10 a qr: 10. Add 4 of a ton to % of an hundred weight, 2516 Ans. 12cwt. lqr. 7 16. 13oz. 11 drs. 11. Suppose I have of a ship worth $6000, and that I buy another person's share of her, which is ' ; what part of her belongs to me then, and what is it worth ? Ans. I have it, and it is worth $4125. SUBTRACTION OF VULGAR FRACTIONS. RULE.—Prepare the fractions as in addition, and the difference of the numerators written above the common denominator, will give the difference of the fractions required. NOTE 1.-In subtracting mixed numbers, when the fraction in the subtrahend is greater than that in the minuend, subtract the numerator of the subtrahend from the denominator, and to the difference add the numerator of the minuend, and carry one to the integer in the subtrahend. If the minuend contain no fraction, proceed in the same way, there being then nothing to add to the difference. Note 2.-In fractions of money, weights, &c. you may find the value of each of the given fractions, by Case 7, in Reduction, and then subtract them in their proper terms. EXAMPLES. 1. From take of 4. of =4 ; and =* =*=* the answer. 2. From Htake. Ans. 115 3. From 711 take 17. Ans. 7073 4. From £take of a shilling. Ans. 16s. 9d. 5. From zoz. take of a pennyweight. 11dwt. 3grs. Ans. 6. From fcwt. take 77 of a pound. Ans. lqr. 27 11. 6oz. 10 drs. 7. From 7 weeks take 9 days you Ans. 5w. 4d. 7b. 12m. 8. From 4 days 74 hours take 1 day 9 hours Tipo Ans. 2d. 22h. 20m. 9. Suppose I own ý of a farm, which is worth $3600, and that I sell of my sbare; what part of it have I left and what is it worth ? Ans. ; and worth $750. MULTIPLICATION OF VULGAR FRACTIONS. Rule.-Reduce compound fractions to single ones, mixed numbers to improper fractions, and those of different integers to the same; then multiply all the numerators together for the numerator, and multiply all the denominators together for the denominator, of the product required. But always, before multiplying, cancel equal numerators and denominators, and divide those that are divisible by the same numbers, both here and in division, agreeably to what is seen in Note 8 of Case 1, in Reduc. tion, page 150, and in Case 5, page 154. Note.—A fraction is best multiplied by an integer, by dividing the denominator by it, if possible, but if that cannot be done, multiply the pumerator by it. = Ans. EXAMPLES. 1. Required the continued product of 21, 5, , of and 2. 1x5 3x6 5X1 X5X2 Then *x*X35X4 2x8x18x1 2. Multiply by ft. 3. Multiply 41 by $. 4. Multiply of 8, by 1 of 5. Ans. 21. 6. Multiply 74 by 91. Ans. 69 6. Multiply 41, of 7, and 184 continually together. Áns. 9140 7. What is the product of 5, 5, 4, of , and 41? Ans. 24: Ans. To Ans. 16 DIVISION OF VULGAR FRACTIONS. Rule.-Prepare the fractions as in Multiplication ; then invert the divisor, and proceed as in Multiplication. The product will be the quotient required. NOTE.—A fraction is divided by an integer, by dividing the numerator by it, if possible, but if it will not exactly divide, then multiply the denominator by it. EXAMPLES. 1. Divide t of 19 by of 1x19 19 } of 19 -, and of ==}; 19 x2 5x1 Ans. 133 3. Divide 99 by 108. Ans. 1906=11 4. Divide of by 1 of 3. Ans. 14 5. Divide 31 by 97. Ans. } 6. Divide ž by 4. 99 Ans. 32 RULE OF THREE IN VULGAR FRACTIONS, Rule.-Prepare the fractions as in Multiplication ; and state the question as directed in the Rule of Three in whole numbers; then invert the first term in the stating, and multiply all the three terms continually together, numerators by numerators and denominators by denominators, and the product will be the answer in the same name as the second or middle term. 15 EXAMPLES. 1. If of a yard cost iz of a pound, what will it of an ell English cost ? 