NOTE. When the operation is performed by Short Division, the several quotients must be placed under their respective denominations. 2. When 10£. 13s. Od. are paid for 9 barrels of flour, what is the cost of one barrel ? 3. If 7bbls. of flour cost 10£. Os. 41d., what cost one barrel ? 4. Paid 3£. 2s. 4d. for 8lbs. of Cayenne pepper; what was the cost of one pound? 5. Divide 54yd. 2qr. 3na. equally among 5 persons. 6. Divide 29cwt. 3qr. 16lb. equally among 9 persons. 7. If 5 pair of oxen in one year consume 37 tons 17cwt. Oqr. 161b. of hay, what quantity would be sufficient for one pair? 8. If one man could perform a piece of labor in 76 days 11h. 53m., how long would it take 10 men to perform the same labor? 9. Divide 151£. 19s. 114d. by 9. 10. Divide 350£. 17s. 3 d. by 519. 11. Divide 225£. 1s. 104d. by 63. 12. Divide 159 £. 4s. 9d. by 12. 13. Divide 75£. 6s. 6d. by 4. 14. Divide 111£. 8s. 11d. by 7. 15. Divide 159£. 4s. 9d. by 12. Ans. 16£. 17s. 91d. Ans. 13s. 6d. Ans. 3£. 11s. 51d. Ans. 13£. 5s. 4 d. Ans. 18£. 16s. 74d. 16. Divide 158lb. 9oz. 1dwt. 21gr. by 9. 17. Divide 111lb. 11oz. 17dwt. 9gr. by 7. 18. Divide 183 tons lcwt. 2qr. 11lb. 4oz. by 11. 19. Divide 105 tons 7cwt. Oqr. 6lb. 8oz. by 8. 20. Divide 19 93 43 27 9gr. by 8. 21. Divide 335yd. 2qr. Ona. 0 in. by 7. 22. Divide 215m. 7fur. 1rd. 14ft. 6in. by 12. 23. Divide 149deg. 8m. Ofur. Ord. lyd. 1ft. 6in. by 9. 24. Divide 97deg. 55m. 7fur. 35rd. 4ft. 2in. 1bar. by 6. 25. Divide 181A. 3R. 11p. 6yd. 4ft. 41in. by 11. 26. Divide 31 cords 83ft. 1332in. by 4. 27. Divide 209hhd. 55gal. 3qt. Opt. 1gi. by 7. 28. Divide 35 tuns 3hhd. 4gal. 2qt. by 9. 29. Divide 56hhd. 45gal. by 8. 30. Divide 118bu. 1pk. 5qt. by 6. 31. Divide 255ch. 24bu. 3pk. 1qt. by 7. 32. Divide 110y. 343d. 3ḥ. 41m. 12sec. by 8. 33. A man divides his farm of 214A. 3R. 12p. equally among his 9 sons; how much does each receive? 34. If one man perform a certain piece of labor in 3da. 16h. 54m., how long would it take 12 men to perform the same work? 35. A farmer has 29 bushels of rye, which he wishes to pu into 8 bags; how much must each bag contain? CASE II. When the divisor is a composite number, proceed as in the following EXAMPLE. 36. If 42 yards of cloth cost 14£. 3s. 6d., what is the value of 1 yard? Ans. 6s. 9d. £. S. d. 7)14 3 6 6)2 0 6 06 9 In this question, we find the component parts of 42 are 6 and 7; we therefore first divide the price by 7, and then divide the quotient by 6. From the above, we deduce the following RULE. Divide the dividend by one of the component parts, and the quotient thence arising by the other, and the last quotient will be the answer. NOTE. To find the true remainder; multiply the last remainder by the first divisor, and to the product add the first remainder. 37. If 16 yards of velvet cost 2£. 18s. 8d., what will 1 yard cost? 38. If 72 yards of broadcloth cost 71£. 14s. Od., what is the value of 1 yard ? 39. If 84 yards of cotton cost 8£. 1s. Od., what will 1 yard cost? 40. If 90 hogsheads of sugar weigh 56T. 13cwt. 3qr. 10lb., what is the weight of 1 hogshead? 41. What will be the price of 1 sheep, if 18 cost 5£. 4s. 3d.? 42. If 21 yards of cloth cost 10£. 8s. 3d., what is the price of 1 yard? 43. What is the value of 1 hat, when 22 cost 12£. 13s. Od. ? 44. When 96 shares of a certain stock are valued at 1290£. 4s. Od., what would be the cost of 1 share? 45. If 120 spoons weigh 32lb. 9oz. 15dwt., what does 1 weigh? 46. If a man in 1 month travel 746m. 5fur. Ord., how far does he go in 1 day? 47. If the earth revolve 15° on its axis in 1 hour, how far does it revolve in 1 minute? 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 365į 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. Ina. Oțin., 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. 111⁄2ft., 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. ls. 330d. 61. Divide 275ch. 19bu. 2pk. equally among 17 men. Ans. 4T. 15cwt. 2qr. 12,2 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, 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=1. 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 8) 16(2 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 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 |