5. The multiplier which turns nines into sevens.
6. The ratio of 7 to 9, or the proportion of 7 to 9.
7. The multiplier which alters a number in the ratio of 9 to 7.
8. The 4th proportional to 9, 1, and 7.
The first two views are in the text. The third is deduced thus: If we divide 9 into 9 equal parts, each is 1, and 7 of the parts are 7; consequently the fraction which 7 is of 9 is ⁷/₉. The fourth view follows immediately: For a time is only a word used to express one of the repetitions which take place in multiplication, and we allow ourselves, by an easy extension of language, to speak of a portion of a number as being that number taken a part of a time. The fifth view is nothing more than a change of words: A number reduced to ⁷/₉ of its amount has every 9 converted into a 7, and any fraction of a 9 which may remain over into the corresponding fraction of 7. This is completely proved when we prove the equation ⁷/₉ of a = 7 times a/9. The sixth, seventh, and eighth views are illustrated in the chapter on proportion.
When the student comes to algebra, he will find that, in all the applications of that science, fractions such as a/b most frequently require that a and b should be themselves supposed to be fractions. It is, therefore, of importance that he should learn to accommodate his views of a fraction to this more complicated case.
| Suppose we take | 2½ | . |
| 4³/₅ |
We shall find that we have, in this case, a better idea of the views from and after the third inclusive, than of the first and second, which are certainly the most simple ways of conceiving ⁷/₉. We have no notion of the (4³/₅)th part of 2½,
| nor of 2 | 1 | ( | 4 | 3 | ) |
| 2 | 5 |
of a unit; indeed, we coin a new species of adjective when we talk of the (4³/₅)th part of anything. But we can readily imagine that 2½ is some fraction of 4³/₅; that the first is some part of a time the second; that there must be some multiplier which turns every 4³/₅ in a number into 2½; and so on. Let us now see whether we can invent a distinct mode of applying the first and second views to such a compound fraction as the above.