by the quotient, and the last quotient will be the anfwer. 4. Divide the rft term by the 3d, divide the 2d by the quotient, and the last quotient will be the answer. (87.) EXAMPLE I. To find the value of 14 oz. 8 dwt. of gold, at 31. 19s. 11d. an ounce. (88.) EXPLANATION. Having flated the three terms by the general rule, as here annexed, the 2d term is reduced to pence, and the 3d to dwts. these being their loweft denominations, as above directed. The 1ft term is alfo reduced to dwts. that it may agree with the 3d. The 2d term is then multiplied by the 3d, and the product divided by the ift, according to the general rule, when the anfwer comes out 13809 pence, and 12 remaining over; which remainder being reduced to farthings, and thefe divided by the fame divifor 20, the quotient is 2 farthings, and 8 remaining. Laftly, the pence are divided by 12 to reduce them to fhillings, and these again by 20 for pounds; when the final fum comes out 571. 10s. 9d. 2q. for the answer. is of the fame kind with the answer, for the middle term. 2. Take one of the other two terms of fuppo fition, and of the demanding terms, both of a kind; and from the direction given in the RULE OF THREE, Confider which places they would poffefs if a ftating were made of them and the middle term only, and place them accordingly; do the fame with the other term of fuppofition and its correfpondent demanding one, writing the terms under each other which fall on the right and left of the middle term. (92.) METHODS of OPERATION. 1. BY TWO OPERATIONS. Take the two upper terms and the middle term, in the fame order as they ftand, for the firft ftating of the rule of three; then take the fourth number refulting from the firft ftating, for the middle term, and the two un der terms in the general stating, in the fame order as they ftand, for the extreme terms of the fecond ftating; and the fourth term refulting from it will be the answer. 2. BY ONE OPERATION. Multiply together the terms of which the one is above the other, on both fides of the middle term; then account the twɔ products and the middle term, as they ftand, the three terms of a rule of three ftating, and the fourth term thence refulting will be the answer. It is generally best to work by the latter method, viz. by one operation. And after the stating, and before commencing the operation, if one of the two firft terms, and either the middle term or one of the two laft terms will exactly divide by one and the fame number, let them be divided, and the quotients ufed inftead of them; which will much fhorten the work. (90.) This rule is so called, because that in it there are five numbers or terms given, to find a fixth. It is often named the double rule of three, because its questions are fometimes performed by two operations of the rule of three. Of the five given numbers, three contain a fuppofition, and the other two a demand; one of the terms of fuppofition being of the fame kind with the number required, and the other two of the fame kind as the demanding terms. (91.) RULE for STATING the FIVE given NUMBERS. 1. Write down the term of fuppofition which SECT. VII. ARITHMETIC. found, as in compound multiplication, and the product will be the price required. (96.) If the muliplier exceed 12, it is commony beft to multiply fucceffively by its component parts, as in fimple multiplication. But if the multiplier cannot be exactly produced by the mul tiplication of fmall numbers, find the nearest to it, either greater or lefs, which can be fo produced; then after having multiplied continually by the component parts of this number, to or from the laft product, add or subtract the product of many as it is lefs or greater than the given number. (97.) EXAMPLE. What is the value of 38 cwt. at 1l 11s 4d per ewt? 497 vide fucceffively by its component parts, as in (101.) EXAMPLE. If 22 cwt. coft 241 4s, what is I? £ 11)12 2 price of 11 cwt. I 2 of I The rent of 172 Acres is 1971 168, what is the 172) 197 16(1 3 per Acre 7 2763 10 10 7000 355 6 3 9:0 31 II 8 82 I 19 5 7985 3274 at ros. or 41. is 1637 3 4d or 1. is 545 13 68 4 5 or of 3 4d. is 4 2 6 Ans. 3142 8 2 (100.) RULE II. When the PRICE of fome CERTAIN NUMBER is given, to find the PRICE of the Divide the given price by its INTEGER, or 1. number, as in compound divifion, and the quotient will be the price of 1 as required. When VOL. II. PART II. If there be POUNDS in the PRICE, multiply the given quantity by the num ber of them: and if there be alfo fome ODD MONEY, find its produce by the former rules, and add them together. Rrr (107.) Ex (112.) GROSS WEIGHT of any commodity is its own weight together with that of its package, whether it be cafk, cheft, or any thing elfe. (113.) TARE is the weight of the package, or an allowance made inftead of it. What remains after the tare is taken from the grofs, may be called tare futtle, if there be more deductions. (114.) TRET is an allowance of 4 lb. upon every 104lb. of tare futtle, on account of duft or other wafte. What remains after tret is deducted, may be called tret futtle, if there be any following deduction. (15.) CLOFF is an allowance of 2 lb. for every 3 cwt. and fome fay, for every 100 lb. of tret futtle, to make the weight hold good, when fold by retail. (116.) When all the deductions are made, the laft remainder is called NEAT or NET WEIGHT. When the tare is at fo much per cwt, it will be beft to divide it into aliquot parts of it, as in the PART III. VULGAR FRACTIONS. SECT. I. DEFINITIONS. (118.) A FRACTION, or broken