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CHAPTER XV.

RATIO AND PROPORTION.

286. Ratio is the relation which one quantity bears to an. other with respect to magnitude. Ratio is denoted by two points like the colon (:) placed between the quantities compared. Thus the ratio of a to b is written a:b.

The first quantity is called the antecedent of the ratio, and the second the consequent. The two quantities compared are called the terms of the ratio, and together they form a couplet. The quantities compared may be polynomials; nevertheless, each quantity is called one term of the ratio.

287. A ratio is measured by the fraction whose numerator is the antecedent and whose denominator is the consequent of the ratio. Thus the ratio of a to v is measured by

a

288. A compound ratio is the ratio arising from multiplying together the corresponding terms of two or more simple ratios. Thus the ratio of a to b compounded with the ratio of c to d becomes ac to bd.

The ratio compounded of the ratios 3 to 5 and 7 to 9 is 21 to 45.

289. The duplicate ratio of two quantities is the ratio of their squares. Thus the duplicate ratio of 2 to 3 is 4 to 9; the duplicate ratio of a to b is ato 12.

290. The triplicate ratio of two quantities is the ratio of their cubes. Thus the triplicate ratio of a to b is a3 to 23.

291. If the terms of a ratio are both multiplied or both divided by the same quantity, the value of the ratio remains unchanged. The ratio of a to b is represented by the fraction 6, and the value of a fraction is not changed if we multiply or divide both numerator and denominator by the same quantity. Thus

a та
mbi

a b
a:b=ma: mb

or

пп

PROPORTION.

292. Proportion is an equality of ratios. Thus, if a, b, c, d are four quantities such that a when divided by b gives the same quotient as c when divided by d, these four quantities are called proportionals. This proportion may be written thus,

a:b::c:d,
a:b=cid,

or

or

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6 đ In either case the proportion is read a is to b as c is to d.

293. The terms of a proportion are the four quantities which are compared. The first and fourth terms are called the extremes, the second and third the means. The first term is called the first antecedent, the second term the first consequent, the third term the second antecedent, and the fourth term the second consequent.

294. When the first of a series of quantities has the same ratio to the second that the second has to the third, or the third to the fourth, and so on, these quantities are said to be in continued proportion, and any one of them is a mean proportional between the two adjacent ones. Thus, if

a:6::6:c::c:d::d:e, then a, b, c, d, and e are in continued proportion, and b is a mean proportional between a and c, c is a mean proportional between b and d, and so on.

295. Alternation is when antecedent is compared with antecedent and consequent with consequent. Thus, if

a:b::c:d, tben, by alternation, a:c:: b:d. See Art. 301.

296. Inversion is when antecedents are made consequents, and consequents are made antecedents. Thus, if

a:b::c:d, then, inversely,

b:a::d:c. See Art. 302.

297. Composition is when the sum of antecedent and consequent is compared with either antecedent or consequent. Thus, if

a:b::c:d, then, by composition, a+b:a::c+d:c, and

a+b:6::c+d:d. See Art. 304.

298. Division is when the difference of antecedent and consequent is compared with either antecedent or consequent. Thus, if

a:b::c:d, then, by division, a-b:a::c-dic, and

a-6:6::c-d:d. See Art. 305.

299. If four quantities are in proportion, the product of the extremes is equal to the product of the means. Let

a:b::c:d. Then

s =

Art. 292.
Multiplying each of these equals by bd, we have ad=bc.

с

Let

300. Conversely, if the product of two quantities is equal to the product of two other quantities, the first two may be made the extremes, and the other two the means of a proportion.

ad=bc. Dividing each of these equals by bd, we have

с

6 đ or

a:0::c:d, Art. 292.

EXAMPLES. 1. Given the first three terms of a proportion, 24, 15, and 40, to find the fourth term.

2. Given the first three terms of a proportion, 3ab3, 4a2b>, and 9a'b, to find the fourth term.

3. Given the last three terms of a proportion, 4a365, 3a365, and 20%b, to find the first term.

4. Given the first, second, and fourth terms of a proportion, 5y*, 7x02y3, and 21x@y, to find the third term.

5. Given the first, third, and fourth terms of a proportion, a+b, a2–62, and (a−b)?, to find the second term.

Which of the following proportions are correct, and which are incorrect?

6. 3a+46: 9a +86 :: 0-20:3a-4b. 7. 9a2-462 : 15a-25ab+852 :: 15a2+25ab+862: 25a--1662. 8. a'+63: a +62 :: a?-12:a-b. 9. a3 +63: a+b::a- a*b+a%b2-a2b3 + ab* -65:a3-13.

301. If four quantities are in proportion, they will be in proportion when taken alternately. Let

a:b::c;d;

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C

302. If four quantities are in proportion, they will be in proportion when taken inversely. Let

a:6::c:d; then

= Divide unity by each of these equal quantities, and we have

hd

c'
b:a::d: C.

K

a

Or

303. Ratios that are equal to the same ratio are equal to each other. If a:b::m:n,

(1.) and c:d::m:n,

(2.) then

a:b::c:d

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304. If four quantities are proportional, they will be proportional by composition. Let

a:b::c:d;

a с

then

ūrā Add unity to each of these equals, and we have

(+1=å +1; that is,

a+b c+d

7 d)
a+b:b::c+d:d.

or

305. If four quantities are proportional, they will be proportional by division. Let

a:b::c:d;

a

с

then

Ö=ă ă
Subtract unity from each of these equals, and we have

7-1=;-1;

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