3x4x1 First of a yard=of 4 of }= -= of an ell. 5X1 X5 Then : 2 :: 15: Note.--Here, 25 and 15 are divisible by 5, giving 5 and 3 for quotients; and 6 and one of the 12s. are divisible by 6, giving 1 and 2 for quotients; then the numerators, are 5, 7, and 1, whose product is 35; and the denominators 12, 2, and 3, whose product is 72. And thus always 6 a proceed, in multiplying vulgar fractions, by cancelling equal numerators and denominators, or dividing those that are divisible by the same number. 2. If of a yard cost of a dollar, what will 404 yds. come to ? Ans. $59,42cts. 711m. 3. If 50% bushels of wheat cost $5924, what is it per bushel ? Ans. $1,16cts. 6fm. 4. If I buy 100 16. of butter for $104, how many hundred can I have for $10511 ? Ans. 104 hundred. 5. If 1 of 1 of a yard of lace cost 30cts., how much is it per yard? Ans. $2,40cts. 6. If of a gallon of wine cost of a dollar, what will 8 of a tun cost ? Ans. $140. 7. A younger brother received $2200, which was just is of his elder brother's fortune ; and 3} times the elder's money was į as much again as the father was worth ; what was the father worth? Ans. $11000. 8. When the days are 13 hours long, a traveller performs his journey in 35,1 days; in how many days will he perform the same journey, when the days are 11 hours long? Ans. 40% 5 days. 9. A schoolmaster being asked how many scholars he had, answered, If I had as many, and 4 as many, and I as many, I should have 99;-how many had he ? Ans. 36. RULE OF THREE IN DECIMALS. RULE.—Reduce vulgar fractions to decimals, and compound numbers either to decimals of the higher names, or to integers of the lower, as also the first and third terms to the same name; then state the question and proceed as in whole numbers; the fourth term will be the answer. EXAMPLES. 1. If of a yard cost of a pound, what will of an ell English cost ? f=,375yd. 5,4£. ell=fryd. 2,3125yd. Yd. £. Yd. £. 2. If lcwt. Iqr. 164 tt. of sugar cost $10,9cts., what will 9cwt. 3qr. cost, at the same rate ? As 1,4 : 10,09 :: 9,75 : 70,269+=$70,26cts: 9ın. + Ans. 3. A man bought 5,8 tuns of oil, for $266, but by accident, lost 50,9 gallons ; how must he sell the residue per gallon, so as not to lose in his bargain ? Ans. 18cts. 8m. + 4. If 12 oxen graze down 16,25 acres of grass in 20 days, how much of the same pasture will suffice 24 oxen 100 days? Ans. 162,5 acres. 5. By how many men would 417,6 acres be mowed in 12 days, if 5 men mow } of that quantity in { the time ? Ans. 20 men. 6. Two men bartered ; A had 24,07 yards of linen, for which B gave him 25,6 ells of Holland, at 45cts. per ell; what was the linen per yard ? Ans. 47cts. 8m. + 7. A man bought a piece of broadcloth for $85,20cts., at the rate of $4,26cts. per yard; how many yards did it contain ? Ans. 20yds. 8. If, during tbe tide of ebb, a boat should set out from Hallowell for Bath, and at the same instant another should put off at Bath for Hallowell, taking the distance by water at 25 miles; the tide forwards one, and retards the other, say 23 miles an hour; the boats are equal in size and form, and equally laden, the rowers equally good, and in the common way of working, in still water, would proceed at the rate of 5 miles an hour ; where in the rive ; er would the two boats meet ? Ans. They will meet 182 miles from Hallowell. 9. Bought 12 pieces of Holland for $40418, at $130 per ell Flemish ; how many yards were in each piece ? Ans. 22 yds. DOUBLE RULE OF THREE IN VULGAR FRACTIONS. RULE.-State the question as in the same rule in whole numbers; invert the fractions which stand in and under the first term ; then proceed as in the Rule of Three in Vulgar Fractions, |