the integer. An integer divided by a fraction equals an integer multiplied by the reciprocal of the fraction.

3. Calculation rule of number divided by fraction: number divided by fraction equals the reciprocal of number multiplied by fraction.

4. Calculation rule of fractional division: A divided by B (except 0) equals the reciprocal of A multiplied by B..

Division of two numbers is also called the ratio of two numbers. The quotient obtained by dividing the former term by the latter term is called the ratio. From the application point of view, the ratio can be divided into homogeneous ratio and heterogeneous ratio; The ratio of the same kind indicates the multiple relationship, and the front and back items of the ratio must be in the same unit; The results of different category ratios produce new quantities, and the units of the former and the latter are different.

6. Ratios are usually expressed in fractions, decimals and integers.

7. The last item of the ratio cannot be 0.

8. Compared with division, the former term of ratio is equivalent to dividend, the latter term is equivalent to divisor, and the ratio is equivalent to quotient;

9. According to the relationship between fraction and division, the former term of ratio is equivalent to numerator, the latter term is equivalent to denominator, and the ratio is equivalent to the value of fraction.

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

1 1. In industrial and agricultural production and daily life, it is often necessary to allocate a quantity according to a certain proportion. This method is usually called proportional distribution.

12. When a number (except 0) is divided by a true fraction, the quotient is greater than itself.

13. When a number (except 0) is divided by a false fraction, the quotient is less than or equal to itself.

14. When a number (except 0) is divided by a band fraction, the quotient is less than itself.

Know what the score of a number is, find this number and calculate it by division; Corresponding quantity ÷ corresponding score = unit "1"

Basic arithmetic

1. The order of fractional elementary arithmetic is the same as that of integer elementary arithmetic. In the calculation of primary operation and secondary operation, the secondary operation should be calculated first and then the primary operation, that is, multiplication and division first and then addition and subtraction. In the operation at the same level, it should be calculated from left to right.

2. In fractional elementary arithmetic, arithmetic can be applied to make the calculation simple.

The algorithm includes: the commutative law of addition, the associative law of addition, the commutative law of multiplication, the associative law of multiplication and the distributive law of multiplication.

per cent

1. Definition of percentage: A number indicating that one number is a percentage of another number is called a percentage. Percentages are also called percentages or percentages.

Percent indicates the proportional relationship between two numbers, not the specific quantity, and there is no unit name.

2. Meaning of percentage: It means that one number is the percentage of another number. For example, 25% means that one number is 25% of another.

3. Percentages are usually not written in fractional form, but expressed by adding "%"after the original molecule. The molecular part can be a decimal or an integer, which can be greater than 100, less than 100 or equal to 100.

4. Rules for decimal and percentage exchange:

To convert decimals into percentages, just move the decimal point two places to the right, followed by hundreds of semicolons;

To convert percentages to decimals, simply remove the percent sign and move the decimal point two places to the left.

5. Reciprocity rule of percentage and score:

When a fraction is converted into a percentage, it is generally converted into a decimal (except for three decimal places) and then converted into a percentage;

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.

14. deposit types: deposits are divided into demand, lump-sum deposit and withdrawal, and lump-sum deposit and withdrawal.

15. Principal: Money in the bank is called principal.

16. Interest: The excess money paid by the bank when withdrawing money is called interest.

17. According to national regulations, deposit interest should be taxed at a certain rate. There is no tax in debt interest.

18. interest rate: the ratio of interest to principal is called interest rate.

19. Calculation formula of after-tax interest of bank deposits: after-tax interest = principal × interest rate × time ×( 1- tax rate)

circle

1. Definition of a circle: a curve graph on a plane.

2. Fold a circular piece of paper twice, and the point where the crease intersects the center of the circle is called the center of the circle.

The center of the circle is generally represented by the letter O, and its distance to any point on the circle is equal.

3. Radius: The line segment connecting the center of the circle and any point on the circle is called radius.

The radius is generally represented by the letter R. If the two feet of a compass are separated, the distance between the two feet is the radius of the circle.

4. The center of the circle determines the position of the circle and the radius determines the size of the circle.

5. Diameter: The line segment whose two ends pass through the center of the circle is called diameter. The diameter is usually indicated by the letter d.

6. In the same circle, all radii are equal and all diameters are equal.

7. The same circle has countless radii and countless diameters.

8. The diameter of the same circle is twice the radius, and the radius is half the diameter. Expressed in letters: d = 2r or r =

9. Circumference: The length of the curve around a circle is called circumference.

1 1. The circumference formula of a circle: C= πd or c = 2 π r.

12. Area of the circle: The area occupied by the circle is called the area of the circle.

13. Divide the circle evenly into several parts, and then cut them, and you can make an approximately rectangular figure. The length of this rectangle is equivalent to half the circumference (=πr), and the width of the rectangle is equivalent to the radius (r) of the circle, so the area of the rectangle is equal to the area of the circle, so the area of the circle is πr×r=πr2.

18. Ring circumference = outer circumference+inner circumference

19. The circumference of a semicircle is equal to half the circumference plus the diameter. The perimeter formula of a semicircle: c = π d ÷ 2+d or c = π r+2r.

20. area of semicircle = area of circle ÷2 The formula is: s = π R2 ÷ 2.

2 1. The radius of the same circle is expanded or reduced by several times, and the diameter and circumference are also expanded or reduced by the same multiple; The area has expanded or shrunk several times.

For example, the radius, diameter and circumference of the same circle are enlarged by four times, and the area is enlarged by 16 times.

22. The radius ratio of two circles is equal to the diameter ratio and the circumference ratio, and the area ratio is equal to the square of the above ratio.

For example, if the radius ratio of two circles is 2: 3, then the diameter ratio and perimeter ratio of these two circles are both 2: 3, while the area ratio is 22: 32 = 4: 9.

23. The radius of the circle increases by a, and the circumference increases by 2π a; The diameter of the circle increases by a, and the circumference increases by π a.

24. In the same circle, the central angle accounts for a fraction of the central angle, and its sector area accounts for a fraction of the circular area; What fraction of a circle does an arc face?

25. The areas of triangles, parallelograms, rectangles, squares and circles with equal perimeters increase in turn.

The perimeters of triangles, parallelograms, rectangles, squares and circles with equal areas decrease in turn.

26. Sector arc length formula: l = π d ÷ 360× n Sector area formula: S= πr2÷360×n (n is the degree of the central angle of the sector).

27. Axisymmetric figure: If a figure is folded in half along a straight line and the figures on both sides can completely overlap, it is an axisymmetric figure. Where the crease is located

This straight line is called the axis of symmetry.