From the beginnings of the human race, we’ve long compared one quantity with another, a comparison that is called a ratio in mathematics. “Their tribe has twice as many cattle as ours” or “Two baskets of wheat cost 12 ducats” are examples of ratios that ring from distant times. Indeed, the concept of a ratio cannot be assigned to any one individual or class of individual. In his *History of Mathematics*, D. E. Smith writes:

It is rather profitless to speculate as to the domain in which the concept of ratio first appeared. The idea that one tribe is twice as large as another and the idea that one leather strap is only half as long as another both involve the notion of ratio; both are such as would develop early in the history of the race, and yet one has to do with ratio of numbers and the other with the ratio of geometric magnitudes. Indeed, when we come to the Greek writers we find Nicomachus including ratio in his arithmetic, Eudoxus in his geometry, and Theon of Smyrna in his chapter on music.

Examples and applications of ratios are limitless: speed is a ratio that compares changes in distance with respect to time, acceleration is a ratio that compares changes in speed with respect to time, and percentages compare the part with the whole. We’ve already studied one classic ratio, the ratio of the circumference of a circle to its diameter, which gives us the definition of \(π\).

One of the most famous ratios in history involves the division of a line segment *AB* into two segments *AC* and *CB* by selecting a point *C* on the segment *AB*.

The idea is to select a point *C* on the segment *AB* so that

\[ \frac{AB}{AC} = \frac{AC}{CB}.\nonumber \]

This ratio has a special name, the **Golden Ratio**, and has an exact value equal to \((1+ \sqrt{5})/2\). The Golden Ratio has been known since the time of Euclid. Ancient and modern architects have long held that the most pleasing rectangular shape is the one whose ratio of length to width is equal to the Golden Ratio.

The comparison of two ratios, such as *AB*/*AC* = *AC*/*CB*, is called a *proportion*. Proportions are used in a number of practical ways. For example, if 5 cans of tomato sauce cost 2 dollars, we can find the number of cans that can be purchased with 10 dollars by comparing two ratios in a proportion:

\[ \frac{ \text{5 cans of tomato sauce}}{\text{2 dollars}} = \frac{ \text{x cans of tomato sauce}}{\text{10 dollars}}\nonumber \]

Any discussion of ratio involves comparing two quantities, so the units of each quantity become extremely important. Two different systems of units are used when measuring length, capacity, and time: the *American* system of units and the *metric* system of units. In this chapter we will discuss both systems and explain how to convert quantities measured in one system to quantities measured in the other system.

Let’s begin the journey.