48. Divide 1275A. 2R. 16p. 22yd. 8ft. 32in. equally among 32 men. 49. If a man walk round the earth in 2y. 68d. 19h. 54m., how long would it take him to walk 1 degree, allowing 365 days to a year? The following questions are to be performed as the second example of this section. 50. If 53 tons of iron cost 1001£. 9s. 7d., what is the value of 1 ton? 51. If 57 gallons of wine cost 23£. 11s. 51d., what cost 1 gallon? 52. Divide 3419A. 2R. 23p. by 29. 53. If 89 pieces of cloth contain 3375yds. 3qr. 1na. Oin., how much does 1 piece contain ? 54. If 59 casks contain 44hhd. 52gal. 2qt. 1pt. of wine, what are the contents of 1 cask? 55. If a man travel in 1 year (365 days) 6357m. 5fur. 14rd. 111ft., how far is that per day? 56. When 175gal. 2qt. of beer are drunk in 52 weeks, how much is consumed in 1 week? 57. When 17 sticks of timber measure 15T. 38ft. 1074in., how many feet does 1 contain ? 58. Divide 132 cords 2ft. by 17. 59. Divide 89hhd. 52gal. 3qt. 1pt. by 39. 60. Divide 179bu. 3pk. 5qt. Opt. 1gi. by 53. Ans. 2. 1s. 330d. 61. Divide 275ch. 19bu. 2pk. equally among 17 men. Ans. 4T. 15cwt. 2qr. 12,26 lb. 66. Divide 916m. 3fur. 30rd. 10ft. 6in. by 47. Ans. 19m. 3fur. 39rd. 13ft. 22 in. 67. Divide 718A. 3R. 37p. by 29. Ans. 9A. 1R. 19p. 1398 ft. 69. Divide 144A. 3R. 18p. 3yd. 1ft. 36in. by 11. Ans. 13A. OR. 27p. 3yd. Oft. 45 in. 70. Divide 6718£. 19s. 11d. by 47. Ans. 142£. 19s. 13 d. 71. Divide 1237£. 17s. 4d. by 86. Ans. 14£. 7s. 1044d. 72. Purchased 18T. 17cwt. 3qr. 20lb. of copperas, at 4 cents per pound. I sold 4T. 6cwt. 1qr. 14lb. at 5 cents per pound, and 7T. lcwt. 3qr. 10lb. at 6 cents per pound. Moses Atwood purchased one fourth of the remainder at 6 cents per pound. One half of what then remained I sold to J. Gale at 10 cents per pound. The remaining quantity I sold to J. Smith at 12 cents per pound; but he has become a bankrupt, and I lose half my debt. What have I gained by my purchase? Ans. $1001.34. QUESTIONS TO BE PERFORMED BY ANALYSIS. 1. If 7 pair of shoes cost $8.75, what will one pair cost? what will 20 pairs cost? Ans. $25.00. 2. If 5 tons of hay cost $85, what will 1 ton cost? what will 17 tons cost? Ans. $289.00. 3. When $0.75 are paid for 3gal. of molasses, what is the value of 1gal.? What cost 37gal. ? Ans. $9.25. 4. Gave $1.92 for 4lbs. of tea; what cost 1lb. ? what cost 37lbs. ? Ans. $17.76. 5. For 12lbs. of rice I paid $ 1.08; what was paid for 1lb.; and what must I give for 25lbs. ? 6. Gave S. Smith $ 63.00 for 9 tubs of butter; what was the cost of 1 tub? Ans. $189.00. What cost 27 tubs ? Ans. $2.25. 7. T. Swan can walk 20 miles in 5 hours; how far can he walk in 1 hour? How long would it take him to walk from Bradford to Boston, the distance being in a straight line 28 miles? Ans. 7 hours. 8. If a hungry boy would eat 49 crackers in 1 week, how many would he eat in 1 day? how many would be sufficient to last him 19 days? Ans. 133 crackers. 9. Gave $20 for 5 barrels of flour; what cost 1 barrel? what cost 40 barrels ? Ans. 160.00. 10. For 3lbs. of lard there were paid 36 cents; what was the cost of 37lbs. ? Ans. $4.44. 11. Paid F. Johnson 72 cents for 9 nutmegs; how many cents were paid for 1 nutmeg; and what should be charged for 37 nutmegs ? Ans. $2.96. 12. Paid 2. 17s. 5d. for 52lbs. of sugar; what cost 1lb.? what cost 76lbs. ? 13. Paid 4£. 3s. 11d. for 76 pounds of 52lbs. ? Ans. sugar; what cost Ans. 14. If 52lbs. of sugar cost 2£. 17s. 5d., how many pounds can be purchased for 4£. 3s. 11d. ? Ans. 15. When 4£. 3s. 11d. are paid for 76lbs. of sugar, how ny pounds can be obtained for 2£. 17s. 5d. ? Ans. 16. Bought 20 bushels of wheat for 8£. 3s. 11d.; what cost 1 bushel? what cost 200 bushels? Ans. Ans. 17. Paid E. Bradley 81£. 19s. 2d. for 200 bushels of wheat; what cost 20 bushels? 18. Mr. Day paid 3£. 4s. 2d. for 10yds. of cloth; what should he have paid for 97yds.? Ans. 31£. 2s. 5d. 19. If 8 barrels of flour cost 2£. 12s., what cost 29 barrels ? Ans. 9£. 8s. 6d. 20. If 17 bushels of wheat cost 6£. 13s. 2d., what cost 101 bushels? Ans. 39£. 11s. 2d. 21. Gave 10£. 4s. 3d. for 19 yards of cloth; what cost 97 yards? Ans. 52£. 2s. 9d. SECTION XVI. VULGAR FRACTIONS. FRACTIONS are parts of an integer, or whole number. An integer is any whole number or quantity, as 1, 7, 11, &c:, or a pound, a yard. VULGAR FRACTIONS are expressed by two numbers, called the Numerator and Denominator; the former above, and the latter below a line. The Denominator shows into how many parts the integer, or whole number, is divided. The numerator shows how many of these parts are taken, or expressed by the fraction. 1. A proper fraction is one whose numerator is less than the denominator; as §. 