EXERCISES ON CHAPTER VIIJ. 1. A farmer sold at one time 24 cwt., at another 3 qr., and at another 18 cut. of hemp. Find the entire quantity that he sold, in cut. 2. A grocer bought, at different times, 3 lb. of soap of which he has sold 2 cwt. 1 qr. tity remaining, in lb. Ans. 213 cut. cwt., 24 gr. and 49 Find the quanAns. 218 lb. 3. A merchant had 3 pieces of cloth, containing each 29 yd. 3 qr.; of which he has sold, to different persons, 5 yd., 3 gr., and 10 yd. 11⁄2 qr. Find the remainder in yd. Ans. 723 yd. 4. A brewer bought of A 13 bu. 3 pk., of B 52 bu., and of C 11 bu. 1 pk. 1 qt. of barley. Having consumed 94 bu., 3 pk., and 5 bu., how many bu. has he on hand? Ans. 148 bu. 5. How many yards of cloth, at 30 s. per yard, may be purchased for 9£; and what quantity of iron, at 6£ per ton, may be purchased for 40 s.? Ans. 6 yd.; and of a ton. 6. How many days will a ship be in sailing 750 L., at the rate of 10 miles per hour; and to what part of a common year will the time required be equivalent ? 25 1022 Ans. 81 da.; 8 of a y. 7. A wine merchant has in one cask 74 gal. 3 qt., and in two others each 60 gal. 2 qt. 1 pt., of wine. What quantity has he altogether? Ans. 196 gal. 8. A traveler who set out on a journey of 600 m. 5 fur., proceeded for 12 days at the rate of 34 m. 2 fur. 20 rd. per day. What distance remained to be traveled? Ans. 188 m. 7 fur. 9. A merchant has 5 pieces of cotton, containing 33 yd. 3 gr. each; which he sold in equal portions to ten customers. What quantity was bought by each customer? Ans. 16 yd. 3 gr. 10. A planter has a tract of land containing 1250 A. 3 R. 25 P., and an adjoining one containing 2400 A. 1 R. 39 P. If the whole be divided into 10 plantations, what will be the average size of each ? Ans. 365 A. 0 R. 22 P. 11. A stage coach is running at the rate of 7 m. 3 fur. 30 rd. per hour. How long will it be in running 75 m. 7 fur.? Ans. 10.158' hr. 12. An agriculturist raised 500 bu. 3 pk. 4 qt. of oats from one field, and he found the produce to be at the rate of 29 bu. 1 pk. 1 qt. per acre. How many acres did the field contain? Ans. 17.105' A. 13. From a tract of land containing 560 A. 2 R. 20 P., a farmer sold to one of his neighbors 7 A. 3 R., and to another 100 A. 25 P. What part of the tract did he sell to each? Ans. .0138'; and .178'. 14. How many yards of carpeting, one yard in width, will be required to cover the floor of a room measuring 19 ft. 5 in. long, and 10 ft. 6 in. wide? Ans. 22.652' yd. 15. How many cords of wood are there in a pile which is 19 ft. 10 in. in length, 4 ft. in breadth, and 3 ft. 11 in. in height, -there being 128 cubic feet in a cord? Ans. 2.427' cords. 16. How many cords of wood in five loads each 8 ft. 10 in. in length, 3 ft. 6 in. wide, and 3 ft. 4 in. high? Ans. 4.025' cords. 17. How many square yards of painting on the walls of a room measuring 11 ft. in height, and 52 ft. 6 in. in compass,— deducting four windows each 7 ft. by 5 ft. 2 in.? Ans. 48.092' sq. yd. 18. Bought a load of wood which measured 6 ft. 4 in., 3 ft. 3 in., and 2 ft. 10 in.; another measuring 7 ft., 3 ft. 11 in., and 3 ft.; and another measuring 8 ft., 4 ft., and 3 ft. Required the number of cords purchased. Ans. 1.849' cords REMARKS. The last two Chapters, it may be remarked, contain but little more than an application of the essential principles of Arithmetic, to the various arbitrary measures of quantities which have been established in the commerce of society. Facility of operation on such quantities will depend very much on a familiarity with the Tables of Weights and Measures—of which, with the several Rules in these Chapters, the student should be master, before he is permitted to advance. CHAPTER IX. ALIQUOT PARTS-ANALYSIS-CANCELLATION-RATIO-PROPOR TION. ALIQUOT PARTS. $208. An aliquot part of a quantity is an exact half, third, or fourth, and so on, of the quantity. Thus 10 s. is an aliquot part of 1£, being 1⁄2 of 1£. What aliquot part is 6 gr. of 1 dwt.? 5 dwt. of 1 oz.? 1 qt. of 