<P> As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors . It defines three terms p, q and r to be in proportion when p: q:: q: r . This is extended to 4 terms p, q, r and s as p: q:: q: r:: r: s, and so on . Sequences that have the property that the ratios of consecutive terms are equal are called geometric progressions . Definitions 9 and 10 apply this, saying that if p, q and r are in proportion then p: r is the duplicate ratio of p: q and if p, q, r and s are in proportion then p:s is the triplicate ratio of p: q . If p, q and r are in proportion then q is called a mean proportional to (or the geometric mean of) p and r . Similarly, if p, q, r and s are in proportion then q and r are called two mean proportionals to p and s . </P> <P> In general, a comparison of the quantities of a two - entity ratio can be expressed as a fraction derived from the ratio . For example, in a ratio of 2: 3, the amount, size, volume, or quantity of the first entity is 2 3 (\ displaystyle (\ tfrac (2) (3))) that of the second entity . </P> <P> If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2: 3, and the ratio of oranges to the total number of pieces of fruit is 2: 5 . These ratios can also be expressed in fraction form: there are 2 / 3 as many oranges as apples, and 2 / 5 of the pieces of fruit are oranges . If orange juice concentrate is to be diluted with water in the ratio 1: 4, then one part of concentrate is mixed with four parts of water, giving five parts total; the amount of orange juice concentrate is 1 / 4 the amount of water, while the amount of orange juice concentrate is 1 / 5 of the total liquid . In both ratios and fractions, it is important to be clear what is being compared to what, and beginners often make mistakes for this reason . </P> <P> Fractions can also be inferred from ratios with more than two entities; however, a ratio with more than two entities cannot be completely converted into a single fraction, because a fraction can only compare two quantities . A separate fraction can be used to compare the quantities of any two of the entities covered by the ratio: for example, from a ratio of 2: 3: 7 we can infer that the quantity of the second entity is 3 7 (\ displaystyle (\ tfrac (3) (7))) that of the third entity . </P>

What is the ratio of 60 to 40