| ap + cq |
| bp + dq |
lies between the two last, it also lies between the two first; that is, if p and q be any numbers or fractions whatsoever,
| ap + cq |
| bp + dq |
lies between a/b and c/d.
201. By the last article we may often form some notion of the value of an expression too complicated to be easily calculated. Thus,
| 1 + x | lies between | 1 | and | x | , or 1 and | 1 | ; |
| 1 + xx | 1 | xx | x | ||||
| ax + by | lies between | ax | and | by | , | ||
| axx + bbyy | axx | bbyy | |||||
that is, between 1/x and 1/by. And it has been shewn that (a + b)/2 lies between a and b, the denominator being considered as 1 + 1.
202. It may also be proved that a fraction such as
| a + b + c + d |
| p + q + r + s |
always lies among