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SECTION XI.

MENTAL OPERATIONS IN FRACTIONS.

NOTE. — The pupil is requested, before he commences the mental operations of Fractions, to examine carefully the definitions on page 78.

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1. If į of a barrel of cider cost 2 dollars, what will a barrel cost ?

2. Bought of a barrel of flour for 2 dollars, what would be the price of a barrel ?

3. If { of a yard of cloth cost 3 dollars, what will cost ? What will cost ? What will a yard cost ?

을 4. If í of a barrel of flour cost 2 dollars, what will cost ? $? ?? ?? $? ?? ?? ?

5. If of a pound of pepper cost 3 cents, what will 3 ? ?

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6. Bought of a barrel of sugar for 2 dollars, what was the price of i? ? ? ?

7. If a man bought of a firkin of butter for 7 cents, what did the whole firkin cost ?

8. Bought in of an acre of land for 5 dollars, what was the price of an acre ? Of 2 acres? Of 10 acres ? Of 20 acres ?

9. Bought of a pound of ginger for 4 cents, what did a pound cost ?

10. Bought of a chest of tea-for 5 dollars, what did i cost ? ? ? ? 18 ?

11. Bought I of a load of hay for 3 dollars, what did the load cost ? What did 2 loads cost ?

12. If į of a cask of wine cost 9 dollars, what was that a cask ? 13. 2 is of what number? 4? 5? 6? 8? 9? 10? 12?

} 14. 4 is į of what number ? 5? 6? 8? 9? 11? 15? 16? 15. 6 is of what number? 77 8? 9? 11? 13?18? 20? 16. 7 is of what number? 4? 5? 6? 8? 11? 15? 25? 17. 9 is of what number? 7? 2? 8? 11? 12? 13? 17? 18. 8 is of what number? 7? 11? 12? 15? 10? 11? 25? 19. 10 is į of what number? 6? 7? 8? 9? 11? 15? 18? 20. 12 is 4 of what number? 14? 15? 16? 17? 18? 19? 21. 11 is of what number? 6? 3? 2? 1? 7? 18? 14? 22. 14 is of what number? 11? 12? 13? 15? 16? 17? 23. 15 is of what number? 21? 18? 13? 14? 17? 19?

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24. 7 is to of what number? 10? 11? 12? 16? 18? 13?
25. 8 is of what number? 6? 7? 6? 5? 2? 7? 18?
26. If of a pound of sugar cost 4 cents, what will of a

} pound cost? What will a pound cost? 4 is of what number?

27. If of a pound of nutmeg cost 6 cents, what will of a pound cost? If of a pound cost 2 cents, what will a

1 pound cost? 6 is of what number?

을 28. If} of a cwt. of starch cost 10 dollars, what will of a cwt. cost? What will a cwt. cost? 10 is of what number?

29. If of a barrel of molasses cost 4 dollars, what is the value of } of a barrel? If } of a barrel cost 2 dollars, what will a barrel cost? 4 is į of what number?

30. If} of a pound of candles cost 15 cents, what will } of a pound cost? If of a pound cost 3 cents, what will a pound cost? 15 is į of what number?

31. Bought of a pound of mace for 20 cents, what was the cost of of a pound? If of a pound cost 4 cents, what will a pound cost? 20 is of what number?

es 32. If it of a pound of opium cost 55 cents, what will i of a pound cost? If i pf a pound cost 5 cents, what will a pound cost? 55 is 1 of what number?

33. 6 is 3 times what number?. 9? 12? 15? 18? 21? 24? 34. 15 is 3 times what number? 20? 25? 35? 65? 70? 35. 20 is 4 times what number? 4? 8? 12? 16? 40? 50? 36. 21 is 7 times what number? 14? 28? 56 ? 63? 70? 37. 24 is 2 times what number? 26? 30? 37? 39? 40? 38. 25 is 5 times what number? 80 ? 85? 95? 100? 200? 39. 3 is 6 times what number?18? 21? 28? 31? 40? 60? 40. 18 is 9 times what number? 6? 7? 18? 30? 37? 28?

41. If 6 be of some number, what will be of the same number? If 3 be of a number, what is that number?

} 42. 3 is of what number? 43. Iff of a pound of sugar cost 12 cents, what will of a

自 pound cost? If of a pound cost 2 cents, what will a pound cost?

44. If i of a pound of nutmegs cost 20 cents, what will ji cost? If i cost 5 cents, what will a pound cost? 20 is ii of what nurnber?

45. Ify of a quarter of veal cost 64 cents, what will } of a pound cost? If y of a pound cost 8 cents, what will a pound cost ? 64 is of what number?

46. Bought of a ton of potash for 84 dollars, what did |

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of a ton cost? If of a ton cost 12 dollars, what did a ton cost? 84 is of what number?

47. Bought } of a barrel of pork for 14 dollars, what was the cost of of a barrel? If 4 of a barrel cost 2 dollars, what was the price of a barrel? 14 is į of what number?

48. Bought of a barrel of fish for 18 dollars, what did of a barrel cost? If of a barrel cost 9 dollars, what was the value of a barrel? 18 is į of what number?

