This is sometimes expressed by the notation a: b::c: d, which is read "a is to b as c is to d." The first and fourth of four quantities in proportion, are sometimes called the extremes, and the second and third of the quantities are called the means. 208. If the four quantities a, b, c, d are proportional, we have by definition, a Ъ C d Multiply each of these equals by bd; then ad = bc. Thus the product of the extremes is equal to the product of the means. Conversely, if ad=bc, then a, b, c, d will be propor tional. = are all true, provided that ad bc. Hence the four proportions are all true when any one of them is true. S. A. 16 Ex. If a b=c : d, then will a+b: a-b=c+d: c-d. This has already been proved in Art. 113: it may also be proved as follows: if that is, if or, if a+ba-b=c+d:c- d, (a+b) (cd)=(a - b) (c+d), But bc is equal to ad, since a: b=c : d. 209. Quantities are said to be in continued proportion when the ratios of the first to the second, of the second to the third, of the third to the fourth, &c., are all equal. Thus a, b, c, d, &c. are in continued proportion if If a bbc, then b is called the mean proportional between a and c; also c is called the third proportional to a and b. If a, b, c be in continued proportion, we have Thus the mean proportional between two given quantities is the square root of their product. Thus, if three quantities are in continued proportion, the ratio of the first to the third is the duplicate ratio of the first to the second. 210. The definition of proportion given in Euclid is as follows: Four quantities are proportionals, when if any equimultiples whatever be taken of the first and the third, and also any equimultiples whatever of the second and the fourth, the multiple of the third is always greater than, equal to or less than the multiple of the fourth, according as the multiple of the first is greater than, equal to or less than the multiple of the second. If the four quantities a, b, c, d satisfy the algebraical test of proportionality, we have a C = b d a; therefore for all Hence mcnd, according as ma nb. Thus a, b, c, d satisfy also Euclid's test of proportionality. Next, suppose that a, b, c, d satisfy Euclid's definition of proportion. If a and b are commensurable, so that a: b = m : n, where m and n are whole numbers; then Thus a, b, c, d satisfy the algebraical definition. If a and b are incommensurable we cannot find two whole numbers m and n such that a: b =m: n. But, if we take any multiple na of a, this must lie between two consecutive multiples, say mb and (m + 1) b of b, so that na > mb and na < (m + 1) b. Hence by the definition, nc > md and nc < (m + 1) d. α d 1 Thus the difference between and is less than; and as this is the case however great n may be, a b d n must be equal , for their difference can be made less than any assignable difference by sufficiently increasing n. Ex. 1. For what value of x will the ratio 7+x: 12+x be equal to the ratio 5: 6? Ex. 2. If 6x2+6y2=13xy, what is the ratio of x to y? Ans. 18. Ans. 2 3 or 3: 2. Ex. 3. What is the least integer which when added to both terms of the ratio 5 9 will make a ratio greater than 7 : 10? Ans. 5. Ex. 4. Find x in order that x+1: x+6 may be the duplicate ratio of 3: 5. Ex. 5. Shew that, if a b c d, then 29 Ans. 16' (i) a2+ab+b2 : c2+cd+d2 :: a2 - ab+b2 : c2 - cd+d2. (ii) a+bc+d :: √ (2a2 – 3b2) : √(2c2 – 3d2). (iii) a2+b2+c2+d2 : (a+b)2+(c+d)2 :: (a+c)2 + (b + d)2 [See Art. 113.] Ex. 6. If a b c : (a+b+c+d)2. d, then will ab + cd be a mean proportional between a2+ c2 and b2+ d2. VARIATION. 211. One magnitude is said to vary as another when the two are so related that the ratio of any two values of the one is equal to the ratio of the corresponding values of the other. Hence the measures of corresponding values of the two magnitudes are in a constant ratio. The symbol ∞ is used for the words varies as: thus A B is read 'A varies as B'. If a b, the ratio a: b is constant; and if we put m for this constant ratio, we have To find the constant m in any case it is only necessary to know one set of corresponding values of a and b. a For example, if a ∞ b, and a is 15 when b is 5, we have =m= ... a=3b. 15 5; 212. One quantity is said to vary inversely as another when the first varies as the reciprocal of the second. Thus a varies inversely as b if the ratio a and therefore ab = m. 1 : b is constant, One quantity is said to vary as two others jointly when the first varies as the product of the other two. Thus a varies as b and c jointly if a ∞ bc, that is if a = mbc, where m is a constant. One quantity is said to vary directly as a second and inversely as a third when the ratio of the first to the product of the second and the reciprocal of the third is constant. |