Fractional multiplication 1. Fractional multiplication of integers, like integer multiplication, is a simple operation to find the sum of several identical addends.
2. The calculation rule of multiplying a fraction by an integer: the product of multiplying a fraction by an integer is a numerator, and the denominator remains unchanged. In order to simplify the calculation, what can be reduced must be reduced first and then multiplied. )
Note: When multiplying with a fraction, the fraction should be converted into a false fraction before calculation.
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. (In order to simplify the calculation, you can divide points first and then multiply them. ) Note: When multiplying with a fraction, the fraction should be converted into a false fraction before calculation.
5. The commutative law, associative law and distributive law of integer multiplication are also applicable to fractional multiplication.
Multiplicative commutative law: a×b=b×a
Law of multiplicative association: (a×b)×c=a×(b×c)
Multiplication and distribution law: (a+b) × c = AC+bcac+BC = (a+b) × c.
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 the unit "1" (hereinafter referred to as "standard quantity") and the unit "1"before the rate in the rate sentence; Or "yes", "occupation", "proportion" and "equivalent"
(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.
Find several times a number: a number × several times;
Find the fraction of a number: a number × a few.
The meaning of reciprocal knowledge: 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.
Percent knowledge points 1. Meaning of percentage: The number that indicates that one number is the percentage of another number is called percentage. Percentages are also called percentages or percentages, and percentages cannot have units.
Note: Percent is specially used to express the special ratio relation, which indicates the ratio of two numbers.
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. A fraction with denominator of 100 is not a percentage, and the denominator must be written as "%",so it is wrong to say that a fraction with denominator of 100 is a percentage. 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, attendance rate, etc. Is to find 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 of A is greater than B: (a-b) B.
How much is B less than A: (a-b) A.
3. Find the percentage of a number. A number (unit "1") × percentage.
4. What percentage of a number is known? Find this number.
Partial quantity ÷ percentage = a number (unit "1")
5, discount, discount means: a few fold is a few tenths, that is, dozens of percent.
Discount, percentage = fraction, percentage, decimal
20% discount = 20% discount = 8/ 10 = 20% discount =0.8
15% discount = 15% discount = 15% discount or more = 15% discount =0.85.
50% off = 50% off = 5/ 10 = 50% off =0.5= half price.
6. 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.
7. Classification of percentage application problems
What percentage of (1)B is A-(A-B) × 100%= what percentage?
(2) What percentage of A is greater than B-(a-b) ÷ b×100%.
(3) How much is A less than B-(B-A) ÷ B × 100%?
The above are some related materials of mathematical knowledge points, I hope to help you.