Where a is called the former ratio and b is called the latter ratio. The quotient of a÷b is called the ratio of A ∶ B. 。

Among the meanings and properties of cognitive ratio, the significance of cognitive ratio is the key. In the practice of associative meaning of ratio, the basic properties of ratio are obtained. The core of cognitive ratio is to summarize the definition of ratio.

The definition of generalization rate is divided into three steps:

The first step is to use the existing knowledge to solve the example. For example,

1 The truck has a load capacity of 5 tons and the pickup truck has a load capacity of 2 tons.

(1) How many times is the load capacity of large trucks more than that of small trucks?

(2) What is the load capacity of a small truck?

There are 25 boys and 20 girls in a class.

① How many times are there more boys than girls?

(2) What percentage of girls are boys?

The second step is to convert Example 1 and Example 2 into ratios.

Example 1① The load ratio of large trucks to small trucks is 5: 2, which is recorded as 5: 2.

(2) The ratio of the load capacity of small trucks to large trucks is 2: 5, which is recorded as 2: 5.

The ratio of the number of boys to the number of girls is 25: 20, which is 25: 20 = 4: 5.

② The ratio of the number of girls to the number of boys is 20: 25, which is 20: 25 = 4: 5.

The third step is to summarize the definition of ratio in the practice of comparing the first step and the second step:

The division of two identical quantities A and B is called the ratio of A and B..

Understand the meaning of ratio:

(1) Significance of analysis ratio

② Understanding of defining elements.

A÷b is called a∶b, which means that the ratio belongs to another form of "division" and mainly represents the relationship between two numbers.

The division of two similar quantities refers to the division of numbers with the same unit name and the division of two numbers without unit name. If the dividend and divisor are divided into different quantities, it can also be called ratio as long as the division relationship between the two numbers is studied.

Divide the same amount. Find the number of copies in the relationship between total number and number of copies. Find the multiple in the multiple relationship; Divide different kinds of quantities, find each part in the relationship between the total number and the number of copies, and double it in the relationship between multiples.

Students learn by consulting "small materials" provided by textbooks:

In A: B, A is called the former term of the ratio, and B is called the latter term of the ratio (the latter term of the ratio cannot be 0).

The quotient obtained by dividing the former term of the ratio by the latter term of the ratio is called the ratio.

According to the definition of ratio, carry out association exercises:

According to the understanding of the definition of contrast, the definition of ratio is extended to: division of two numbers, also known as the ratio of two numbers.

② The relationship among ratio, fraction and division.

The difference between ratio, division and fraction, ratio is considered by comparing the relationship between two numbers (quantities), except that it is an operation, and fraction represents a number.

③ According to the relationship between ratio and fraction (or division), the basic properties of ratio are obtained:

The size of this value remains the same.

The first and last items of the ratio are multiplied or divided by the same number (except zero), and the ratio remains unchanged.

At the same time, several other properties of the ratio are derived from the division equation and the "change of quotient":

According to "dividend = divisor × quotient":

The former term of the ratio = the latter term of the ratio × the ratio.

According to "divisor = dividend quotient", the latter ratio = the former ratio.

According to "the dividend is expanded (or reduced) several times, but the divisor remains unchanged, and the quotient is also expanded (or reduced) by the same times", it is concluded that "the former term of the ratio is expanded (or reduced) several times, and the latter term of the ratio remains unchanged, and the ratio is also expanded (or reduced) by the same times. That is, if a: b = q, then (a× m): b.

According to "the dividend is constant, the divisor is enlarged (or reduced) several times, and the quotient is reduced (or expanded) by the same times." The conclusion is that if the former term remains unchanged and the latter term is enlarged (or reduced) several times, the ratio will be reduced (or expanded) by the same multiple. That is, if a: b = q, then a:(b×m)= 1

According to dividend > divisor, quotient > 1. Dividend = divisor, quotient = 1. Divider b, then q > 1. On the other hand, if q < 1, then a < b；; If q= 1, then a = b；; If q > 1, then a > b.

④ According to the definition of ratio, write the method of calculating ratio.

The former term of the ratio and the latter term of the ratio = ratio.

⑤ Simplify the proportion according to its basic properties.

Ratio, divided from the range of component quantity, is divided into the following three forms:

Integer ratio: The first term and the last term of the ratio are integers, which is called integer ratio.

Decimal ratio: the ratio of the first and last items to decimals, or a decimal and an integer, is called decimal ratio.

Fractional ratio: the ratio in which the first and second terms of a ratio are both fractions, or one is a fraction and the other is an integer is called fractional ratio.

The number of projects is divided into:

Single ratio, the ratio of two quantities, is called single ratio. For example, 2: 3.

Even ratio, the ratio of three or more numbers, is called even ratio. Even ratio is not even division. For example, A: B: C means that the ratio of A to B is A: B, and the ratio of B is B: C. 。

Ratio simplification refers to simplifying the front and back terms of a ratio to the simplest integer ratio.

The simplest ratio is the ratio of prime numbers, called the simplest ratio.

Simplified method of ratio:

(1) integer ratio: divide the front and back terms of the ratio by their greatest common divisor (or common divisor) until it becomes the simplest ratio.

② Decimal ratio: First rewrite the decimal ratio into an integer ratio, and then simplify it by simplifying the integer ratio.

③ Fractional ratio: First rewrite the fractional ratio into an integer ratio, and then simplify it by simplifying the integer ratio.

Compare simplified ratio with calculated ratio.