at 64331 dollars, as follows: 5630 dollars to his wife, 1245 dollars for charitable purposes, the remainder to be divided equally among his seven children; how much did each child receive? 5. The salary of the President of the United States is 25000 dollars a year; how much will he save in one year, if he spend 50 dollars a day? 6. A merchant bought a ship for 27342 dollars, and paid for cargo 37564 dollars; he sold the ship and cargo in California for 164000 dollars; how much did he gain, after deducting 2560 dollars for the expenses. of the voyage? 7. If you were to count 2 every second for twelve hours a day, how long would it take to count a million? 8. What is the difference. between 644X46 and 61543313? 9. A gentleman left his estate, valued at 16596 dollars, to be divided between his wife and three children in the following manner: his wife was to have one third of the whole estate; the oldest child was to receive one third of what was left; the remainder was to be equally divided between the two youngest; what did each receive? 10. The sum of two numbers is 1496, one of which is 984; what is the other number? 11. If a man's income be 1650 dollars a year, and he expends 24 dollars a week, how much will he have left at the end of the year? 12. The difference between two numbers is 965, the greater is 1875; what is the less number ? 13. A merchant owes 14560 dollars, which is less than what is due him by 9560 dollars; what is the sum due him ? 14. The difference between two numbers is 564, and the less number is 896; what is the greater number? 15. The product of two numbers is 1296, and one of the numbers is 9; what is the other? 16. If the quotient be 345, and the divisor 64, what must the dividend be? 17. If the remainder of a sum in division be 20, the quotient 423, and the divisor the sum of the quotient and remainder, plus 19, what will be the dividend? 18. There are two numbers, the greater of which is 37 times 45, their difference 4 times 19; what is their sum and product? 19. If the dividend be 1728, and the quotient 24, what will be the divisor? 20. There are three numbers, whose continued product is 3456; one of the numbers is 12, another is 18; what is the other number? 21. The sum of two numbers is 1296, their difference. is 144; what are the numbers ? 22. The sum of three numbers is 640, the difference between the least and the greatest is 220, and the difference between the middle number and the sum is 460; what are the numbers? 23. A grocer bought a hogshead of molasses, containing 133 gallons, at 27 cents a gallon; but 19 gallons having leaked out, he sold the remainder at 39 cents a gallon; did he gain or lose, and how much? 24. What is the difference between 19 times 144 and 6732 divided by 17? EXAMPLES IN THE PRECEDING RULES, WITH SIGNS. The signs of addition, multiplication, subtraction, and division have already been explained, Art. 27. 43. A vinculum, 9 -, or parenthesis, is used to collect several quantities into one. Thus 5+4X 6, or (5+4) X.6, signifies that the sum of 5 and 4 is to be multiplied by 6, which is 54. Without the vinculum What is a vinculum? 5+4×6, only 4 is multiplied by 6, and the 5 is added to the product, making 29. Also, 12-3x3, or (12—3) × 3, signifies that the difference between 12 and 3 is to be multiplied by 3, which is 27. Without the vinculum, the product of 3 by 3 is subtracted from 12, which leaves 3. 25. 6+5×4 ? 32. 84×6+8×9÷÷÷12— ? 39. Write the sum of the products of 8 into 9, and 7 into 4. 40. Write the difference of the products of 8 into 9, and 7 into 4. 41. Write the product of the sum of 8 plus 9, and 7 plus 4. 42. Write the product of the difference of 9 minus 8, and 7 minus 4. 43. Write 24 divided by the product of 4 into 2. 44. Write the quotient of 24 for a dividend, and 3 for a divisor. 45. Write the quotient of 24 divided by the product of 3 and 2. THE DIVISIBILITY OF NUMBERS. 44. The divisibility of a number is its property of being exactly divided by another. Thus 36 is divisible by 2, 3, 4, 6, 9, 12, and 18, because each of these numbers will divide it without a remainder. What is the divisibility of a number? 45. Every number divisible by another is also divisible by each one of its factors. Thus 48 is divisible by 12, also by 4 and 3, and 2 and 6, factors of 12. 46. Every number divisible by another is called a multiple of that number. If it be divisible by two or more numbers, it is a common multiple of those numbers. Thus 24 is the common multiple of 3, 4, 6, 8, and 12. 47. The least number divisible by two or more numbers is the least common multiple of those numbers. Thus 12 is the least common multiple of 3, 4, and 6. 48. Every number that will divide exactly another number is called the measure of that number. Thus 4 is the measure of 8. 49. Every number that will divide exactly two or more numbers is their common measure. Thus 6 is the common measure of 12 and 18. 50. The greatest number that will divide exactly two or more numbers is their greatest common measure, Thus 8 is the greatest common measure of 24 and 56. 51. Every number that will divide exactly another number will also divide any multiple of that number. Thus 6 will divide 12; it will also divide 24, 36, 48, &c. 52. Every number that will divide exactly two other numbers will also divide their sum, their difference, and their product. Thus 3 will divide 6 and 15; it will also divide 6+15=21, 6×15-90, and 1.5-6-9. 53. Every number divisible by 2 is called an even. number; when not divisible by 2, an odd number. 54. Every number is divisible by 3 when the sum What is a multiple? What is a common multiple? What is the least common multiple? What is the measure of a number? What is the common measure? What is the greatest common measure? What is an even number? What is an odd number? What numbers are divisible by 3? D* of its figures, considered as units, is divisible by 3. Thus 321 is divisible by 3, since 3+2+1-6 is divisible by 3. 55. Every number is divisible by 4 when its two right hand figures are divisible by 4. Thus 112, 116, 128, are divisible by 4, since 12, 16, 28, are divisible by 4. 56. Every number is divisible by 5 when its right hund figure is 0 or 5. Thus 10, 15, 20, 25, are all divisible by 5. 57. Every even number is divisible by 6 which is divisible by 3. Thus 12, 18, 24, 30, are all divisible by 6. 58. Every number is divisible by 9 when the sum of its figures, considered as units, is divisible by 9. Thus 8172 is divisible by 9, since 8+1+7+2=18 is divisible by 9. PRIME AND COMPOSITE NUMBERS. 59. Whole numbers are either prime or composite. 60. A prime number is one which cannot be formed by multiplying together any two or more whole numbers greater than a unit, as 1, 2, 3, 5, 7, 11, 13. 61. A composite number is one which may be formed by multiplying together two or more whole numbers greater than a unit, as 4, 6, 9, 8, 10, 12. Овs. A prime number cannot be divided by another number, except itself and a unit. 62. Every composite number may be resolved into prime numbers, which are called prime factors. 63. To find the prime factors of any composite. number, - RULE. Divide the given numbers in succession by What numbers are divisible by 4? What by 9? What are prime numbers? bers? What are prime factors? What by 5? What by 6? |