Examples for practice.* Reduce 0,18 to the form of a vulgar fraction. 4995 Reduce 0,2418 to the form of a vulgar fraction. Ans. 13. Reduce 0,275463 to the form of a vulgar fraction. Reduce 0,916 to the form of a vulgar fraction. Ans.. To form a correct idea of the nature of these fractions it is sufficient to consider the fraction 0,999..... In trying to discover its original value we find that it answers to 9 divided by 9, that is, to unity; nevertheless, at whatever number of figures we stop in its expression, it will never make an unit. If we stop at the first figure, it wants of an unit; if at the second, it wants ; if at the third, it wants, and so on; so that we can arrive as near to unity as we please, but can never reach it. Unity then in this case is nothing but a limit, to which 0,999 . . tinually approaches the nearer the more figures it has.† con 99. The preceding part of this work contains all the rules absolutely essential to the arithmetic of abstract numbers, but to apply them to the uses of society it is necessary to know the different kinds of units, which are used to compare together, or ascertain the value of quantities, under whatever form they may present themselves. These units, which are the measures in use, have varied with times and places, and their connexion has been formed only by degrees, accordingly as necessity and the progress of the arts and sciences have required greater exactness in the valuation of substances, and the construction of instruments. * In these examples, the better to distinguish the period, a point is placed over it, if it be a single figure, and over the first and last figure, if it consist of more than one. See note on continued fractions at the end of this treatise. Tables of Coin, Weight, and Measure. Denominations of Federal money, as established by act of Congress, April 2, 1792.† †The act above referred to provides, that the money of account of the United States shall be expressed in dollars or units, and dimes or tenths of a dollar, cents or hundredths of a dollar, and mills or thousandths of a dollar. The coins of gold, silver, and copper, shall be of the following denominations, namely; 1. Gold-eagle, half-eagle, quarter-engle. 2. Silver-dollar, half-dollar, quarter-dollar, dime, half-dime. 3. Copper-cent, half-cent. The standard for all gold coin shall be eleven parts of pure gold and one part alloy in twelve parts of the coin. The alloy is to be of silver and copper, the silver not exceeding one half in the alloy. The eagle shall contain 247 grains of pure gold, or 270 grains of standard gold. In the other gold coins the weight of pure or standard gold shall be in proportion to their values. The standard for all silver coins shall be 1485 parts of pure silver to 179 parts of alloy; and the alloy shall be pure copper. The dollar shall contain 3711 grains of pure silver, or 416 grains of standard silver. In the other silver coins the weight shall be in proportion to their respective values. The copper coins shall be pure copper. The cent shall contain 11 pennyweights of copper; and the half-cent half this weight of copper. In practical treatises on arithmetic, may be found rules for reducing the Federal Coin, the currencies of the several United States, and those of foreign countries, each to the par of all the others. It may be sufficient here to observe respecting the currencies of the several states, that a dollar is considered as 6s. in New England and 24= 1 02 24 Grains make 1 Pennyweight dwt 20 Pennyweights 1 Ounce 12 Ounces 1 Pound 480= 20= 1 lb lb 5760 240 = 12 = 1 By this weight are weighed gold, silver, and jewels. Virginia; 8s. in New York and North Carolina; 7s. 6d. in New Jersey, Pennsylvania, Delaware, and Maryland; and 4s. 8d. in South Carolina and Georgia; the denomination of shilling varying its value accordingly. 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. 256 16= 1 gr 1 crot 7168= 4 = 1 ton 448= 28= 28672 = 1792 = 112 57344035840 = 2240 80 20 = 1 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 dut gr Note, that 1lb Avoirdupois = 14 11 15 Troy. Hence it appears that the pound Avoirdupois contains 69991 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 Avoirdupois is to the ounce Troy as 175 to 192. |