Positive proportion, or proportional for short, refers to two related quantities. One quantity changes, and the other changes with it.If the ratio (or quotient) of the two corresponding quantities is fixed, the two quantities are called proportional quantities, and their relationship is called proportional relationship.[1]
Chinese name
Positive proportion
Foreign name
proportional
Overview
One quantity changes, and the other also changes
Relational expression
Y/x=k (certain) (k ≠ 0)
Related contacts
The similarities are transformed into each other
Example
The perimeter and side length of a square (ratio 4)
）The relationship between the two variables that meet the relationship y=k · x (k is a certain quantity) is said to be in direct proportion。
Obviously, if y is in direct proportion to x, then y/x=k (k isconstant）And vice versa.
For example, in the travel problem, if the speed is fixed, the distance is proportional to the time;In engineering problems, if the work efficiency is certain, the total amount of work is in direct proportion to the working time.
Note: k cannot be equal to 0.[2]
Related contacts
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The relationship with reverse proportion is as follows:
Similarities
1. There are two variables in the relationship between things, one is quantitative.
2. In the two variables, when one variable changes, the other variable also changes.
3. The product or quotient of the corresponding two variables is certain.
Mutual transformation
When the x value (the value of the independent variable) in the inverse scale is also converted to its reciprocal, the inverse scale is converted to the positive scale;When the value of x in the positive proportion (the value of the independent variable) is converted to its reciprocal, the positive proportion is converted to the negative proportion.
give an example
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(1) The perimeter and side length of a square (ratio: 4).
(2) Circumference and diameter of the same circle (ratio: π).
(3) The total price of purchase and the quantity of purchase (ratio: unit price).
(4) With a certain speed, the distance is proportional to the time;Time is fixed, distance is proportional to speed.
Solution: in aX=Y, if a is unchanged, then X and YProportional。One variable changes with another variable.
(5) Is the circumference of a circle proportional to the radius?Why?
Solution: Because the circumference of a circle divided by the radius of a circle=2 π, the circumference of a circle is in direct proportion to the radius.
(6) Error prone question: area of circle (S): radius (R)=π R
Solution: This ratio is wrong. It does not belong to the positive ratio.Because (S: R=π R) according to the above, the ratio must be a constant quantity, and the front and rear terms of the ratio must be changeable quantities. If R changes, the ratio will also change, so the area of the circle is not in direct proportion to the radius.
(7) Error prone question: area of circle (S): π=R · R (certain)
Solution: This is a wrong proportion, because the ratio is a constant quantity, and the former and the latter should change with the change of one. Here, the ratio is a fixed quantity, and π is also a fixed quantity, and the former cannot change, so the ratio becomes a fixed proportion, which does not conform to the above that the former and the latter must be changeable quantities.
(8) Error prone question: in the area and side length of a square, S: A=A
Solution: It can be seen from the above that the ratio is a variable, which cannot be the same as any item of the ratio, so the ratio is not positive.
But if the area of a circle (S): (R · R) (R squared)=π, it can be regarded as a positive proportion, which is S in direct proportion to (R · R).
↑ One kind of quantity
(9) Common error: The perimeter of a rectangle is fixed, and the length is in direct proportion to the width.
The specific proof can be derived from the formula: 2 (a+b)=C, a+b=C/2 (certain). It is found that the sum of length and width is certain, but the product is not certain, so the length and width are not in direct proportion.It can also be calculated with specific data. For example, when the perimeter is 10 cm, the product of length 9 width 1 is 9, the product of length 8 width 2 is 16, the product of length 7 width 3 is 21, and the product of length 6 width 4 is 24. It is found that the product is not necessarily, so it is not in direct proportion.
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→ A quantity
The positive scale image is on a ray passing through the origin.
It refers to the development from the intersection of abscissa and ordinate of the statistical table along the diagonal from the lower left corner to the upper right corner, extending out of the table. In the sense of positive proportion, it can extend downward, so it is considered as a straight line.
Application examples
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For example, if the maximum speed of a car is X km/h and it takes Y hours to go to place A, the time required to go to place B can be calculated by using the positive ratio.
