each of these ratios (pa" +qc"+re"+. pb"+qda+rf" + where p, q, r, n are any quantities whatever. a с e 1 n a e Let to a = then a=bk, c=dk, e=fk,...; whence pan=pbrk”, qon=qdrk, reř=rfnk", ...; pa" +qc* +re" +... _ pbakn +qdrkn +rfnk" +... pon+qda +rf" +... pon+qd"+rf"+... =kn; par +qon +ren + =*==;= .... pon+qda +rf" +.../ By giving different values to p, q, r, o many particular cases of this general proposition may be deduced; or they may be proved independently by using the same method. For instance, a+cte if each of these ratios b b+d+fi a result which may be thus enunciated : In a series of equal ratios the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. 3 5x – 3y 7x +2y 3 4 3 21 29" + 2 4 Example 2. Two numbers are in the ratio of 5:8. If 9 be added to each they are in the ratio of 8:11. Find the numbers. Let the numbers be denoted by 5x and 8.x. 5x+9 8 Then 8.x +9 11 Hence the numbers are 15 and 24. 15 Example 3. If A :B be in the duplicate ratio of A +x:B+, prove that x'=AB. A +30 A B * (A - B)=AB (A - B); .. x2= AB, since A - B is, by supposition, not zero. EXAMPLES XXXV. a. Find the ratio compounded of 1. The duplicate ratio of 4 : 3, and the ratio 27 : 8. 2. The ratio 32 : 27, and the triplicate ratio of 3 : 4. 3. The subduplicate ratio of 25 : 36, and the ratio 6 : 23. 4. The ratio 169 : 200, and the duplicate ratio of 15 : 26. 5. The triplicate ratio of x : y, and the ratio 2y2 : 3x2. 6. The ratio 31 : 46, and the subduplicate ratio of 64 : a4. 7. If x : y=5 : 7, find the value of x+y: Y - X, x – 3y 8. If =3}, find the value of y 2x – 57 9. If b:a=2 : 5, find the value of 2a -36 : 36 - a. 3 5 3ax – by 10. If and find the value of y 4by - Tax 11. If 7x – 4y : 3x+y=5 : 13, find the ratio x : y. 2a2 – 312 2 12. If find the ratio a :b. a” +62 41 13. If 2x : 3y be in the duplicate ratio of 2.x in : 3y – m, prove that ma=6xy. 14. If P: Q be the subduplicate ratio of P-x:Q-x, prove PQ P+Q 15. If ū df prove that each of these ratios is equal to 2a-c+3c3e + 4cc 20-d + 3d3e + 4f2d' a that x= a с e 16. Two numbers are in the ratio of 3 : 4, and if 7 be sub tracted from each the remainders are in the ratio of 2 : 3. Find them. 17. What number must be taken from each term of the ratio 27 : 35 that it may become 2 : 3? 18. What number must be added to each term of the ratio 37 : 29 that it may become 8 : 7? show that p+q+r=0. -b' 19. If -C C-a a e ace a 21. If show that the square root of is equal to bdf" 22. Prove that the ratio la+mc+ne : 16+nd+nf will be equal to each of the ratios a : b,c:d, e : f, if these be all equal; and that it will be intermediate in value between the greatest and least of these ratios if they be not all equal. 2+y 23. If then will each of these fracby - ax 2+2 tions be equal to unless b+c=0. y 2x – 3y 24. If x +32 -Y 3z+y ,, prove that each of these ratios is equal to hence show that either x=y, or Z=x + y. ข cx - az cy - az 2y – 3x) PROPORTION. a 325. DEFINITION, When two ratios are equal, the four quantities composing them are said to be proportionals. Thus if d' then a, b, c, d are proportionals. This is expressed by saying that a is to b as c is to d, and the proportion is written a :b :: 0 :d ; a : b = c : d. The terms a and d are called the extremes, b and c the meanz. or a с с a с с 326. If four quantities are in proportion, the product of the extremes is equal to the product of the means. Let a, b, c, d be the proportionals. b ài whence ad=bc. Hence if any three terms of a proportion are given, the ad fourth may be found. Thus if a, c, d are given, then b= Conversely, if there are any four quantities, a, b, c, d, such that ad=bc, then a, b, c, d are proportionals; a and d being the extremes, b and c the means; or vice versâ. 327. DEFINITION. Quantities are said to be in continued proportion when the first is to the second, as the second is to the third, as the third to the fourth; and so on. Thus a, b, c, d, are in continued proportion when b ū đ If three quantities a, b, c are in continued proportion, then a : b=b : 0; [Art. 326.] In this case b is said to be a mean proportional between a and c; and c is said to be a third proportional to a and b. 328. If three quantities are proportionals the first is to the third in the duplicate ratio of the first to the second. Let the three quantities be a, b, c; then ū b a? Now to 72; that is, a:c=a2 : 62. 329. The products of the corresponding terms of two or more proportions form a proportion. If a : b=c:d and e:f=g: h then will ae : bf=cg : dh. For and сд or ae : bf=cg: dh. b h bf an .. a Х с a Х с с = e ae Cor. If a :b=c:d, and b : x=d : y, then a : x=C : Y. This is the theorem known as cx æquali in Geometry, 330. If four quantities, a, b, c, d form a proportion, many other proportions may be deduced by the properties of fractions. The results of these operations are very useful, and some of them are often quoted by the annexed names borrowed from Geometry. (1) If a : b=c : d, then b : a=d : c. [Inversion.] 20 à i b d that is a ; a с or [Alternation.] b:ard :c. (2) If a : b=c : d, then a : c=b : d. ad bc For ad=bc; therefore cd cii b that is, di a с or a that is, ; a:c=b:d. (3) If a : b=c: d, then a+b : b=c+d : d. [Composition.] For b cl i +1; d a+b c+d a+b:b=c+d:d. [Division.] Ī 1; - 6 that is, C-d b or a-b:b=c-d:d. or a с a с For a |