Chapter 24: Variation

Variation is an abbreviated method of expressing certain functional relationships. When we say one quantity varies as another, we are defining a specific proportional connection between them that remains constant even as the values change.

24.1 Direct Variation

One quantity is said to vary directly as another when their ratio remains constant. If y varies directly as x, we write y ∝ x. This is expressed as the equation:

y = kx

The letter k is called the constant of variation.

Example: If y varies directly as x, and y = 10 when x = 2, find y when x = 6.

Step 1: Find k.
10 = k(2) → k = 5.

Step 2: Use the constant to find the new value.
y = 5(6) = 30.

Result: y = 30.

24.2 Inverse Variation

One quantity varies inversely as another when it varies as the reciprocal of that quantity. If y varies inversely as x, we write y ∝ 1/x, or:

y = k/x

24.3 Joint Variation

A quantity varies jointly as two or more other quantities when it varies as their product. If y varies jointly as x and z, then:

y = kxz

The Combined Law: In many physical problems, multiple types of variation occur at once. For example, the force of gravity (F) varies jointly as the masses (m¹, m²) and inversely as the square of the distance (r²):

F = k(m¹m²)/r²

Doctor's Pro-Tip: Variation is the language of science. Whether you are calculating the strength of a radio signal over a distance or the pressure of a gas in a cylinder, you are using the principles found in this chapter.