PRACTICE. 174. A concrete number which is contained an exact number of times in another, is said to be an ALIQUOT PART of that other. Thus, 28. 6d. is an aliquot part of 10s., but not of 128.; 5 cwts. is an aliquot part of 15 cwts., but not of 14 cwts.; 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"-the only difference being that the former expression is used when the numbers are concrete, and the latter when they are integral and abstract. (See p. 133.) 66 175. 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., whenas is almost invariably the case in commercial transactions the given price is that of some unit. 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 1, 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., (b) between 1d. and 18., (c) between 18. and 1, or (d) more than 1. The instances in which the given price is 1d., 18., or £1 need not be noticed because the merest child can tell that, at Id., 18., or 1 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 d. each. At 1d. each, 78 apples would cost 78d.; therefore, at d. (of 1d.) each, 78 apples must cost of 78d.-that is, (78d.÷2) 39d., or 38. 3d.: (a) EXAMPLE II.-Find the price of 234 oranges, at d. each. Regarding d. as the difference between 1d. and 4d., we see that 234 oranges at id. (of 1d.) each, would, at id. each, cost 234d.; of 234d.—that is, (234d.÷4=) 58d.; and at 2d. each, 1754d., or 148. 72d.,* the difference between 234d. and 58d.: (b) EXAMPLE III.-Find the price of 57 lbs. of sugar, at 4d. a pound. As 4d. is of 18., and the price at 1s. a pound would be 578., the price at 4d. a pound must be of 578.—that is, (573) 198. : * In 234d. (the price of 234 oranges at id. each) there would be an overcharge of d. for each orange, or an overcharge of 584d. for the 234 oranges. (6) EXAMPLE IV.-Find the price of 146 gallons of milk, at 10d. a gallon. The given price is not, but the difference (1d.) between it and is. is, an aliquot part of 18. We therefore regard 10d. as the difference between 18. and 1d. The price of 146 gallons, at is. a gallon, would be 146s.; and at id. (of Is.) a gallon, of 146s.—that is, (146s.÷8=) 188. 3d.; so that the price at rod. a gallon must be 1278. 9d., or £6 78. 9d.—the difference between 1468. and 188. 3d.: (b) EXAMPLE V.-Find the price of 85 yards of ribbon, at 9 d. a yard. Neither the given price, nor the difference between it and Is., being an aliquot part of 18., we break up 9åd. into 6d. (an aliquot part of 1s.), 3d. (an aliquot part of 6d.), and id. (an aliquot part of 3d.). Dividing 85s., the price at is. a yard, by 2, we find the price at 6d. ( of 18.) a yard to be 42s. 6d.; dividing this amount by 2, we find the price at 3d. (of 6d.) a yard to be 218. 3d.; dividing this last amount by 4, we find the price at åd. (of 3d.) a yard to be 58. 3d.; and, adding together the price at 6d., the price at 3d., and the price at 2d. a yard, we find the price at (6d.+3d.+d.=) 9 d. a yard to be (428. 6d. +2 18. 3d.+58. 3ąd.=) 69s. o3d., or £3 98. 0ąd.: -we subtract This result is more easily obtained when-regarding 9åd. as the difference between 10d. and 4d.the price at d. a yard from the price at 10d. a yard : (c) EXAMPLE VI.-Find the price of 92 lambs, at 148. each. 92 @ 148. each. Seeing that the given price is an even number of shillings, we take half this number for multiplier, and 92 for multiplicand. The resulting product, when the righthand figure is "cut off," expresses the pounds (£64) of the answer; and by doubling the cut-off figure (4) we obtain the shillings (8s.) of the answer. 7 92 £64-* 88. This is easily explained. Expressed as a fraction of £1, the given price is £, or io. Now, at £1 each,' 92 lambs would cost £92; therefore, at 1 of 1 each, the price must be of £92-that is, 92 X 7 £ = £ 1 3 × 7 =£9°2×7=£64°4=(§ 89) £64 88. 10 (c) EXAMPLE VII.-Find the price of 69 articles, at 28. 6d. each. As 69 articles at £1 each would cost £69, and as 2s. 6d. is of £1, the price at 2s. 6d. each must be of £69— that is, 69÷8=) £8 12s. 6d. : (c) EXAMPLE VIII.-Find the price of 392 articles, at 16s. 8d. each. The given price is not, but the difference (38. 4d.) between it and £1 is, an aliquot part of £1. We therefore regard 16s. 8d. as the difference between £1 and 38. 4d. The price of 392 articles at 1 each would be 392; and at 38. 4d. ( of £1) each, of £392-that is, (£392÷6=) £65 6s. 8d.; so that, at 16s. 8d. each, the price must be 326 138. 4d.—the difference between the price at each and the price at 3s. 4d. each : (c) EXAMPLE IX.-Find the price of 108 articles, at 138. 74d. each. Neither the given price, nor the difference between it and 1, being an aliquot part of £1, we regard 138. 71d. as the sum of 10s. (an aliquot part of £1), 2s. (an aliquot part of 10s.), IS. (an aliquot part of 2s.), 6d. (an aliquot part of 18.), Id. (an aliquot part of 6d.), and d. (an aliquot part of id.) Price of 108 articles at 1 each=108; at 10s. (of 1) each of 108 £54; at 28. (of 10s.) each= of £54-10 168.; at 1s. ( of 28.) each of 10 168. =£5 88.; at 6d. (of 18.) each of £5 8s.=£2 148.; at Id. (of 6d.) each of £2 148.=98.; at d. (of Id.) each of 98.-28. 3d.; and at (10s. +28. + Is.+6d. + Id.+d.) 138. 74d. each (£54+£10 168.+£5 88.+ £2 148.+98.+2s. 3d.=) £73 98. 3d. : |