Note 1. If the fractions are of different denominations, reduce them to their proper quantities, &c. as in addition, and then take their difference. 2. In subtracting mixed numbers, when the lower fraction is greater than the upper, subtract the numerator of the lower fraction from the denominator of the upper, and to their difference add the numerator of the upper fraction, carrying one to the unit's place of the lower whole number. 3. When a fraction is to be subtracted from an unit, subtract the numerator from the denominator; the remainder will be the numerator to be placed over the denominator. 4. When a proper fraction is to be subtracted from any whole number, subtract the numerator from the denominator for the numerator of the remainder, which must be annexed to the whole number, made less by 1. (2.) What is the difference between and (3.) What is the difference between 3 (4.) What is the difference between and ? and of { ? 49 343 ? 97 145 (7.) Subtract (8.) What is the difference between 3 of 15 and of 72? (9.) To what fraction must I add that the sum inay be &? (10.) What number is that to which if 73 be added the sum will be 17}? (11.) What number is that from which if you subtract of of an unit, and to the remainder add} of } of an unit, the sum will be 9? (12.) What is the difference between of a £ and & of a shilling? (13.) From of a lb. troy take of an ounce. (14.) From of ton take of of a lb. (15.) From of 3 of a hhd, of wine take of of a pint. (16.) From of a league take g of a mile. (17.) From of 365 days take of of an hour. (18.) A pound avoirdupois is equal to 14oz. 11dwts. 16 grains troy; what is the difference (in troy-weight) between the ounce avoirdupois and the ounce troy * ? MULTIPLICATION OF VULGAR FRACTIONS. RULE. Reduce mixed numbers to improper fractions, and complex fractions to simple ones. Then multiply all the numerators together for a new numerator, and all the deno minators together for a common denominator. Note 1. The work may be abbreviated by striking out such multipliers as are found both in the numerators and denominators. 2. To multiply a fraction by an integer, divide the denominator of the fraction by the integer, (if possible ;) but, if that cannot be done, multiply the numerator of the fraction by it. 3. If a proper fraction be multiplied by a proper fraction, the product will be less than either the multiplier or multiplicand. And, if any number, either whole or mixed, be multiplied by a proper fraction, the product will always be less than the multiplicand. It seems rather paradoxical that the name multiplication should be applied to a work which really diminishes; when the word, strictly speaking, signifies the increasing of a number by repetition. But this apparent paradox will vanish, when we reflect that the multiplication of a fraction must necessarily increase the number of the parts into which the whole thing is divided, and consequently the value of each of these parts will be diminished. Examples. 3급 (1.) Multiply 3§, 3, and 3 of %, together. Troy-weight has its name from Troyes, a town in the province of Champagne in France, now in the department of Aube, and was introduced into England by William the Conqueror. The English were dissatisfied with this weight, because the pound did not weigh so much. as the pound in use at that time in England. Hence arose the term, Avoir du Poids, which was a medium between the Erench and ancient, English weights. 37 84 Then × × × = 3278, product. of 2? 4 73 (7.) What is the continued product of15' and (8.) What is the product of of of 15, and of 118? (9.) Multiply 7ft. 9in. by 3ft. 11in. and that product by 5ft. 3in. (10.) If a board be 12ft. 9in. long, and 5ft. 7in. broad, how many square feet does it contain? (11.) If a closet be 17ft. 9 in. round, and 9ft. 9in. high, how many square feet does it contain? DIVISION OF VULGAR FRACTIONS. RULE. Reduce mixed numbers to improper fractions, complex and compound fractions to simple ones. Then invert the divisor, and proceed exactly as in multiplication. Note 1. When it can be done, divide the numerator of the dividend by the numerator of the divisor, and the denominator by the denominator for the quotient. 2. To divide a fraction by an integer, divide the numerator of the fraction by the integer, it possible; but, if that cannot be done, multiply the denominator of the fraction by it. 3. If the denominators are equal, place the numerator of the dividend over the numerator of the divisor for the quotient. 4. If a proper fraction be divided by a proper fraction, the quotient will be greater than either the divisor or dividend. And, if any whole, or mixed, number be divided by a proper fraction, the quotient will be greater than the dividend; but, if a proper fraction be divided by a whole, or mixed, number, the quotient will be less than the dividend, See the third note in multiplication. 5. If any whole number, greater than 2, be divided by itself less 1, the quotient will be a mixed fraction; and if this mixed fraction be added to and multiplied by the whole number, the sum and product will be equal. (10.) Divide by 95 149* (11.) Divide of 3 of 5 by 1⁄2 of of 19. (12.) What number multiplied by the product. (13.) What part of 108 is will give 15 for of an unit? of an unit? (14.) What number is that, which, if multiplied by of of 15, will produce only THE RULE OF THREE DIRECT IN VULGAR FRACTIONS. RULE. State the question as in the Rule of Three in whole numbers. Reduce mixed numbers to improper fractions, complex and compound fractions to simple ones, and the first and third terins to the same denomination. Then invert the first term of the stating, and multiply the three terms together, and the product will be the answer. Examples. (1.) If g of a yard cost of a £. what will of an ell English cost? 11 First of a yard = { of = of an ell. 2×3×7=23£. 10s. 27d. answer. (2.) If of an English ell cost 10s. 2d. what will { of a yard cost? Answer 8 shillings. (3.) If of a lb. cost 7s. 9d. what will 543lb. cost? (4.) If of of 15 ells of holland cost 2. what will of 175 yards cost at that rate? (5.) Bought 5 pieces of silk, each containing 35 ells English, at 5s. 33d. per ell, what is the value of the whole quantity? (6.) Bought 14,5, tuns of wine at 3s. 3§d. per quart, how much did I pay for the whole? (7.) If 3 of 3 of a yard of cloth cost of § of a £. what will 179 English ells cost? (8.) At 7 d. per lb. what will 11hhds. of sugar amount to, each hhd. weighing 4cwt. 3qr. 15gib.? THE RULE OF THREE INVERSE IN VULGÀR FRACTIONS. RULE. State the question as in whole numbers. Reduce mixed numbers to improper fractions, complex and compound fractions to simple ones, and the first and third terms to the same denomination. Then invert the third term of the stating, and multiply the three terms together. |