648 a higher denomination? How will you reduce a fraction of one denomination to an equivalent fraction of a lower denomina tion? How will you find the value of a fraction in the known parts of the integer? How will you reduce a given quantity to the fraction of any greater denomination of the same kind? How will you reduce a whole number to an equivalent fraction, having a given denominator? How will you reduce a given fraction to another equivalent one, having a given numerator? How will you reduce a given fraction to an equivalent one, having a given denominator? What is the rule for performing addition of vulgar fractions? What will you do when a mixed number is given? What is to be observed when different denominations are given? when all the denominations are alike? Add of a yard to 5 of an inch. Ans. 23 yards, or 14, inches. Add of a cwt. 85 lbs. and 3 oz. together. Ans. 2 qrs. 17 lbs. 1254 oz. How do you perform subtraction of vulgar fractions? when the numerator is the greater? What is to be done when the fractions are of different denominations? From of a league take of a mile. Ans. 1 m 2 fur. 16 po. Repeat the rule for performing multiplication of vulgar fractions. How will you prepare the given terms? Required the product of 4, of, and 183. Ans. 9140. What is the rule for per2 forming division of vulgar fractions? Divide 141 of by 31 of 6. Ans. 13. How will you prepare the fractions for stating in simple proportion? How will you then proceed? What is the rule? How will you prepare the fractions for stating in compound proportion? Repeat the rule. The questions may be continued by the teacher. 1440 1? MISCELLANEOUS MATTER IN 1. If 7 lbs. of coffee cost of a dollar, what is that per lb.? 7 of a dollar. Ans. 2. If 6 bushels of wheat cost 47 dollars, what is it per bush47=28=13=81 cents Ans. 3. How many times is 3 contained in 4621? Ans. 154. 4. If a man spend § of a dollar per day, how much will he spend in 8 days? x8=40-5 dollars. Ans. A fraction can be multiplied by multiplying the numerator, without changing the denominator. 5. At of a dollar for 1 pound, what will 12 pounds cost? x12=60=8=3 of a dollar. Ans. 5. 80 6. What will 913 tons of hay come to at 17 dollars per ton? Ans. 164 7. At of a dollar per yard, what cost 7 yards? dollars. Ans. 61 dollars. 8. Reduce,, and, to fractions having the least common denominator, and add them together. Thus, 12+2+32=11 =111 amount. 9. "What is the amount of of of a yard, 3, and of 2 yards? Reduce the compound fraction to a simple fraction, thus:of 3=3, and of 2=2; then, +3+3=123=153 yds. 10. Reduce of a hogshead to the fraction of a gallon. 11. Rednce 8 of a bushel to the fraction of a quart. of a day to the fraction of a minute. 13. Reduce 10 of a cwt. to the fraction of a pound. 12. Reduce 504 240 504 1260 14. Reduce of a hogshead to the fraction of a pint. 15. Reduce of a furlong to the fraction of a yard. 128 16. Reduce of a gallon to the fraction of a hogshead. 17. Reduce 8 2 of a quart to the fraction of a bushel. 18. Reduce of a minute to the fraction of a day. 240 480 504 19. Reduce of a pound to the fraction of a cwt. 20. Reduce of a pint to the fraction of a hhd. 21. At 63 dollars per barrel for flour, what will rel cost? of a bar. 10 Ans. 22 dollars 128 22. At D2 per yard, what cost 6 yards? Ans. D1432. 23. At D42 per yard, how many yards bought for D37? of cloth may be Ans. 8 yards. 24. How many times is 36 contained in 6 ? Ans. of 1. 25. If 43 pounds of butter serve a family 1 week, how many weeks will 36 pounds serve them? 26. How many times is contained in 47? 27. Multiply Ans. 8 weeks. of 2 by of 43. Ans. Multiply 1000000 by =5555555. Ans. 28. A man bought 27 gallons, 3 quarts, 1 pint, of molasses; what part is that of a hogshead? 29. From 4 days 7 hours, take 1 day 9 hours. Ans. 2 days, 22 hours, 20 minutes. 30. If 6 yards of cloth cost 3 dollars, what cost 91 yards? Ans. D4.269 31. If oz. of silver cost of a dollar, what cost 1 oz. ? Ans. D1.283, 32. If of a ship cost 251 dollars, what is of her worth? Ans. D53.785. 