2. An improper fraction is one whose numerator exceeds or is equal to the denominator; as org. 3. A single or simple fraction consists of but one numerator and one denominator; as . 4. A compound fraction is a fraction of a fraction, connected by the word of; as 7 of & of § of §. 5. A mixed number is an integer with a fraction; as 7, 58. 6. A complex fraction is a fraction having a fraction or a mixed number for its numerator or denominator, or both; as, 77 £ 1 81 ช "or 97 7. The terms of a fraction are the numerator and denominator; the numerator being the upper term, and the denominator the lower. 8. The greatest common measure of two or more numbers is the largest number that will divide them without a remainder. 9. The least common multiple of two or more numbers is the least number that may be divided by them without a remainder. 10. A fraction is in its lowest terms, when no number but a unit will measure both its terms. 11. A prime number is that which can be measured only by itself or a unit; as 7, 11, and 19. 12. Numbers are said to be prime to each other, when only a unit measures or divides them both without a remainder; thus, 7 and 11 are prime to each other. 13. Prime factors of numbers are those factors which can be divided by no number but by themselves or a unit; thus the prime factors of 21 are 7 and 3. 14. An even number is that which can be divided into two equal whole numbers. 15. An odd number is that which cannot be divided into two equal whole numbers. 16. A square number is the product of a number multiplied by itself. 17. A cube number is the product of a number multiplied by its square. 18. A composite number is that produced by multiplying two or more numbers together. 19. The factors of a number are those whose continued product will exactly produce the number. 20. An aliquot part is that which is contained a precise number of times in another. 21. An aliquant part is such a number as is contained in another a certain number of times with some part or parts over. 22. A perfect number is that which is equal to the sum of all its aliquot parts, or is equal to the sum of all the numbers that will divide it without a remainder; thus 6 is a perfect number, because it can be divided by 3, 2, and 1; and the sum of these numbers is 6. But 12 is not a perfect number, because its aliquot parts are more than 12; thus 6+4+3+1 14. 8 is not a perfect number, because its aliquot parts are less than 8; thus 4+2+1=7. But 28, 496, and 8128 are perfect numbers. The chief use of a knowledge of these numbers is in the higher branches of mathematics. 23. A fraction is equal to the number of times the numerator will contain the denominator. 24. The value of a fraction depends on the proportion which the numerator bears to the denominator. 25. Ratio is the relation which two numbers or quantities of the same kind bear to each other, and may be found by dividing one number by the other. For example, the ratio of 12 to 4 is 3, because 12÷4=3; and the ratio of 4 to 8 is, because 4 by 8 = 4. CASE I. To find the greatest common measure of two or more numbers, or to find the greatest number that will divide two or more numbers, without a remainder. RULE. Divide the greater number by the less, and, if there be a remainder, divide the last divisor by it, and so continue dividing the last divisor by the last remainder until nothing remains, and the last divisor is the greatest common measure. If there be more than two numbers, find the greatest common measure of two of them, and then of that common measure and the other numbers. If it should happen that 1 is the common measure, the numbers are prime, to each other, and are incommeasurable. The above rule may be illustrated and demonstrated by the following example. Let it be required to find the greatest common measure or divisor of 24 and 88. . According to the rule, we first divide 88, the greater number, by 24, the less; for it is evident that no number greater than the less of two numbers can measure or divide those numbers. As therefore 24 will exactly measure or divide itself, if it will also divide 88, it will be the greatest common divisor sought. OPERATION. 24)88(3 72 16)24(1 16 Now we find that 24 will not exactly measure or divide 88, but there is a remainder, 16. 24, therefore, is not the common divisor of the two numbers. Now as 72, the number which we subtracted from 88, is an exact multiple of 24, we know that any number 8)16(2 which will exactly measure or divide 24 will also divide 72; and as 16, the remainder of the division of 88 by 24, is that part of 88 16 |