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the simplest type of functional relation. Quantities may be directly or inversely proportional.
Two variable quantities are said to be directly proportional, or proportional, if their ratio is constant—that is, if the multiplication or division of one of the quantities by some number means that the other is multiplied or divided by the same number. If the quantities x and y are directly proportional, this is described analytically by the equation y = kx, where k is called the proportionality factor, or proportionality constant. The relation between x and y is represented graphically by the line or half line passing through the origin whose slope is equal to the proportionality factor.
Two variable quantities x and y are said to be inversely proportional if one of them is proportional to the inverse of the other—that is,
or xy = k. An equilateral hyperbola (or one of its branches) is the graph of such a functional relation.
Proportional relations are encountered very often. For example, the distance S traveled by a body in uniform motion is proportional to the time t: S = kt, where k is the speed of the body. The weight P of a homogeneous body is proportional to the body’s volume v: P = kv, where k is the body’s specific weight. The time required for a given quantity of soil to be excavated is inversely proportional to the amount excavated per man-hour.