33. Add together E. 3D. and 1cts. 34. Take 33cts. from of 21 dollars. Ans. 7D. 53c. 26m Ans. 43 cts. Note. The reason of the rules, both for Addition and Subtraction of Fractions, is manifest. If the given fractions have the same denominator and are of the same denomination, the sum of the numerators written over the given denominator, will be the sum of the fractions. A fraction is subtracted from a whole number, by taking the numerator of the fraction from its denominator, and placing the remainder over the denominator, then taking one from the whole number, thus :- 41 113 Note. To multiply by a fraction, whether the multiplicand be a whole number or a fraction, as before observed, you can divide by the denominator of the multiplying fraction, and multiply the quotient by the numerator, thus: 20 multiplied by 3, the product is 15. To divide by a fraction, whether the dividend be a whole number or a fraction, we multiply by the denominator of the dividing fraction, and divide the product by the numerator, thus: 20 divided by is 263, and 12 divided by 3 is 16, &c. You will observe in multiplication, the multiplier being less than unity, or 1, will require the product to be less than the multiplicand; and in division, the divisor being less than unity or 1, it will be contained a greater number of times, and consequently will require the quotient to be greater than the dividend, to which it will be equal when the divisor is 1, and less when the divisor is more than 1. By Vulgar and Decimal Fractions. 35. If 2 bushels sow an acre, how many acres will 22 bushels sow? Thus, 23×4+3= improper fraction; 22 x 4÷ 11=8 acres Ans. Or, 22-2.75-8 Ans. 36. At 4 dollars per yard, how many yards of cloth may be bought for 37 dolls.? Thus, 4×5+2=22; 37×5÷22=822 yards Ans. D37÷4.40 8 yards, 1 quarter, 1 nail+Ans. 37. If 143 yards cost 75 dollars, how much per yard? Thus, 14x8+3=115; 75x8-600-115-5,25 Ans. Or, D75-14.375 yards = D5.21.7 Ans. 38. How many times is contained in 746? 23 Thus, 746x8-3-19891; 746-375-1989123 Ans. 39. Divide by 1; divide by 1-3; divide by 1=2. 40. If of a yard cost 5 dollars, how much per yard? 41. If of a bushel cost 3 dollars, how much for 3 bushels ? how much for a bushel? 8 3 4 5 8 9 42. Add , }; add fe, 16, 16, f=14; add 12, 1, 18, 2, {, }=}; 16, 16 24, together. 43. There are 3 pieces of cloth, one containing 73 yards, another 13 yards, and the other 15 yards, how many yards in all? First reduce the fractional parts to their least common denominator, thus :-- Adding together all the 64ths, namely, 48, 32, and 56, you have 136, that is 136-2; write down the fractions 64 under the other fractions, and carry the two integers to the other integer, and you have 371. Ans. Or, thus: 7.75+13.5+15.875=37.125. The last method is the most convenient. 44. From of an ounce take of a pwt. Ans. 11 pwt. 3 grs. 45. From 4 days, 7 hours, take 1 day 9 hours. 46. A man purchased he purchase? Ans. 2 days, 22 hours, 20 min. of 7 cwt. of sugar, how much did 47. 14 hours, 48 minutes, 54 seconds, is what part of a day? 48. Add lb. troy to of an oz. Ans. 6 oz. 11 pwt. 16 grs 49. At 35 dollars per cwt., what will 63 lbs. cost? 50. If of a yard cost cost? 51. Add together D., 52. Add of a yard, 53. Add of a week, minute, together. 3 of a dollar, what will of a yard cents, mills. of a foot, and 12 of a mile, together. of a day, 1 of an hour, and of a 54. From 5 weeks, 21 days, 9 hours, take 73 days, 147 hours. 55. Multiply of 5, by 4 of 3 of 113. of, and 3? Ans. 21 1 20 57. In a certain school of the pupils study Greek, o Latin, arithmetic, read and write, and 20 attend to other studies; required the number of pupils. Ans. 100. 58. In 342 gallons, how many of a gallon? Ans. 1371 of a gallon, or 1371 quarts. NOTE. Any whole number may be made an improper frac- . tion by drawing a line under, and putting a unit or 1 for a denominator, as 5 may be expressed thus, f, and 10, 0, &c. Reduce .275 to a vulgar fraction. Thus: 275 -25 com. m.=11. Ans. SECOND REVIEW OF VULGAR FRACTION. What do you mean by common divisor? by the greatest common divisor? How can you multiply a fraction by a whole number? How will you multiply a mixed number? When the multiplier is less than a unit, what is the product compared with the multiplicand? How do you multiply a whole number by a fraction? How can you multiply one fraction by another, and can you do it in more than one way? How do you multiply a mixed number by a mixed number? How can you divide a fraction by a whole number? What is understood by common denominator? the least common denominator? Do cases frequently occur in vulgar fractions, where the terms may be reduced, and solved by decimals? What can you say of the importance of a correct knowledge of all kinds of fractions? PRACTICE. PRACTICE is a contraction of simple proportion, when the first term happens to be a unit, or one, and has its name from its daily use among merchants and tradesmen; being an easy and concise method of working many questions which occur in trade and business; also, many questions in simple interest may be readily and easily solved by aliquot parts, after the pupil has made himself thoroughly acquainted with the tables, which he should commit to memory. One number is an aliquot, or even part of another when it forms an exact part of it, as 25c. is an aliquot part of a D=1; 75c.; 10c.=1; 5c.=2%, &c. 56 lb. is an aliquot part of a cwt.; 28 lbs.; 14 lbs.; 7 lbs =, &c 20 |