This document discusses proportional relationships between variables. It defines direct and inverse proportionality, where two variables are directly proportional if changing one causes the other to change by the same factor. Variables are inversely proportional if one increases as the other decreases while their product remains constant. Examples are given like distance being directly proportional to time at a constant speed. Properties are described, like the graph of a direct proportional relationship being a straight line through the origin. Other concepts covered include proportionality constants, hyperbolic coordinates, and exponential/logarithmic proportionality.

1. MADE BY : SAKSHI SHRUTI

3. Geometric illustration
 When the duplication of a given rectangle preserves its shape, the ratio of the
large dimension to the small dimension is a constant number in all the copies,
and in the original rectangle. The largest rectangle
of the drawing is similar to one or the other rectangle with stripes. From
their width to their height, the coefficient is A ratio of their dimensions
horizontally written within the image, at the top or the bottom, determines
the common shape of the three similar rectangles.
 The common diagonal of the similar rectangles divides each rectangle into two
superposable triangles, with two different kinds of stripes. The four striped
triangles and the two striped rectangles have a common vertex: the center
of an homothetic transformation with a negative ratio −k or , that transforms
one triangle and its stripes into another triangle with the same stripes, enlarged
or reduced. The duplication scale of a striped triangle is the proportionality
constant between the corresponding sides lengths of the triangles, equal to
a positive ratio obliquely written within the image:
or

4. In the proportion , the terms a and d are called
the extremes, while b and c are the means,
because aand d are the extreme terms
of the list (a, b, c, d), while b and c are in the middle
of the list. From any proportion, we get another
proportion by inverting the extremes or the means.
And the product of the extremes equals the product
of the means. Within the image, a double arrow
indicates two inverted terms
of the first proportion.

5. Symbols
 The mathematical symbol ∝ (U+221D in Unicode)
is used to indicate that two values are
proportional. For example, A ∝ B means the
variable A is directly proportional to the variable B.
 Other symbols include:
 ∷ - U+2237 "PROPORTION"
 ∺ - U+223A "GEOMETRIC PROPORTION"

6. Direct proportionality
 Given two variables x and y, y is directly
proportional to x (x and y vary directly, or x and
y are in direct variation) if there is a non-zero
constant k such that
 The relation is often denoted, using the ∝ symbol,
as
 and the constant ratio
 is called the proportionality
constant or constant of proportionality

7. Examples
 If an object travels at a constant speed, then the distance traveled
is directly proportional to the time spent traveling, with the
speed being the constant of proportionality.
 The circumference of a circle is directly proportional to
its diameter, with the constant of proportionality equal to π.
 On a map drawn to scale, the distance between any two points
on the map is directly proportional to the distance between the
two locations that the points represent, with the constant of
proportionality being the scale of the map.
 The force acting on a certain object due to gravity is directly
proportional to the object's mass; the constant of proportionality
between the mass and the force is known as gravitational
acceleration.

8. Properties
 Since
 is equivalent to
 it follows that if y is directly proportional to x, with
(nonzero) proportionality constant k, then x is also directly
proportional to y with proportionality constant 1/k.
 If y is directly proportional to x, then the graph of y as
a function of x will be a straight line passing through
the origin with the slope of the line equal to the constant of
proportionality: it corresponds to linear growth.

9. Inverse proportionality
 The concept of inverse proportionality can be contrasted
against direct proportionality. Consider two variables said to be
"inversely proportional" to each other. If all other variables are
held constant, the magnitude or absolute value of one inversely
proportional variable will decrease if the other variable increases,
while their product (the constant of proportionality k) is always
the same.
 Formally, two variables are inversely proportional (also
called varying inversely, in inverse variation, in inverse
proportion, in reciprocal proportion) if one of the variables is
directly proportional with the multiplicative Inverse (reciprocal)
of the other, or equivalently if their product is a constant. It
follows that the variable y is inversely proportional to the
variable x if there exists a non-zero constant k such that

10. Hyperbolic coordinates
 The concepts of direct and inverse proportion lead to
the location of points in the Cartesian plane
by hyperbolic coordinates; the two coordinates
correspond to the constant of direct proportionality
that locates a point on a ray and the constant of inverse
proportionality that locates a point on a hyperbola.

11. Exponential and logarithmic proportionality
 A variable y is exponentially proportional to a
variable x, if y is directly proportional to
the exponential function of x, that is if there exist non-
zero constants k and a
 Likewise, a variable y is logarithmically
proportional to a variable x, if y is directly
proportional to the logarithm of x, that is if there exist
non-zero constants k and a.

12. Ratio
 A ratio is a relationship between two numbers of the same
kind (e.g., objects, persons, students, spoonfuls, units of
whatever identical dimension), usually expressed
as "a to b" or a:b, sometimes expressed arithmetically as a
dimensionless quotient of the two that explicitly indicates
how many times the first number contains the second (not
necessarily an integer).

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