Ratio and Proportion

The design of mathematical investigations, is to arrive at the knowledge of particular quantities, by comparing them with other quantities, either equal to, or greater or less than those which are the objects of inquiry. The end is most commonly attained by means of a series of equations and proportions. When we make use of equations, we determine the quantity sought, by discovering its equality with some other quantity or quantities already known.

We have frequent occasion, however, to compare the unknown quantity with others which are not equal to it, but either greater or less. Here a different mode of proceeding hecomes necessary. We may inquire, either how much one of the quantities is greater than the other; or how many times the one contains the other. In finding the answer to either of these inquiries, we discover what is termed a ratio of the two quantities. One is called arithmetical and the other geometrical ratio. It should be observed, however, that both these terms have been adopted arbitrarily, merely for distinction's sake. Arithmetical ratio, and geometrical ratio are both of them applicable to arithmetic, and both to geometry.

As the whole of the extensive and important subject of proportion depends upon ratios, it is necessary that these should be clearly and fully understood.

338. Arithmetical ratio is the difference between two quantities or sets of quantities. The quantities themselves are called the terms of the ratio, that is, the terms between which the ratio exists. Thus 2 is the arithmetical ratio of 5 to 3. This is expressed, by placing minus between the quantities thus, 5 - 3. Indeed the term arithmetical ratio, and its notation by points, are almost needless. For the one is only a substitute for the word difference and the other for the sign -.

339. If both the terms of an arithmetical ratio be multiplied or divided by the same quantity, the ratio will, in effect, be multiplied or divided by that quantity.
Thus if      a - b = r
Then mult, both sides by A, (Ax. 3.) ha - hb = hr
And dividing by A, (Ax. 4.) $\frac{a}{h}-\frac{b}{h}=\frac{r}{h}$

340. If the terms of one arithmetical ratio be added to, or subtracted from, the corresponding terms of another, the ratio of their sum or difference will be equal to the sum or difference of the two ratios.
If a - b
And d - h,
are the two ratios,
Then (a + d) - (b + h) = (a - b) + (d - h). Foreach = a + d - b - h.
And (a - d) - (b - h) = (a - b) - (d - h). For each = a - d - b + h.
      Thus the arith. ratio of 11 - 4 is 7
      And the arith. ratio of 5 - 2 is 3
The ratio of the sum of the terms 16 - 6 is 10, the sum of the ratios.
The ratio of the difference of the terms 6 - 2 is 4, the difference of the ratios.

341. Geometrical ration is that relation between quantities which is extressed by the QUOTIENT if the one divided by the other.
Thus the ratio of 8 to 4, is 8/4 or 2. For this is the quotient of 8 divided by 4. In other words, it shows how often 4 is contained in 8.

In the same manner, the ratio of any quantity to another may be expressed by dividing the former by the latter, or, which is the same thing, making the former the numerator of a fraction, and the latter the denominator.
            Thus the ratio of a to b is $\frac{a}{b}$
            The ratio of d + h to b + c, is $\frac{d+h}{b+c}$.

342. Geometrical ratio is also expressed by placing two points, one over the other, between the quantities compared,
Thus a:b expresses the ratio of a to b; and 12:4 the ratio of 12 to 4. The two quantities together are called a couplet, of which the first term is the antecedent, and the last, the consequent.

343. This notation by points, and the other in the form of a fraction, may be exchanged the one for the other, as convenience may require; observing to make the antecedent of the couplet, the numerator of the fraction, and the consequent the denominator.
Thus 10:5 is the same as $\frac{10}{5}$ and b:d, the same as $\frac{b}{d}$.

344. Of these three, the antecedent, the consequent, and the ratio, any two being given, the other may be found.

Let a= the antecedent, c= the consequent, r= the ratio.
By definition $r=\frac{a}{c}$; that is, the ratio is equal to the antecedent divided by the consequent.
Multiplying by c, a = cr, that is, the antecedent is equal to the consequent multiplied into the ratio.
Dividing by $r$, $c=\frac{a}{r}$, that is, the consequent is equal to the antecedent divided by the ratio.

Cor. 1. If two couplets have their antecedents equal, and their consequents equal, their ratios must be equal.

Cor. 2. If, in two couplets, the ratios are equal, and the antecedents equal, the consequents are equal; and if the ratios are equal and the consequents equal, the antecedents are equal.

