week, respectively, for pasturing the sheep, cows, oxen and horses. What were the net profits of this farm, supposing that he paid $32 taxes, and that the cost of cultivating and harvesting the potatoes, corn, wheat, rye, oats and hay was $35, $35, $33, $25, $15 and $6 per acre, respectively? Ans. $1070.283. 53. A merchant, owning of a ship, sold # of his share for Ans. $12000. 54. The cargo of a certain ship is worth $48000, and § of the value of the cargo is the value of the ship; what is the ship worth? Ans. $12000.

$3000; what was the value of the ship?

55. In a certain school the scholars study arithmetic, algebra, geometry, and the remainder of the school, viz. 10 scholars, study surveying; how many scholars are there in the school?

56. (++−3+} +1)× 51=? 57. ++ ( − 3 + } + 1) × 51=? 58.13÷2+1×4 —↓÷÷÷— X}=?

59. ∗ ÷ (2 + 1) – ♀ ÷ 12 – ?

7/10 54

Ans. 200. Ans. 121.

Ans. 6247.

Ans. 1783.

Ans.

1

60. of + ÷ 1 4 − 3 × 8 + († of ‡ + b × }) ÷ 14} 71

-

Ans. 17.

1qr. what fraction of a pound?

=

61. £3 + 4s. + d. what fraction of a shilling? penny? farthing?

1st. Ans. £54317. 134400

62. 27 is a divisor, and 53 is the quotient; what is the dividend?

Ans. 148.

63. The sum of two numbers is 873, and one of the numbers is 183; what is the other?

Ans. 68

64. The difference of two numbers is 173, and the less number is 181; what is the greater? Ans. 351 65. 48 is a dividend, and 24g is the quotient; what is the divisor.

Ans. 2.

66. 47% is the product of two factors, and 123 is one of those factors; what is the other? Ans. 31.

67. 173 is a dividend, and 153 is the divisor; what is the quotient?

Ans. 1

31

234

68. The factors of a certain number are 324, 154 and 193; what is of 3 of of the number? Ans. 322341.

69. What is the value of of a barrel of flour at $7 per barrel?

70. How much cloth that is of a yard line a cloak containing 84 yards which is

71. A can build 33 rods of wall in 24 hours per day; in how many days of 9 times as many rods?

Ans. $5. wide will it take to of a yard wide? Ans. 12 yds. days by laboring 12 hours will he build 1

72. A garden whose breadth is 10 rods, and whose length is 1 times its breadth, has a wall 3 feet thick around it; what was the cost of digging a trench 23 feet deep, in which to lay this wall, at of a cent per cubic foot? Ans. $62.943.

73. What will be the cost of digging a ditch around the abovementioned garden, within and adjacent to the wall, 3 feet wide and 2 feet deep, at 3 of a cent per cubic foot?

74. A can perform a piece of work in 6 days of 10 hours each, and B can do the same in 8 days of 11 hours; in how many days of 11 hours can A and B together do the work?

Ans. 337

75. A sold an ox for $62.50, and received in payment 121 yards of broadcloth at $34 per yard and the balance in sugar at 12 cents per pound; how many pounds did he receive?

76. Bought a pair of oxen and a horse for $340 and a wagon for the price of the horse. The oxen cost the price of the horse; what was the cost of each?

77. From a piece of land that is 73 rods long and 74 rods wide, take 3 square rods and 3 rods square, and what will remain ? Ans. 3733 square rods.

78. A owns of a field, and B, the remainder; the difference between their shares is 7a. 3r. 153rd. What is B's share? 79. A boy, having a number of marbles, gives to one schoolfellow of them; to another of the remainder; loses of what then remains; and sells 24 times as many as he loses, when he has but 6 marbles left. How many had he at first?

80. If a family of 5 persons eat of a barrel of flour in 43 weeks, how much will be sufficient for 243 weeks, if the family is increased by its former number?

81. A regiment of 1024 men are to be clothed with cloth that is of a yard wide. Now, if it takes 23 yards of this cloth for each soldier, how many yards of cloth that is of a yard wide will be sufficient to line all the garments?

82. What number is that which, being increased by of of 10 and the sum diminished by 7, will give a remainder of 9?

§ 11. DECIMAL FRACTIONS.

141. A DECIMAL* FRACTION is a fraction whose denominator is a unit, with one or more ciphers annexed.

142. The denominator of a Vulgar Fraction may be any number whatever, and the FORM OF THE DENOMINATOR of a Decimal Fraction is its DISTINGUISHING CHARACTERISTIC.

143. Every principle and every operation in Vulgar Fractions is equally applicable to Decimals; † but the peculiar form of the denominator gives facilities for operating in Decimals that do not exist in Vulgar Fractions.

144. The denominator of a decimal fraction is not usually expressed, since it can be easily determined, it being 1 with as many ciphers annexed as there are figures in the given decimal.

145. A decimal fraction is distinguished from a whole number by a point, called the decimal point or separatrix, placed before the decimal; the first figure at the right of the point is tenths; the second, hundredths; the third, thousandths; etc.; thus, .6 25.042 =2009

.25

=

*Decimal, from the Latin decem, ten.

† By the term decimal we usually mean a DECIMAL FRACTION.

146. Since whole numbers and decimal fractions both decrease by the same law from left to right, they may be expressed together in the same example, and numerated as in the following

147. The integral number is numerated from the separatrix

toward the left, and the fraction from the same point toward the right, each figure, both in the whole number and decimal, taking its name and value by its distance from the decimal point.

148. In reading a decimal, we may give the name to each figure separately, or read it as a whole number and give the name of the right hand figure only; thus, the expression .23 may be read and 73, or it may be read, for 1 and 13

149. Since multiplying both terms of a fraction by the same number does not alter its value (133, a, Note 1), annexing one or more ciphers to a decimal does not affect its value; thus, fo= etc.; i. e. .2 10009 = .20 = .200, etc.

20 100

=

200

150. Prefixing a cipher to a decimal, i. e. inserting a cipher between the separatrix and a decimal figure, diminishes the value of that figure to its previous value; for it removes the figure one place farther from the decimal point (147); thus, .3, but .03 = only, which is but

of.

What is the effect of prefixing 2, 3 or more ciphers to a decimal?

151. A vulgar fraction is sometimes annexed to a decimal; thus, .24. This is equivalent to the complex fraction The

24

10

vulgar fraction is never to be counted as a decimal place, but it is always a fraction of a unit of that order represented by the preceding decimal figure; thus, in .234, the is half of a thousandth.

NOTATION OF DECIMAL FRACTIONS.

152. Let the pupil express in figures the following numbers:

1. Twenty-seven hundredths.

2. Thirteen thousandths.

3. Eighty-nine tenths of millionths.

Ans. .27.

NOTE.-An ambiguity often arises in enunciating a whole number and a decimal in the same example; thus, .203 is two hundred and three thousandths, and 200.003 is two hundred, and three thousandths. This ambiguity may, however, be avoided by placing the word decimal before the fraction; thus, 200.003 may be read two hundred and decimal three thousandths.

4. Write the decimal two hundred and fifty-two thousandths. 5. Decimal six hundred and sixty-three tenths of thousandths. 6. Five hundred and decimal five thousandths.

7. Three thousand and decimal three thousandths.

8. Twelve hundred and fifty and six-tenths.

9. Decimal seven hundred and seventy-seven thousandths. 10. Eight thousand and decimal eighteen millionths.

153. Let the following numbers be written in words, or read