40. If 3500 is the minuend and 2687 the subtrahend, what is the remainder? 41. The salary of the President of the United States is $ 50000 a year; what sum may he expend each year, and yet save $ 75584 in 4 years, his term of office? 42. If the divisor is 19, the quotient 37, and the remainder 11, what is the dividend? 43. The difference of two numbers is 367854, and the greater number is 897325; what is the smaller? What is the sum of the two numbers? 44. A field of 12 acres yields 16.5 bushels of wheat an acre. The labor and seed cost $164.25, and the wheat brought $1.12 a bushel. How much profit was made from the field? 45. A cattle dealer deposited in the bank $10481 Monday and drew out $4550; on Tuesday he deposited $ 2900 and drew out $5875; on Wednesday he deposited $13470 and drew out $8645; how much of this money had he remaining in the bank? 46. If a town containing 23040 acres is divided into farms of 80 acres each, how many farms will there be, and what will they be worth at $65 an acre? 47. A drover bought 164 cows at $52 each, 95 at $59 each. He paid two dollars a head for transportation and sold the lot for $15457.50. Did he gain or lose, and how much? 48. The area of Massachusetts is 8315 square miles, of Texas 265780 square miles, and of Spain 191367 square miles. How many States as large as Massachusetts could be made out of what would remain of Texas after taking away an area as large as Spain? 49. If a man's annual income is $2250, and he puts in the Savings Bank $700, spends $ 350 for rent, $ 427.50 for provisions, $211.75 for servant, $75 for coal and wood, $200 for clothing, $76.75 for travelling expenses, $153.50 for other expenses, and the balance in books at an average price of $1.50 apiece, how many books does he buy? 50. How many pounds of butter at 23 cents a pound must be given for 5 pounds of raisins at 11 cents a pound, 2 pounds of tea at 63 cents a pound, and a barrel of sugar at $9? 51. The area of Ireland is 32481 square miles, and 5023984 acres are uncultivated. How many more acres are cultivated than three times the number uncultivated, there being 640 acres in a square mile? 52. If a man earns $ 29 a week and pays $3 a week for the board of each of his 2 sons, how much in the course of a year, or 52 weeks, does he have for other expenses? 53. A farmer sold 23 tons of hay at $18 a ton, and received in payment a yoke of oxen at $125, 5 cows at $30 each, 15 sheep at $4 each. Then, to balance the account, he agrees to take 2 cows more at the same price, and the rest as far as possible in sheep at $4 each, and the balance in money. How many more sheep and how much money did he receive? 54. A flour dealer paid $4914 for 819 barrels of flour, and sold it at $6.50 a barrel. Did he gain or lose, and how much? 55. Taking the areas as given in Ex. 100, page 26, how many States as large as Massachusetts could be made out of Maine, and how many square miles would be left? 56. How many States as large as Rhode Island could be made out of Maine, and how many square miles would be left? 57. Ohio contains 41060 square miles. How many States like Massachusetts could be made out of it, and how many square miles would be left? 58. A merchant bought 5 barrels of sugar, each barrel containing 235 pounds, at an average price of 9 cents a pound. He sold one barrel at 8 cents a pound, one at 9, two at 10, and one at 11. How much did he gain? 59. In a certain High School building one room will seat 75, 2 will seat 54 each, 4 will seat 48 each, 2 will seat 36 each, and 1 more will seat 40. How many pupils in all can be seated in the building? PROPERTIES OF NUMBERS. 85. When in division there is no remainder, the dividend is said to be divisible by the divisor, and the divisor is called an exact divisor. When there is a remainder, the dividend is said to be indivisible by the divisor. 86. An Even Number is a number that is divisible by 2; as, 2, 4, 8, 12. 87. An Odd Number is a number that is not divisible by 2; as, 1, 3, 5, 11, 19. 88. A Prime Number is a number that is not divisible by any whole number except itself and one; as, 1, 2, 3, 5, 7, 11, 19. NOTE. by 2. Two is the only even prime number, for all even numbers are divisible 39. Numbers are mutually prime (that is, prime to each other) when no whole number but one will divide all of them without remainder; thus, 8 and 9 are mutually prime, although neither 8 nor 9 is absolutely prime. 90. A Composite Number is a number that is divisible by other numbers besides itself and one; thus, 6 and 33 are composite, as 6 = 3 x 2, and 33 = 3 x 11. FACTORING. 91. Oral Exercises. 1. What two numbers multiplied together give 6 9 12? 15? 18? 2. What numbers multiplied together give 22? 25? 38? 39 ? 44? 3. 51 is the product of what numbers? 57? 63? 81? 93? 4. 76 is the product of what numbers? 87? 91? 111? 121? 92. The Factors of a number are those numbers whose continued product is the number; thus, 3 and 7 are the factors of 21; 3 and 6, or 3, 3, and 2, are the factors of 18; etc. NOTE. Every number is a factor of itself, the other factor being 1. In speaking of the factors of a number, we shall exclude the number itself and 1. 93. The prime factors of a number are those prime numbers whose continued product is the number; thus, the prime factors of 12 are 2, 2, and 3; the prime factors of 36 are 2, 2, 3, and 3; etc. 94. To factor a number is to separate it into its factors. In separating a number into its factors, The following facts will be found convenient : (a.) Every number whose unit figure is 0, or an even number, is divisible by 2. (b.) Any number is divisible by 3 when the sum of its figures is divisible by 3; thus, 4257 is divisible by 3, as the sum of its figures, 4+2+5+7=18, is divisible by 3. (c.) Any number is divisible by 4 when 4 will divide the number expressed by the two right-hand figures; thus, 4 will divide 32, hence it will divide 7532. (d.) Any number whose unit figure is 0 or 5 is divisible by 5; as, 90, 1740, 35, 34975, etc. (e.) Any even number which is divisible by 3 is also divisible by 6; thus, 3528 is divisible by 3 and hence by 6. (f.) For 7 no general rule is known. (g.) Any number is divisible by 8 when 8 will divide the number expressed by the three right-hand figures; thus, 8 will divide 816, hence it will divide 175816. (h.) Any number is divisible by 9 when the sum of its figures is divisible by 9; thus, 7146 is divisible by 9 because the sum of its figures, 7+1+4+6=18, is divisible by 9. (i.) Any number whose unit figure is 0 is divisible by 10. NOTE. For explanation of these statements, see Appendix, page 336. Also, for table of prime numbers, see Appendix, page 337. Divide the given number by any prime number greater than one, that will divide it; divide the quotient thus found by any prime number greater than one that will divide it, and so on till the quotient is prime. The several divisors and last quotient will be the prime factors sought. 6. What are the prime factors of 40? 7. Separate 84 into its prime factors. 8. Separate 250 into its prime factors. 9. What are the prime factors of 735? 10. What are the prime factors of 9800 ? Ans. 2, 2, 2, and 5. |