345. If the two quantities compared are equal, the ratio is a unit, or a ratio of equality. The ratio of 3.6:18 is a unit, for the quotient of any quantity divided by itself is 1.

If the antecedent of a couplet is greater than the consequent, the ratio is greater than a unit. For if a dividend is greater than its divisor, the quotient is greater than a unit. Thus the ratio of 18:6 is 3. This is called a ratio of greater inequality.

On the other hand, if the antecedent is less than the consequent, the ratio is less than a unit, and is called a ratio of less inequality. Thus the ratio of 2:3, is less than a unit, because the dividend is less than the divisor.

346. Inverse or reciprocal ratio is the ratio of the reciprocals of two quantities.
Thus the reciprocal ratio of 6 to 3, is ⅙ to ⅓, that is ⅙:⅓.
The direct ratio of $a$ to $b$ is $\frac{a}{b}$, that is, the antecedent divided by the consequent.
The reciprocal ratio is $\frac{1}{a}:\frac{1}{b}$ or $\frac{1}{a}.\frac{b}{1}=\frac{b}{a}$.
that ie the consequent b divided by the antecedent a.

Hence a reciprocal ratio is expressed by inverting the fraction which expresses the direct ratio; or when the notation is by points, by inverting the order of the terms.
Thus a is to b, inversely, as b to a.

347. Compound ratio is the ratio of the products, of the correspondind terms of two or more simple ratios.
            Thus the ratio of          6:3, is 2
            And the ratio of           12:4, is 3
The ratio compounded of these is          72:12 = 6.

Here the compound ratio is obtained by multiplying together the two antecedents, and also the two consequents, of the simple ratios.
So the ratio compounded,
      Of the ratio of         a:b
      And the ratio of            c:d
      And the ratio of            h:y
      Is the ratio of         $ach:bdy=\frac{ach}{bdy}$.
Compound ratio is not different in its nature from any other ratio. The term is used, to denote the origin of the ratio, in particular cases.

Cor. The compound ratio is equal to the product of the simple ratios.
      The ratio of         $a:b$, is $\frac{a}{b}$
      The ratio of         $c:d$, is $\frac{c}{d}$
      The ratio of         $h:y$, is $\frac{h}{y}$
And the ratio compounded of these is ach/bdy, which is the product of the fractions expressing the simple ratios.

348. If, in a series of ratios, the consequent of each preceding couplet, is the antecedent of the following one, the ratio of the first antecedent to the last consequent, is equal to that which is compounded of all the intervening ratios.
Thus, in the series of ratios
            a:b
            b:c
            c:d
            d:h
the ratio of a:h is equal to that which is compounded of the ratios of a:b, of b:c, of c:d, of d:h. For the compound ratio by the last article is $\frac{abcd}{bcdh}=\frac{a}{h}$, or a:h.

In the same manner, all the quantities which are both antecedents and consequents will disappear when the fractional product is reduced to its lowest terms, and will leave the compound ratio to be expressed by the first antecedent and the last consequent.

349. A particular class of compound ratios is produced, by multiplying a simple ratio into itself or into another equal ratio. These are termed duplicate, triplicate, quadruplicate, etc. according to the number of multiplications.

A ratio compounded of two equal ratios, that is, the square of the simple ratio, is called a duplicate ratio.

One compounded of three, that is, the cube of the simple ratio, is called triplicate, etc.

In a similar manner, the ratio of the square roots of two quantities, is called a subduplicate ratio; that of the cube roots a subtripicate ratio, etc.
      Thus the simple ratio of a to b, is a:b
      The duplicate ratio of a to b, is a:b2
      The triplicate ratio of a to b, is a3:b3
      The subduplicate ratio of a to b, is √a:√b
      The subtriplicate of a to b, is 3a:3b, etc.
      The terms duplicate, triplicate, etc. ought not to be confounded with double, triple, etc.
The ratio of 6 to 2 is      6:2 = 3
Double this ratio, that is, twice the ratio, is   12:2 = 6
Triple the ratio, i.e. three times the ratio, is  18:2 = 9
But the duplicate ratio, i.e.the square of the ratio, is 62:22 = 9
And the triplicate ratio,i.e. the cube of the ratio, is 63:23 = 27

350. That quantities may have a ratio to each other, it is necessary that they should be so far of the same nature, as that one can properly be said to be either equal to, or greater, or less than the other. A foot has a ratio to an inch, for one is twelve times as great as the other. But it cannot be said that an hour is either shorter or longer than a rod; or that an acre is greater or less than a degree. Still if these quantities are expressed by numbers, there may be a ratio between the numbers. There is a ratio between the number of minutes in an hour, and the number of rods in a mile.

