that will allow each owner to lay out all his land. How many rods must there be in a lot? PROBLEM X. Two or more numbers being given, to find their least common multiple; that is, the least number that will contain each of the given numbers a whole number of times. RULE. Divide two or more of the given numbers by any prime number that will measure them, repeat the operation upon the quotients and undivided numbers, and thus continue, till they become prime to each other. Multiply the several divisors, the last quotients, and undivided numbers together; the product will be the least common measure. If, among the numbers to be divided, any number is a measure of another, the measuring number may be rejected; that is, dropped from the operation. It is obvious, that one number is the multiple of another, when the former contains all the factors of the latter. The factors of 6 are 3 and 2, and the factors of 9 are 3 and 3. Now 54 contains all these factors, (3 X2 X3 X 3=54), and 54 is a common multiple of 6 and 9, but it is not their least common multiple-it is 3 times as great as the least, owing to the existence of the factor, 3, in both 6 and 9. Hence we observe, that a common factor of two or more numbers must enter but once into the multiplication, to give the least common multiple. The above rule effects the necessary exclusion. 23. What is the least common multiple of 12, 25, 30, and 45. 3) 12 25 30 45 We find, after dividing twice, that 4 and 2 appear; and, by dropping the 2 because it measures the 4, we avoid another Ans. 900. 3X5X4 X5 X3-900 division. 24. What is the least common multiple of 6, 10, 16, and 20? 25. What is the least common multiple of 25, 35, 60, and 72? 26. What is the least common multiple of 105, 140, and 245 ? 27. What is the least common multiple of 18, 82, 94, 788, and 356 ? 28. Allowing 63 gallons to fill a hogshead, 42 a tierce, and 32 a barrel, what is the smallest quantity of molasses, that can be first shipped in some number of full hogsheads, then discharged and reshipped in some number of full tierces, and again discharged and reshipped in some number of full barrels ? 29. A certain flour dealer, who purchased his flour from a mill on the opposite side of a river, owned four boats, one of which would carry 8 barrels of flour, another 9, another 15, and another 16. What is the smallest number of barrels he could purchase, that would make some number of full freights for either of the boats ? IX. COMPOUND NUMBERS. COMPOUND NUMBERS are those which are employed to express quantities that consist of several denominations; each denomination being denoted separately. Under this head are classed, all the subdivisions of measures; of length, surface, solidity, weights, money, time, &c. The following tables of denominations of compound numbers, show how many units of each lower denomination are equal to a unit of the next higher, and, exhibit each lower denomination as a fraction of the next higher. MONEY, WEIGHTS, AND MEASURES. ENGLISH MONEY. The denominations of English Money are, the pound, £, the shilling, s., the penny, d., and the farthing, qr. = 1 d. 1 qr. 4 farthings 12 pence 1 d. 20 shillings ... ... = of 1 d. TROY WEIGHT. The denominations of Troy Weight are, the pound, lb., the ounce, oz., the pennyweight, dwt., and the grain, gr. 24 grains 1 dwt. 1gr. 1 ... 1 oz. 1 dwt. 24 of 1 dwt. of 1 oz. 12 of 1 lb. AVOIRDUPOIS WEIGHT. The denominations of Avoirdupois Weight are, the ton, T., the hundred-weight, cwt., the quarter, gr. the pound, lb., the ounce, oz., and the dram, dr. 16 drams APOTHECARIES' WEIGHT. The denominations of Apothecaries' Weight are, the pound, b, the ounce, 3, the dram, 3, the scruple, ǝ, ell, Fr. e., the English ell, E. e., the Flemish ell, Fl. e., the yard, yd., the quarter, qr., and the nail, na. The denominations of Dry Measure are, the bushel, bu., the peck, pk., the gallon, gal., the quart, qt., and WINE MEASURE, The denominations of Wine Measure are, the ton, T., the pipe, p., the puncheon, pun., the hogshead, hhd., the tierce, tier., the barrel, bl., the gallon, gal., the quart, qt. the pint, pt. and the gill, gi. 4 gills. = 1 pt. 1 gi. = of 1 pt. 2 pints of 1 qt. 4 quarts 1 gal. 1 qt. of 1 gal. 31 gallons of 1 bl. 42 gallons 63 gallons of 1 tier. of 1 hhd. 84 gallons 126 gallons of 1 pun. 1 gal. 1 of 1 p. 26 2 pipes of 1 T. firkin, fir., the gallon, gal., the BEER MEASURE. The denominations of Beer Measure are, the butt, bt., the hogshead, hhd., the barrel, bl., the kilderkin, kil., the quart, qt., and the pint, pt. 2 pints 1 qt. 1 pt. of 1 qt. 1 gal. 1 qt. of 1 gal. 1 fir. 1 gal. of 1 fir. 9 gallons 2 firkins 2 kilderkins 3 kilderkins 2 hogsheads NOTE. In the United States, the Dry gallon contains 268 cubic inches, the Wine gallon 231 cubic inches, and the Beer gallon 182 cubic inches. By an Act of the British government, however, the distinction between the Dry, Wine, and Beer gallon was abolished in Great Britain, in 1826, and an Imperial Gallon was established, as well for liquids as for dry substances. The Imperial gallon must contain "10 pounds, Avoirdupois weight, of distilled water, weighed in air, at the temperature of 62° of Fahrenheit's thermometer, the barometer standing at 30 inches." This quantity of water will be found to measure 277-274 cubic inches. The same Act establishes the pound Troy at 5760 grains, and the pound Avoirdupois at 7000 grains. LONG MEASURE. The denominations of Long Measure are, the mile, m., the furlong, fur., the rod or pole, r., the yard, yd., the foot, ft., and the inch, in. 12 inches 3 feet 5 yards 40 rods 8 furlongs 1 ft. 1 in. of 1 yd. of 1 r. SQUARE MEASURE. The superficial contents of any figure having four sides and four equal angles, is found in squares, by multiplying together the length and breadth of the figure. The denominations of Square Measure are, the mile, m., the acre, A., the rood, R., the rod, r., the yard, yd., the foot, ft., and the inch, in. 144 inches 9 feet.. 30 yards. 40 rods 4 roods 640 acres The cubical contents of any thing which has 6 sidesits opposite sides being equal-is found in cubes, by multiplying together, the length, breadth and depth. The denominations of Cubic Measure are, the yard, yd., the foot, ft., and the inch, in. 40 feet of round timber, or 50 feet of hewn timber make a ton. 16 cubic feet make a foot of wood, and 8 feet of wood make a cord. TIME. The denominations of Time are, the year, Y., the day, d., the hour, h., the minute, m., and the second, s. |