We thus obtain the Compound Proportion 10: 9 :: 30 : a. 5: 8 12:16 3: 2 And from the resulting Simple Proportion, 10X5X12X3:9X8X16X2 :: 30: a, 1,800=38pieces. 172. Rule for the working of exercises in Compound Proportion: Leaving the fourth place for the answer, write, as the third term, the number which is of the same kind as the answer. Of the remaining numbers, take any two which are of the same kind, and (having, if necessary, reduced them to the same denomination) set them down-one in the first, the other in the second place—as if they and the third term were the only numbers upon which the answer depended. Treat in a similar manner every other pair of numbers of the same kind. Then say-as the product of the numbers in the first place is to the product of those in the second place, so is the third term to the answer; and from the Simple Proportion so obtained, find the answer in the ordinary way ($ 166). Here are the solutions of the preceding examples according to the Unit Method : (I.) A. 20 I 7 8 20 8x7 20 X 5 5 I 8X7 20 X 5 XII 8X7 22 W. D. (III.) lg. bd. 10 in 5 can, of cloth 16 ft. and 2 ft., make 5 2 ,, 16 P. IOX 5 10 X 5 10 X 5 30 x 16x2x9x8 10 X 5 30 x 16 x 2 x 9 x 8 10 x 5 x 12 30 x 16 x 2 x 9 x 8 IO X 5 X 12 X3 66 The Chain Rule.-Certain exercises in Compound Proportion can be worked with great facility by what is called the Chain Rule,” which will be understood from the following illustra tion: If 5 lbs. of tea be worth 9 lbs. of coffee, and 4 lbs. of coffee worth 17 lbs. of sugar, and 8 lbs. of sugar worth 3 lbs. of butter, and 2 lbs. of butter worth 7 lbs. of rice, what quantity of rice is worth 12 lbs. of tea ? 9 coffee 5 tea 8 sugar 7 rice c. Arranging the numbers in the way shown in the margin, we simply divide the product (9 X 17 X 3 X 7 X 12) of lbs. lbs. those in the second column by the product (5X4X8 17 sugar X2) of those in the first 3 butter column. The fact can 2 butter easily be established that ? rice = 12 tea the resulting quotient is the answer. a=9X 17 X3X7X125X4X8X2 =38,556:320=1203! lbs. rice. (I.) If 5 lbs. of tea be worth 9 lbs. of coffee, 12 lbs. of tea must be worth 12 x 9 lbs. of coffee : 5 12 X9 5 (II.) If 4 lbs. of coffee be worth 17 lbs. of sugar, 12X9 lbs. 5 of coffee must be worth 12X9X 17 lbs. of sugar : 5X4 12X9X17 :: 17 : 5 5X4 (III.) If 8 lbs. of sugar be worth 3 lbs. of butter, 12 X9X17 5X4 lbs. of sugar must be worth 12X9X 17 X3 lbs. of butter : b. 12 X9X17 X 3 5X4x8 (IV.) If 2 lbs. of butter be worth 7 lbs. of rice, 12 X 9 X 17 X3 5X4x8 lbs. of butter must be worth 12 X 9 X 17 X 3X7 lbs. of rice : 5x4x8x2 r. 12 x 9 x 17 x 3X7 2 : :: 7 : 5 X 4x 8 5x4x8x2 We thus find 12 lbs. of tea to be worth (I.) 5 12 X9 X 17x3 coffee, or (II.) 5 X4 5 X 4x8 lbs. of butter, or (IV.) 12 X 9 X 17 X 3x7 lbs. of rice. So that 5x4x8x2 r. 12 X 9 lbs. of 12x9x 17 lbs. of sugar, or (III.) the ratio 12 : a is compounded of the four ratios 5:9, 4: 17, 8: 3, and 2 :7 5 : 9 3 :: 12:a 2 : 5X4X8X2: 9X 17 X3X7 :: 12 : a a=12X9X 17 3x7=5* 4x8 x 2. The Chain Rule is employed principally in the working of a class of exercises in ExcHANGE. PRACTICE 173. A concrete number which is contained an exact number of times in another, is said to be an ALIQUOT PART of that other. 66 Thus, 4s. is an aliquot part of 12s., but not of 18s.; 5 cwt. is an aliquot part of 10 cwt., but not of 14 cwt. ; 3 yds. is an aliquot part of 21 yds., but not of 26 yds. ; &c. Aliquot part," therefore, may be said to be synonymous with“ “measure, (see § 89)—the only difference being that the former expression is used when the numbers are concrete, and the latter when the numbers are abstract. 174. PRACTICE teaches us how to employ our knowledge of Fractions* in finding, by means of aliquot parts, the price of any number of articles when the price of one is given; also, the price of any particular quantity of merchandise, &c., when -as is almost invariably the case in commercial transactions—the given price is that of some unit. . So far, however, as Practice is concerned, an extensive knowledge of Fractions is by no means necessary. In the great majority of instances, the only Fractions to be dealt with are the Fractional units (1, , 1, ;, &c.), which, at a comparatively early stage, pupils can easily be taught to read and write the teaching being, of course, accompanied by an explanation of what is meant by a half, a third, a fourth, a fifth, &c., of anything. a As every exercise in Practice can be worked in more ways than one, the pupil, in determining the best solution in each case, must rely upon his judgment and ingenuity, rather than upon any formal Rule which could be laid down. The following examples, however, may assist the pupil, who, in selecting “ aliquot parts,” will naturally employ larger divisors than 12 as seldom as possible : Class I, in which the quantity whose price is required is expressed by a simple number of the same denomination as that of the unit whose price is given. Every example in this class may be regarded as belonging to some one of four sub-classes, according as the given price is (a) less than id., (6) between Id. and IS., (c) between is. and £1, or (d) more than £1. The instances in which the given price is id., 18., or £i need not be noticed : because the merest child can tell that, at id., 18., or £ I each, any number of articles would cost that number of pence, or of shillings, or of pounds—as the case may be. (a.) EXAMPLE I.–Find the price of 78 apples, at id. each. At id. each, 78 apples would cost 78d.; therefore, at £d. (of Id.) each, 78 apples must cost of 78d.—that is, (78d.+2=) 39d., or 38. 3d. : 78 @ jd. each. £d.=} of id. 78d.=price @ Id. each. 390.2 or 38. 3d. S (a.) EXAMPLE II.—Find the price of 234 oranges, at id. each. Regarding įd. as the difference between id. and įd., we see that 234 oranges would, at id. each, cost 234d.; at įd. (1 of id.) each, 1 of 234d.--that is, (234d. -4=) 58£d.; and at gd. each, 1752d., or 148.73d., * the difference between 234d. and 58£d. * In 234d. (the price of 234 oranges at id. each) there would be an overcharge of id. for each orange, or an overcharge of 58 d. for the 234 oranges. |