Ex. 3. that x2 = If A: B be in the duplicate ratio of A+x: B+x, prove By the given condition, (4+ 2)2 = 1; B А. Find the ratio compounded of 1. The duplicate ratio of 4: 3, and the ratio 27: 8. 8. If ba 2 : 5, find the value of 2 a - 3b: 3b. 12. If 2x: 3y be in the duplicate ratio of 2x that m2 = 6 xy. 13. If P: Q be the subduplicate ratio of P xx, prove that m: 3ym, prove 15. Two numbers are in the ratio of 3:4, and if 7 be subtracted from each the remainders are in the ratio of 2:3: find them. 16. What number must be taken from each term of the ratio 27:35 that it may become 2:3? 17. What number must be added to each term of the ratio 37:29 that it may become 8:7? 21. Prove that the ratio la+mc+ne: lb+md+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. 348. DEFINITION. PROPORTION. Four quantities are said to be in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth. The four quantities are called proportionals, or the terms of the proportion. Thus, if then a, b, c, d are proportionals. This is α с b = expressed by saying that a is to b as c is to d, and the proportion is written a:b::c:d, or a: b = c: d. The terms a and d are called the extremes, b and c the means. 349. 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. Hence if any three terms of a proportion are given, the fourth may be found. Thus if a, c, d are given, then b=ad с 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 versa. 350. Continued Proportion. 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 If three quantities a, b, c are in continued proportion. then a: b=b:c; 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.. 351. If three quantities are proportionals, the first is to the third in the duplicate ratio of the first to the second. α Let the three quantities be a, b, c; then b 352. The products of the corresponding terms of two or more proportions form a proportion. If a:bc:d and e:f=g: h, then will 353. Transformations that may be made in a Proportion. 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. For=; therefore 1 + α 1÷ = C b [Inversion.] (3) If a: bc:d, then a+b: b=c+d:d [Composition.] (5) If a:bc: d, then a+b: a-b=c+d: c-d. Several other proportions may be proved in a similar way. (3 a + 6b+c+ 2 d) (3 a − 6 b -c+2d) prove that a, We have = (3a-6b+c-2d) (3 a +6b-c-2d), b, c, d are in proportion. = с 2 d [Art. 349.] α ·6b+c-2d 3a-6b c+ 2 d whence, dividing by x2, which gives a solution x = 0, [Art. 291, note.] |