5800* 3. Divide 7380964 by 23000. Ans. 32020967 4 Divide 2304109 by 5800. Ans. 397509 III. When the Divisor is the exact Product of two or more of the small Numbers not greater than 12 : * Divide by each of those numbers separately, instead of the whole divisor at once. N. B. There are commonly several remainders in working by this rule, one to each division ; and to find the true or whole remainder, the same as if the division had been performed all at once, proceed as follows : Multiply the last remainder by the preceding divisor, or last but one, and to the product add the preceding remainder ; multiply this sum by the next preceding divisor, and to the product add the next preceding remainder ; and so on, till you have gone backward through all the divisors and remainders to the first. As in the example following: EXAMPLES. 1. Divide 31046835 by 56 or 7 times 8. 7) 31046835 6 the last rem. mult. 7 preced. divisor: 8) 4435262-1 first rem. 42 Ans. 55440777 43 whole rem. 2. Divide 7014596 by 72.. 3. Divide 5130652 by 132. 4. Divide 83016572 by 240. Ans. 97424f1. principle on which it is founded is evident : for cutting off the same number of ciphers, or figures, from each, is the same as di. viding each of them by 10, or 100, or 1000, &c. according to the number of ciphers cut off; and it is evident, that as often as the whole divisor is contained in the whole dividend, so often must any part of the former be contained in a like part of the latter. * This follows from the second contraction in Multiplication, being only the converse of it ; for the half of the third part of any thing, is evidently the same as the sixth part of the whole ; and so of any other numbers. The reason of the method of finding the whole remainder from the several particular ones, will best appear from the nature of Vulgar fractions. Thus in the first example above, the first remainder being 1, when the divisor is 7, makes this must be added to the second remainder, 6, making 64 to the divisor 8, or to be divided by 8. But 63 = 6 x7 + 1 43 43 43 :-; and this divided by 8 gives 7 7 7 X 8 56 IV. Common IV. Common Division may be performed more concisely, by omitting the several products, and setting down only the remainders ; namely, multiply the divisor by the quotient figures as before, and, without setting down the product, subtract each figure of it freni the dividend, as it is produced ; alw+ys remembering to carry as many to the next figure as were borrowed before. 88 EXAMPLES. 6056 88 OF REDUCTION. REDUCTION is the changing of numbers from one name or denomination to another, without altering their value.This is chiefly concerned in reducing money, weights, and measures. When the numbers are to be reduced from a higher name to a lower, it is called Reduction Descending ; but when, contrarywise, from a lower name to a higher, it is Reduction Ascending. Before proceeding to the rules and questions of Reduction, it will be proper to set down the usual Tables of money, weights, and measures, which are as follow : grs Of MONEY, WEIGHTS, AND MEASURES. TABLES OF MONEY. 2 Farthings = 1 Halfpenny d = 1 1 12 Pence = 1 Shilling 1 £ 20 Shillings = 1 Pound £ 960 = 240 = 20 = 1 PENCE 48 = 12 = * £ denotes pounds, s shillings, and d denotes pence. 1 denotes i farthing, or one quarter of any thing. denotes a half enny or the half of any thing, i denotes 3 farthings or three quarters of any thing. The 10 Mills (m)= 1 Cent cl Standard Weight. dwt gr 4 Gola The standard for Federal Money of Gold and Silver is 11 parts fine, and I part alloy. A Dollar is equal to 48 and 8d in South Carolina, to 68 in the New-England States and Virginia, to 78 and 6d in NewJersey, Pensylvania, Delaware, and Maryland, and to 88 in New-York, and North-Carolina. TROY SILVER. d dwt gr dwt gr The full weight and value of the English gold and silver coin, is as Value. Weight. d A Guinea 1 1 0 5 91 A Crown 5 0 1981 Half-guinea 0 10 6 2 163 || Half-crown 2 6 9 164 Seven Shillings 0 7 0 1 194 Shilling 1 3 3 21 Quarter-guinea 0 5 3 1 81 | Sixpence 06 1 223 The usual value of gold is nearly 41 an ounce, or 2d a grain ; and that of silver is nearly 5s an ounce. Also the value of any quantity of gold, is to the value of the same weight of standard silver, nearly as 15 to 1, or more nearly as 15 and 1. 14th to 1. Pure gold, free from mixture with other metals, usually called fine gold, is of so pure a nature, that it will endure the fire without TROY WEIGHT". gr .dwe OZ Grains marked gr 24 Grains make 1 Pennyweight dwt 24 = 1 20 Pennyweights! Ounce 480= 20 1 lb 12 Ounces I Pound 16 5760=240=12 By this weight are weighed Gold, Silver, and Jewels. OZ This is the same as Troy weight, only having some different divisions. Apothecaries make use of this weight in compounding their Medicines; but they buy and sell their Drugs by Avoirdupois weight. AVOIR without wasting, though it be kept continually melted. But silver, not having the purity of gold, will not endure the fire like it ; yet fine silver will waste but a very little by being in the fire any moderate time ; whereas copper, tin, lead, &c will not only waste, but may be calcined, or burnt to a powder. Both gold and silver, in their purity, are so very soft and flexible (like new lead, &c), that they are not so useful, either in coin of otherwise (except to beat into leaf gold or silver), as when they are allayed, or mixed and hardened with copper or brass. And though most nations differ, more or less, in the quantity of such allay, as well as in the same place at different times, yet in England the standard for gold and silver coin has been for a long time as follows-viz, Tbat 22 parts of fine gold, and 2 parts of copper, being melted together, shall be esteemed the true standard for gold coin : And that 11 ounces and 2 pennyweights of fine silver, and 18 pennyweights of copper, being melted together, is esteemed the truc standard for silver coin, called Sterling silver. The original of all weights used in England, was a grain or corn of wheat, gathered out of the middle of the ear, and, being well dried, 32 of them were to make one pennyweight, 20 penny weighs Vor., I. AVOIRDUPOIS WEIGHT. marked dr 16 gr i Hundred Weight CWE 20 Hundred Weight 1 Ton ton dr OZ 1 16 gr 28 1 cwt 1 ton 573440 = 35840 - 2240 80 = 20 By this weight are weighed all things of a coarse or drossy nature, as Corn, Bread, Butter, Cheese, Flesh, Grocery Wares, and some Liquids ; also all Metals, except Silver and Gold. oz dwt gr 0 18 51 0 1 3 Hence it appears that the pound Avoirdupois contains 69994 grains, and the pound Troy 5760 ; the former of which augmented by half a grain becomes 7000, and its ratio to the latter is therefore very nearly as 700 to 576, that is, as 175 to 144 ; consequently 144 pounds Avoirdupois are very nearly equal to 175 pounds Troy: and hence we infer that the ounce Avoirdu. pois is to the ounce Troy as 175 to 192. weights one ounce, and 12 ounces one pound But in latter times, it was theuglit sufficient to divide the same pennyweight into 24 equal parts, still called grains, being the least weight now in common use ; and from thence the rest are computed, as in the Tables above. 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