number, is an expreffion of one or more parts of any number. (119.) The number of parts into which the number is fuppofed to be divided, is called the DENOMINATOR; and the number of thofe parts exprefled by the fraction, is called the NUMERATOR. Thefe two numbers are in general named the TERMS of the fraction. (120.) If the number of which the fraction is a part, or parts, be 1, it is called a SIMPLE FRAC TION; and is denoted by the numerator written above the denominator, with a final line between them: So, denotes one third of 1; denotes feven eighths of r. (121.) But if the number be different from I, it is called a COMPOUND, and is denoted by the word of, and the number fubjoined to the nume rator and denominator, expreffed as before. So, of 6, denotes one half of 6; of 8, denotes two thirds of 8; and 4 of , denotes three fourths of five eighths of 1. (122.) Simple fractions, whofe numerators are lefs than their denominators, are called PROPER FRACTIONS. And thofe whofe numerators are equal to, or greater than, their denominators, are called IMPROPER FRACTIONS. (123.) The expreffion formed from an integer and a fraction joined together, is called a MIAT NUMBER. If both the numerator and denomina tor of a fraction be multiplied or divided by the fame number, the fraction will retain its original value. All computations in fractions are four ded on this principle. (124.) The following SIGNS, being frequently ufed to avoid circumlocution, require to be tere explained. (128.) If the fraction must be brought to its leaft terms at one divifion, divide its terms by their greatest common measure, which common meafure is found by dividing the greater term by the lefs, and this divifor by the remainder; and fo on, always dividing the laft divifor by the last remainder, till o remain; then is the laft divifor the greatest common measure required. (129.) EXAMPLE. Reduce at one divifion. First, 246)372(1 126)246(1 to its least terms Then 4 I (4 and 26 8 X 2 = 16 (133.) EXAMPLE. Reduce 7 to a fraction whofe denominator fhall be 4. 7= 7X4 28 (134) RULE IV. To REDUCE a MIXT NUMBER Multo an EQUIVALENT IMPROPER FRACTION. tiply the integer by the denominator of the frac- (135) EXAMPLE.-Reduce 2 to a fraction. 3 2 = 7 7 (136.) RULE V. To REDUCE a COMPOUND FRACTION to an EQUIVALENT SIMPLE ONE. Multiply all the numerators together for the numerator, and all the denominators together for the denominator of the fimple fraction required. If part of the compound fraction be an integer or a mixt number, reduce it to a fraction by one of the for mer cafes. (138) RULE VI. To REDUCE FRACTIONS of DIFFERENT DENOMINATORS to EQUIVALENT FRACTIONS of a COMMON ONE. If the fractions can be conveniently reduced to a common denominator, by applying or dividing their terms, proceed by that method. But, if not, multiply each 246÷6 numerator continually into all the denominators, 372-6 62 except its own, for each new numerator; and multiply all the denominators together for the common denominator. In this and feveral other operations, when any of the propofed quantities are integers, mixt numbers or compound fractions, they must be reduced by their proper rules, to the form of fimple fractions. MINATIONS. Multiply the numerator by the in- denominator, then their fum and a third, and fo teger, and divide by the denominator. 1.= 9X20 9 If a compound whole number be propofed, reluce it all to the lowest denomination mentioned in it, and proceed as before. (144.) RULE. IX. To REDUCE FRACTIONS to EQUIVALENT ONES of a DIFFERENT INTEGER, when a certain number of the LESS IS NOT EXACTLY contained in the GREATER. 1. By the laft, reduce the given fraction to an equivalent one of fuch an integer, of which a certain number are contained in the integer to which the fraction must be brought, or which thall contain a certain number of this. 2. By the laft alfo, reduce this fraction to an equivalent one of the integer required. on. (147.) EXAMPLE. What is the fum of, 7 and of. + 7 + of +71+1=1+#+ 78 the fun. SRCT. IV. SUBTRACTION OF VULGAR FRAC TIONS. (148.) RULE. The fame preparations being made here as in addition, take the difference of the numerators and fet it over the common denominator, for the difference of the fraction required. In fubtracting mixt numbers, when the fraction in the fubtrahend is greater than that in the minuend, fubtract the numerator of the fubtrahend from the denominator, and to the dife rence add the numerator of the minuend; and carry one to the integer in the fubtrahend. (150) RULE. Reduce mixt numbers, if there be any, to fractions; then multiply ali the num. rators together for the numerator, and multipas all the denominators together for the denominater of the product required. A fraction is best mutiplied by an integer, by dividing the denominator by it if poflible, but if that cannot be done, mul tiply the numerator by it. (152.) RULE. Having prepared the terms as in multiplication; take the quotient of the rur rators and of the denominators if they will exclly divide, for the numerator and denominator ef the fraction required; but if that cannot be desk, multiply the dividend by the RECIPROCAL of the divifor, for the quotient required. By the reciprocal of a fraction, is meant the fraction got by inverting its terms: fo the reciprocal et is, and of 5 or is. A fraction is divided by an integer by dividing the numerater by it. poffible; but if it will not exactly divide, the multiply the denominator by it. (153) EXAMPLES. 1. What is the quotient of-by-? |