SECTION VIII. ON THE PROPORTION OF NUMBERS.

199. When the fraction a/b is not equal to c/d, but greater, a is said to have to b a greater ratio than c has to d; and when a/b is less than c/d, a is said to have to b a less ratio than c has to d. We propose the following questions as exercises, since they follow very simply from this definition.

I. If a be greater than b, and c less than or equal to d, a will have a greater ratio to b than c has to d.

II. If a be less than b, and c greater than or equal to d, a has a less ratio to b than c has to d.

III. If a be to b as c is to d, and if a have a greater ratio to b than c has to x, d is less than x; and if a have a less ratio to b than c to x, d is greater than x.

IV. a has to b a greater ratio than ax to bx + y, and a less ratio than ax to bx- y.

200. If a have to b a greater ratio than c has to d, a + c has to b + d a less ratio than a has to b, but a greater ratio than c has to d; or, in other words, if a/b be the greater of the two fractions a/b and c/d,

a + c

b + d

will be greater than c/d, but less than a/b. To shew this, observe that (mx + ny)/(m + n) must lie between x and y, if x and y be unequal: for if x be the less of the two, it is certainly greater than

mx + nx

m + n

or than x; and if y be the greater of the two, it is certainly less than

my + ny

m + n

or than y. It therefore lies between x and y. Now let a/b be x, and let c/d be y: then a = bx, c = dy. Now

bx + dy

b + d

is something between x and y, as was just proved; therefore

a + c

b + d

is something between a/b and c/d. Again, since a/b and c/d are respectively equal to ap/bp and cq/dq, and since, as has just been proved,

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

a	,	b	,	c	, and	d	,
p		q		r		s

that is, is less than the greatest of them, and greater than the least. Let these fractions be arranged in order of magnitude; that is, let a/p be greater than b/q, b/q be greater than c/r, and c/r greater than d/s. Then by (200)

	is less than				and greater than
a + b		a				b	and	c
p + q		p				q	and		r
a + b + c		a + b	and	a		c	and	d
p + q + r		p + q	and	p		r	and	s
a + b + c + d		a + b + c	and	a		d
p + q + r + s		p + q + r	and	p		s

whence the proposition is evident.

203. It is usual to signify “a is greater than b” by a > b and “a is less than b” by a < b; the opening of V being turned towards the greater quantity. The pupil is recommended to make himself familiar with these signs.