49. If i of an acre of land cost 42 dollars, what is the value of is of an acre? If is of an acre cost 6 dollars, what is the value of an acre? 42 is 1a of what number?

50. If no of a bushel of apples cost 35 cents, what will in of a bushel cost? If is of a bushel cost 5 cents, what is the value of a bushel? 35 is to of what number?

51. If į of a bushel of corn cost 63 cents, what is of a bushel worth? If į of a bushel cost 9 cents, what is the value of a bushel? 63 is of what number?

7 52. If 4 of a pound of saleratus cost 8 cents, what is ti of a pound worth? If I of a pound cost 2 cents, what does a pound cost?

53. If 11 of a load of hay is sold for 14 dollars, what is of a load worth? If of a load cost 2 dollars, what will 3 loads cost? 14 is 11 of what number?

54. If 10 cents will buy of a pound of figs, how many cents will buy a pound?

55. If 12 dollars will buy i of a cwt. of sugar, how many dollars will it take to buy a cwt.? 56. 8 is, of what number? 7? 6? 11 12? 15? 19? 20?

1 / 3 57. 9 is of what number? 6? 7? 8? 15? 17?18? 23? 58. 12 is of what number? 13? 16? 17? 19? 25? 27? 59. 4 is of what number? 11 ? ? ? 16? 17? 19? 21?

2 / 4 60. 3 is i of what number? 4? 5? 6? 8? 9? 11? 18? 61. 15 is iš of what number? 16? 18? 20? 25? 26? 30? 62. 18 is of what number? 3? 4? 5? 6?-7? 8? 11? 63. 20 is of what number? 10? -8? 18? 21? 27? 28? 64. 21 is 11 of what number? 27? 28? 29? 33? 37? 38? 65. 25 is ja of what number? 11? 16? 17 ? 18? 27? 36 ? 66. 24 is î of what number? 28? 29? 30? 37? 38? 61? 67. 27 is is of what number? 36? 42? 58 ? 592 61? 72? 68. 30 is of what number? 35? 40? 45? 55? 67? 85? 69. If y of a bushel of salt cost 42 cents, how many pounds of raisins at 6 cents a pound, would it take to purchase a bushel?

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70. If i of a load of hay cost 18 dollars, how many barrels of cider at 3 dollars a barrel would it take to purchase a load?

18 is i of how many times 3? 71. Bought of a yard of cloth for 60 cents ; how many pounds of sugar at 9 cents would pay for one yard? 60 is of how many times 9?

72. A man sold a pair of oxen for 48 dollars, which was of what they cost him ; he had paid for them in calves at 6 dollars a head. How

many calves did it take to pay for the oxen? 73. 28 is of how many times 7? 8? 9? 10? 11? 12? 13?

e 74. 12 is i of how many times 6? 77 8? 12? 15? 16? 75. 32 is z of how many times 10? 12? 13? 16? 17? 18? 76. 60 is of how many times 14? 7? 21? 24? 28? 30? 77. 72 is g of how many times 9? 10? 11? 12? 15? 16? 78. 64 is of how many times 4? 5? 6? 77 8? 9? 12? 79. 50 is of how many times 5? 6? 8? 9? 10? 12? 16? 80. 36 is of how many times 14? 16? 17? 18? 19? 20?

VULGAR FRACTIONS.

Fractions are parts of an integer.

Vulgar Fractions are expressed by two numbers, called the Numerator and Denominator ; the former above, and the latter below a line.

Numerator

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Denominator 11

The Denominator shows into how many parts the integer, or whole number, is divided.

The Numerator shows how many of these parts are taken. 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 i} or, 3. A simple fraction has a numerator and denominator only;

as , 4 4. A cor pound fraction is a fraction of a fraction, connect

ed by the word of; as į of off of . 5. A mixed number is an integer with a fraction; as 7, 5%. 6. A compound mixed fraction is one, whose numerator or denominator, or both, is a mixed number ; as 73 or 4

7. The common measure of two or more numbers is the

largest number, that will divide them without a remainder. 8. The least common multiple of two or more numbers is

the least number, that may be divided by them without a

remainder. 9. A fraction is in its lowest terms, when no number but a

unit will measure both its terms. 10. A prime number is that, which can be measured only by

itself or a unit; as 7, 11, and 19. 11. A perfect number is equal to the sum of all its aliquot

parts ; as 6, 28, 496, &c. 12. A fraction is equal to the number of times the numerator

will contain the denominator. 13. The value of a fraction depends on the proportion, which

the numerator bears to the denominator.

CASE 1.

To find the greatest common measure of two or more numbers, or to find the greatest number, that will divide two or more numbers.

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 incommensurable.

EXAMPLE. 1. What is the greatest common measure of 98 and 114? 98)114(1

By this process, it is found 98

that 2 is the greatest number,

that will divide 98 and 114.
16)98(6
96

NOTE. -- As 2 will divide 16, it will Common measure 2)16(8

also divide 96, because it is a multiple of

16. It will also divide 98, because 98 is 16

the sum of 96 and 2; and, as it will divide

them separate, it will also united. Again 114 is equal to 98 +16 and as 2 will divide both these numbers, it will also that of their sum. Therefore 2 will measure or divide 98 and 114. Q. E. D.

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