4-inch by 6-inch what is the width of 15 inches for a ratio table

This is a ratio problem; the ratio of the length to width is constant (and therefore equal):

4 / 6 = 15 / x

Now, with a ratio, we may do any allowable algebra operation: cross-multiply, invert both sides, multiply or divide both sides by the same amount, etc.

Let's cross-multiply:

4x = (15) (6)

x = 90/4

x = 22.5 in.