351. Having attended to the nature of ratios, we have next to consider in what manner they will be affected, by varying one or both of the terms between which the comparison is made. It must be kept in mind that, when a direct ratio is expressed by a fraction, the antecedent of the couplet is always the numerator, and the consequent the denominator. It will be easy, then, to derive from the properties of fractions, the changes produced in ratios by variations in the quantities compared. For the ratio of the two quantities is the same as the value of the fractions, each being the quotient of the numerator divided by the denominator. (Art. 341.) Now it has been shown, that multiplying the numerator of a fraction by any quantity, is multiplying the value by that quantity; and that dividing the numerator is dividing the value. Hence,

352. Multiplying the antecedent of a couplet by any quantity, is multiplying the ratio by that quantity; and dividing the antecedent is dividing the ratio.
            Thus the ratio of      6:2 is 3
            And the ratio of       24:2 is 12.
Here the antecedent and the ratio, in the last couplet, are each four times as great as in the first.
The ratio of $a:b$ is      $\frac{a}{b}$
And the ratio of $na:b$ is $\frac{na}{b}$.

Cor. With a given consequent, the greater the antecedent, the greater the ratio; and on the other hand, the greater the ratio, the greater the antecedent.

353. Multiplying the consequent of a couplet by any quantity is, in effect, dividing the ratio by that quantity; and dividing the consequent is multiplying the ratio. For multiplying the denominator of a fraction, is dividing the value ; and dividing the denominator is multiplying the value.
      Thus the ratio of 12:2, is 6
      And the ratio of 12:4, is 3.
Here the consequent in the second couplet, is twice as great, and the ratio only half as great, as in the first.
            The ratio of $a:b$ is $\frac{a}{b}$
      And the ratio of $a:nb$, is $\frac{a}{nb}$.

Cor. With a given antecedent, the greater the consequent, the less the ratio; and the greater the ratio, the less the con* sequent.

354. From the two last articles, it is evident that multiplying the antecedent of a couplet, by any quantity, will have the same effect on the ratio, as dividing the consequent by that quantity; and dividing the antecedent, will have the same effect as multiplying the consequent.
      Thus the ratio of         8:4, is 2
      Mult. the antecedent by 2, the ratio of 16:4, is 4
      Divid. the consequent by 2, the ratio of 8:2, is 4.

Cor. Any factor or divisor may be transferred, from the antecedent of a couplet to the consequent, or from the consequent to the antecedent, without altering the ratio.

It must be observed that, when a factor is thus transferred from one term to the other, it becomes a divisor; and when a divisor is transferred, it becomes a factor.
      Thus the ratio of      $3\cdot 6:9 = 2$
Transferring the factor 3,      $6:\frac{9}{3}=2$
the same ratio.

The ratio of      $\frac{ma}{y}:b=\frac{ma}{by}$
Transferring $y$      $ma:by=\frac{ma}{by}$
Transferring $m$,      $a$ is $a:\frac{m}{by}=\frac{ma}{by}$.

355. It is farther evident, from Arts. 352 and 353, that if the antecedent and consequent be both multiplied, or both divided, by the same quantity, the ratio will not be altered.

Cor. 1. The ratio of two fractions which have a common denominator, is the same as the ratio of their numerators.
Thus the ratio of a/n:b/n, is the same as that of a:b.

Cor. 2. The direct ratio of two fractions which have a common numerator, is the same as the reciprocal ratio of their denominators.

356. From the last article, it will be easy to determine the ratio of any two fractions. If each term be multiplied by the two denominators, the ratio will be assigned in integral expressions. Thus multiplying the terms of the couplet a/b:c/d by bd, we have $\frac{abd}{b}:\frac{bcd}{d}$, which becomes ad:bc, by cancelling equal quantities from the numerators and denominators.

