1, multiplication property: if a: b = c: d, then a/c = b/d.

The multiplication property shows that when two ratios are equal, two ratios in the ratio can be multiplied to get equal results.

2. reciprocity: if a: b = c: d, the reciprocal of a: b is b: a, and the reciprocal of c: d is d: C.

Reciprocity shows that when two ratios are equal, two ratios in the ratio can get equal results in turn.

3. Fractional property: If A: B = C: D, then a/(a+b)=c/(c+d).

Fractional properties show that when two ratios are equal, two ratios in the ratio can be divided by their sum to get equal results.

4. Sum and difference properties: If A: B = C: D, then (A+C): (B+D) = A: B = C: D.

The properties of sum and difference show that when two ratios are equal, two ratios in the ratio can be added (or subtracted) to get equal results.

5. Recursive property: If a: b = b: c, then A: B = C: D. ..

The recursive property shows that when two ratios in two proportions are equal, the two ratios in the proportion can be equal in turn and get equal results.

6. Parallel: If a: b = c: d, then a: b and c: d are parallel.

Parallelism means that when two scales are equal, two proportional line segments can be parallel.

7. The nature of the positive-negative ratio: If A: B = C: D, A: B is inversely proportional to D: C. ..

The nature of the positive-negative ratio shows that when the two ratios are equal, the inverse relationship between the two ratios can be obtained.

8. Length ratio property: If A: B = C: D, then the length ratios of a/c and b/d are equal.

The properties of length ratio show that when the two ratios are equal, it can be obtained that the length ratios of the two ratios are equal.

9. Distribution property: If A: B = C: D, then A/(A+C) = B/(B+D) = A: B = C: D.

The distribution property shows that when two proportions are equal, two proportions in the proportion can be divided by their sum to get equal results, and the equal relationship between the two proportions can also be obtained.

Methods of learning mathematics

1. Understand the basic concepts: You must understand the basic concepts and principles of mathematics, not rote learning. Applying concepts to practical problems can deepen understanding.

2. Practice solving problems: Mathematics is a subject that needs to be practiced. Do more problems, keep practicing, and improve the proficiency and speed of solving problems.

3. Make a study plan: define the study goals and plans and allocate time reasonably. For your own weaknesses, you should carry out extra exercises in a targeted manner.