Constants, Variables, and Direct Variation

Constants and variables

The word quantity denotes anything which is measurable. In this sense distance, rate, time, etc., are quantities. Many operations and problems in mathematics deal with numerical measures of quantities, some of which are fixed and others constantly changing.

An abstract number, or the numerical measure of a fixed quantity, is called a constant.

Thus the abstract numbers 1, 3, and -\frac57 are constants. Any definite quantities, as the area of a square whose side is 2, the circumference of a circle divided by its diameter (3.1416 nearly), the time of one revolution of the earth on its axis (23^{\text{h}}\,56^{\text{m}}\,4.09^{\text{s}}), and the velocity of light through space (186{,}330 miles per second), are constants.

The numerical measure of a changing quantity is called a variable.

For example, the distance between a passenger on a moving car and a point on the track either ahead of or behind him is a variable, decreasing in the first instance, increasing in the second. Other examples of variables are one's weight, the height of the mercury in the thermometer, and the distance to the sun.

The equation x = 3y may refer to no physical quantities whatever, yet it is possible to imagine y as taking on in succession every possible numerical value, and the value of x as accompanying every change, and consequently always being three times as great as the corresponding value of y. In this sense, which is strictly mathematical, x and y are variables.

Problems in variation deal with at least two variables so related that any change in one is accompanied by a change in the other. Frequently one variable depends on several others: the number of lines of printing on a page depends on the distance between the lines, the size of the type, and one dimension of the page.

The symbol for variation is \propto, and x \propto y is read x varies directly as y, or x varies as y.

Direct variation

One hundred feet of copper wire of a certain size weighs 32 pounds. Obviously a piece of the same kind 200 feet long would weigh 64 pounds; a piece 300 feet long would weigh 96 pounds, and so on.

Here we have two variables W (weight) and L (length) so related that the value of W depends on the value of L, and in such a way that W increases proportionally as L increases. That is, W is directly proportional, or merely proportional, to L. Hence, if W_1 and W_2 are any two weights corresponding to the lengths L_1 and L_2 respectively,

W_1 : W_2 = L_1 : L_2. \qquad (1)

In the form of a variation (1) becomes W \propto L.

In general, if x \propto y, and x and y denote any corresponding values of the variables, and x_1 and y_1 a particular pair of corresponding values,

\frac{x}{x_1} = \frac{y}{y_1}, \qquad\text{so}\qquad x = \left(\frac{x_1}{y_1}\right)y.

But \dfrac{x_1}{y_1} is a constant, being the quotient of two definite numbers. Call this constant K; then

x = Ky.

That is, if one variable varies as a second, the first equals the second multiplied by a constant. Thus for the copper wire just mentioned, W = \frac{32}{100}L, or \frac{8}{25}L. Here, though W varies as L varies, W is always equal to L multiplied by the constant \frac{8}{25}.

The phrase varies with is often incorrectly used in place of varies as. The latter should be used to denote a proportional change in one variable with respect to a second; the former should not be so used. A boy's height varies with his age, but does not vary as his age. At 3 years the average boy is about 3 feet tall; at 12 years he is about 5 feet. If his height varied as his age from 3 years up to 12 years, he would be 12 feet tall.

Example. If x \propto y, and x = 4 when y is 6, find x when y is 8.

The variation is direct. Therefore \dfrac{x_1}{x_2} = \dfrac{y_1}{y_2}. Substituting, \dfrac{4}{x_2} = \dfrac68. Solving, x_2 = 5\frac13.

(Equally, K = \frac46 = \frac23, so x = \frac23 y, and when y is 8, x = \frac{16}{3}.)