A number of dollars, as 754, may be read 754 dollars, or 75 eagles, 4 dollars; and decimal parts of a dollar, as *365, may be read 3 dimes, 6 cents, 5 mills, or 36 cents, 5 mills, or 365 mills; and others in a similar manner. Addition, subtraction, multiplication and division of federal money are performed just as in decimal fractions; and consequently with more ease than in any other kind of money. EXAMPLES. 1. Add 2 dollars, 4 dimes, 6 cents, 4D. 2d., 4d. 9c., 1E. 3D. 5c. 7m., 3c. 9m., 1D. 2d. 8c. Im., and 2E. 4D. 7d. 8c. 2m. together. coarser than standard, the variation from standard is estimated by carats and grains of a carat in gold, and by penny-weights in sil ver. Alloy is used in gold and silver to harden them. of NOTE. Carat is not any certain weight or quantity, but any weight or quantity; and the minters and goldsmiths divide it into 4 equal parts, called grains of a carat. 7. Multiply 3D. 4d. 5c. 1m. by ID. 2d. 3c. 2m. It has already been observed, that when an infinite decimal repeats always one figure, it is a single repetend; and when more than one, a compound repetend; also that a point is set over a single repetend, and a point over the first and last figures of a compound repetend. It may be further observed, that when other decimal figures precede a repetend, in any number, it is called a mixed repetend : as '23, or 104123: otherwise it is a pure, or simple, repetend : as 3 and 123. Similar repetends begin at the same place: as 3 and 6, or 1341 and 2.156. Dissimilar repetends begin at different places: as 253 and .. 4752. Conterminous repetends end at the same place: as 125 and 009. Similar and conterminous repetends begin and end at the same place: as 2*9104 and 0613. REDUCTION of CIRCULATING DECIMALS. CASE I. To reduce a simple repetend to its equivalent vulgar fraction. RULE.* r. Make the given decimal the numerator, and let the denominator be a number consisting of as many nines as there are recurring places in the repetend. 2. If * If unity, with cyphers annexed, be divided by 9 ad infinitum, the quotient will be I continually; i. e. if be reduced to a decimal, it will produce the circulate i; and since I is the decimal equivalent to 3, 2 will = 3, 3 = 3, and so on till 9== 1. Therefore, every single repetend is equal to a vulgar fraction, whose numerator is the repeating figure, and denominator 9. 2 2. If there be integral figures in the circulate, as many cyphers must be annexed to the numerator, as the highest place of the repetend is distant from the decimal point. EXAMPLES. 1. Required the least vulgar fractions equal to 6 and 123. 6==; and *123== Ans. 999 2. Reduce 3 to its equivalent vulgar fraction. 333 3. Reduce 162 to its equivalent vulgar fraction. Ans.. 4. Required the least vulgar fraction equal to 769230. To reduce a mixed repetend to its equivalent vulgar fraction. RULE.* 1. To as many nines as there are figures in the repeténd, annex as many cyphers as there are finite places, for a denominator. 2. Multiply Again, g, or gå, being reduced to decimals, makes '010101, .. &c. or '001001, &c. ad infinitum =1 or 'oot; that is, or, and Tool; consequently =02, 5=03, &c. and 2 .. =002, 5=003, &c. and the same will hold universally. * In like manner for a mixed circulate; consider it as divisible into its finite and circulating parts, and the same principle will be seen to run through them also: thus, the mixed circulate 16 is divisible into the finite decimal 1, and the repetend 06; but '1=1', and '06 would be, provided the circulation began immediately after the place of units; but as it begins after the place of 1 2. Multiply the nines in the said denominator by the finite part, and add the repeating decimal to the product, for the numerator. 3. If the repetend begin in some integral place, the finite value of the circulating part must be added to the, finite part. EXAMPLES. 1. What is the vulgar fraction equivalent to 138 9×13+8=125= numerator, and 900= the de 2 nominator; ... 138 the answer. 3 2. What is the least vulgar fraction equivalent to 53? Ans. 3. What is the least vulgar fraction equal to 5925? 8 16 Ans. 4. What is the least vulgar fraction equal to '008497133? 83 58 Ans. 이이이에여 5. What is the finite number equivalent to 3162? CASE III. 9 Ans. 31. To make any number of dissimilar repetends similar and con terminous. RULE.* Change them into other repetends, which shall each consist of as many figures as the least common multiple of 6 the tens, it is f of ro, and so the vulgar fraction = 16 is Tot =+=, and is the same as by the rule. * Any given repetend whatever, whether single, compound, pure or mixed, may be transformed into another repetend, that shall consist, |