356. b. A ratio of greater inequality, compounded with another ratio, increases it
Let the ratio of greater inequality be that of      1+n:1
And any given ratio, that of          a:b     
The ratio compounded of these, (Art. 347,) is     a + na:b
Which is greater than that of a:b (Art. 351. cor.)
But a ratio of lesser inequality, compounded with another ratio, diminishes it.
Let the ratio of lesser inequality be that of      1-n:1
And any given ratio, that of          a:b     
The ratio compounded of these is        a - na:b
Which is less than that of a:b.

357. If to or from the terms of any couplet, there be added or subtracted two other quantities having the same ratio, the sums or remainders will also have the same ratio.
      Let the ratio of      a:b
      Be the same as that of       c:d
Then the ratio of the sum of the antecedents, to the sum of the consequents, viz. of a + c to b + d, is also the same.
That is $\frac{a+c}{b+d} = \frac{c}{d} = \frac{a}{b}$.

Demonstration.

1. By supposition,         $\frac{a}{b} = \frac{c}{d}$
2. Multiplying by b and d,         $ad = bc$
3. Adding cd to both sides,         $ad + cd = bc + cd$
4. Dividing by d,          $a+c=\frac{bc+cd}{d}$
5. Dividing by b + d,            $\frac{a+c}{b+d} = \frac{c}{d} = \frac{a}{b}$.

The ratio of the difference of the antecedents, to the difference of the consequents, is also the same.

358. If, in several couplets, the ratios are equal, the sum ofall the antecedents has the same ratio to the sum of all the consequents, which any one of the antecedents has to its consequent.
Thus the ratio
            |12:6 = 2
            |10:5 = 2
            |8:4 = 2
            |6:3 = 2
Therefore the ratio of (12 + 10 + 8 + 6):(6 + 5 + 4 + 3) = 2.

358. b. A ratio of greater inequality is diminished, by adding the same quantity to both the terms.
Let the given ratio be that of      a+b:a or $\frac{a+b}{a}$
Adding x to both terms, it becomes a+b+x:a+x or $\frac{a+b}{a}$.
Reducing them to a common denominator,
The first becomes         $\frac{a^2+ab+ax+bx}{a(a+x)}$
And the latter         $\frac{a^2+ab+ax}{a(a+x)}$.
As the latter numerator is manifestly less than the other, the ratio must be less. (Art. 351. cor.)

But a ratio of lesser inequality is increased, by adding the same quantity to both terms.
Let the given ratio be that of      (a-b):a, or $\frac{a-b}{a}$.
Adding x to both terms, it becomes      (a-b+x):(a+x) or $\frac{a-b+x}{a+x}$
Reducing them to a common denominator,
The first becomes         $\frac{a^2-ab+ax-bx}{a(a+x)}$
And the latter,         $\frac{a^2-ab+ax}{a(a+x)}.\frac{(a^2-ab+ax)}{a(a+x)}$.

As the latter numerator is greater than the other, the ratio is greater.
        If the same quantity, instead of being added, is subtracted from both terms, it is evident the effect upon the ratio must be reversed.

Examples.

1. Which is the greatest, the ratio of 11:9, or that of 44:35?

2. Which is the greatest, the ratio of $(a+3):\frac{a}{6}$, or that of $(2a+7):\frac{a}{3}$?

3. If the antecedent of a couplet be 65, and the ratio 13, what is the consequent?

4. If the consequent of a couplet be 7, and the ratio 18, what is the antecedent.

5. What is the ratio compounded of the ratios of 8:7, and 2a:5b, and (7x+1):(3y-2)?

6. What is the ratio compounded of (x+y):b, and (x-y):(a + b), and (a+b):h?         Ans. (x2 - y2):bh.

7. If the ratios of (5x+7):(2x-3), and $(x+2):\left(\frac{x}{2}+3\right)$ be compounded, will they produce a ratio of greater inequality, or of lesser inequality?      Ans. A ratio of greater inequality.

8. What is the ratio compounded of (x + y):a and (x - y):b, and $b:\frac{x^2-y^2}{a}$?         Ans. A ratio of equality.

9. What is the ratio compounded of 7:5, and the duplicate ratio of 4:9, and the triplicate ratio of 3:2?
              Ans. 14:15.

10. What is the ratio compounded of 3:7, and the triplicate ratio of x:y, and the subduplicate ratio of 49:9?
              Ans. x3:y3.


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