Ratios and Proportions

Proportionality

Proportionality
Two related variable quantities x and y whose ratio is constant are called proportional or directly proportional, written y : x = m.
That is,   y = m · x ,   where m (m > 0) is called the constant of proportionality, related quantities x and y are proportional if their corresponding values are a constant multiple of each other.
Or, by whatever ratio one quantity changes, the other changes in the same ratio.
It is also called linear relationship.
Graphic representation of proportionality
If x and y are proportional quantities, i.e.,  y : x = m  then, their corresponding values x1, y1 and x2, y2 are also proportional alternately
x1 : x2 = y1 : y2   that is,  x1 is to xas is  yto  y2 .
Given four related variable quantities are Cartesian-coordinates of the points P1 and P2 of a straight line shown below: Thus,

Proportion practice problems, use of the rule of three
Linear relationship of two variable quantities whose ratio is constant, show the proportions
x1 : y1 = x2 : y2      or      x1 : x2 = y1 : y2
we use to solve practice problems.
Two quantities are in direct proportion, or in linear relationship, when they increase or decrease in the same ratio. Or, by whatever ratio one quantity changes, the other changes in the same ratio.
If any three terms in a proportion are given, the fourth may be found using the principles of solving equations.
Example:   A lift reaches the third floor in seven seconds. When will reach the 18th floor?
Given quantities make the proportion
 x1 : y1 = x2 : y2, that is, 3 : 7 = 18 : x => seconds.
We could obtain the same result by setting up the corresponding values of the proportional physical quantities in a table.
Values of the same physical quantity (the same units) put in the first colon and the corresponding values of another quantity in the second colon. The arrows show the sequence of the terms of the proportion.
 x1 y1 x2 y2 or x1 : x2 = y1 : y2
Thus, for the values from the above example follows:
 3rd floor 7 sec. 18th floor x sec. or 3 : 18 = 7 : x
 => seconds.
Example:   To the central angle a = 30°, of a circle of radius r, corresponds the arc length equal p/6 r
What arc length corresponds to the central angle a = 320°?
Beginning Algebra Contents B