1 pk.? Of 2 oz.? 2 fur. of 1'm.? Of 4 m.? 2 R. of 1 A.? Of 5 A.? It is often convenient to regard the lower orders of units in a polynomial multiplier, as aliquot parts of one or more higher units. ANALYSIS. $209. ANALYSIS, in Arithmetic, consists in determining the operations to be performed in the solution of questions, without the aid of special Rules. EXAMPLE Of Analysis by Aliquot Parts. To find the value of 2 lb. 5 oz. 12 dwt. of silver ware, at $45.12 per lb. We first find the value of 2 lb., by multiplying, and then of 5 oz. 12 dut. by taking aliquot parts, as folows: The values of the several parts, 2 lb., 4 oz., 1 oz., 10 dwt., 2 dut. added together, make the value of the whole quantity 2 lb. 5 oz. 12 dwt. For convenience in dividing, the aliquot parts should be apportioned, when practicable, so that no divisor, exclusive of Os annexed, shall exceed 12. EXERCISES In Analysis by Aliquot Parts. 1. Find the value of 3 lb. 4 oz. 17 dwt., of jewelry, at $50.50 per lb. After multiplying the given price by 3, aliquot parts may be taken thus: 4 oz. is of 1 lb., 10 dwt. is of 4 oz., 5 dwt. is dut., and 2 dwt. is of 10 dwt. of 10 Ans. $171.909'. 2. Find the sum that should be paid for 3 T. 10 cwt. 3 qr. of iron, at $30.371⁄2 per ton. Ans. $107.45'. 3. Find the sum that should be paid for 13 cwt. 2 gr. 15 lb. of soap, at $3.62 per cwt. Ans. $49.422'. 4. A farmer sold 125 bu. 3 pk. 1 qt. of wheat, at $0.87 per bushel. What did the whole amount to? Ans. $110.058'. 5. A market man sold 3 pk. 3 qt.of cherries, at $0.183. per peck. What did his cherries amount to? Ans. $0.632'. 6. A merchant sold 15 yd. 3 qr. 3 na. of silk, at $1.37 per yard, and 5 yd. 2 qr. of lace, at $2.50 per yard. What did the whole amount to? Ans. $35.661'. 7. Find what would be the amount of profit or loss on 139 yd. 2 gr. of cloth, if purchased at $5 a yard, and sold at the rate of $1.564 per qr. Ans. $174.531' profit. 8. A townsman bought a lot of ground containing 3 A. 2 R. 25 P., at $75 per acre. What did he pay for the lot? Ans. $274.218'. 9. Find what would be the expense of putting up 150 rd. 4 yd. of fencing, at $.50 per rod, and 230 rd. 21 yd. of another kind at the rate of 62 cts. per rod. Ans. $219.397. 10. A speculator purchased 15 A. 2 R. 30 P. of land, at $5 per acre, which he divided into town lots, and sold at the rate of $20 per rood or quarter acre. What did he gain by the speculation? Ans. $1176.563'. CANCELLATION. $ 210. CANCELLATION, in Arithmetic, consists in rejecting equal factors from a dividend and its divisor;-which is equivalent to dividing the dividend and divisor by the same number (§ 69), and therefore does not alter the required quotient. (§ 57). $211. In the Analysis of questions, the given numbers may often be put as factors in a dividend and divisor, and cancellation then applied. EXAMPLE Of Analysis and Cancellation. Allowing of a yard of cloth to cost $10, what will § of a yard cost at the same rate? Analysis. Since 3 fourths of a yd. costs $10, 1 fourth of a yd. will cost of $10, which is and 1 yd. will cost 4 times as much as yd., Again; yd. will cost as much as 1 yd. will, and yd. will cost 5 times as much as § yd., $10 3 $10X4 3 $10X4 3 X 8' $10×4×5 3X8 Cancellation. Dividing the numerator and denominator, in other words, the dividend and divisor, both by 4, we have $; $5X5 $25 3 = =$81. dividing these again by 2, we have If the factors of the dividend be set, one under another, on the right of a line, and those of the divisor on the left, the sign of multiplication may be omitted. In canceling, the rejected factors may be crossed, and those to be substituted set respectively right and left. By this method the result of the preceding Analysis and Cancellation, will be presented thus: Canceling 4 and 8, we substitute 1 and 2; canceling 2 and 10, we substitute 1 and 5. The factor 1 is omitted in multiplying. But observe that when each factor of the dividend becomes 1, the 1 must be retained as the numerator of a fraction. |