70. A speculator sold 840 bushels of wheat for $2180, which was $500 more than he gave for it: what did it cost him a bushel?

71. A farmer sold a grocer 30 bushels of potatoes, at 371 cents a bushel, for which he received 6 gallons of molasses, at 45 cents a gallon; 60 pounds of mackerel, at 61 cents a pound; and the remainder in sugar, at 10 cents a pound: how many pounds of sugar did he receive ?

72. If a man travels 12 mi. 3 fur. 20 rd. in one day, how long will it take him to travel 174 mi. 1 fur. at the same rate?

73. If a man sells 2 bar. 12 gal. 2 qt. of beer in one week, how much will he sell in 12 weeks?

74. A liquor merchant had 550 pint bottles, 400 quart bottles, 350 two-quart bottles, 375 three-quart bottles, and 150 jugs holding a gallon each: how many barrels of wine will fill them?

75. How many yards of carpeting, one yard wide, will it take to cover the floors of two parlors, each 18 feet long and 16 feet wide; and what will it cost, at $1.33 a yard?

76. How many rolls of wall-paper, each 10 yards long and 2 feet wide, will it take to cover the sides of a room 22 feet long, 16 feet wide, and 9 feet high?

77. Two persons are 1 mi. 4 fur. 20 rd. apart, and are travelling the same way. The hindmost gains upon the foremost 5 rods in travelling 25 rods: how far must he travel to overtake him?

78. A man sold 500 bushels of wheat at $1.75 a bushel, and took his pay in sugar at 5 cents a pound. He afterwards sold one-half of the sugar: what quantity had he left?

79. A man bought 7 barrels of sugar, at $12.87 a barrel; he kept two barrels for his own use, and sold the remainder for what the whole cost him: what did he receive per barrel ?

80. A flour merchant bought a quantity of flour for $18750, and sold the same for $26250, by which he gained $3 a barrel : how many barrels were there?

81. Three men rented a farm, and raised 964 bush. 2 pk. 4 qt. of grain, which was to be divided in proportion to the rent paid by each. The first was to have one-half the whole; the second, one-third the remainder; and the third what was left : how much did each have?

82. A vessel, in longitude 70° 25' east, sails 105° 30′ 56′′ west hen 46° 50' east, then 10° 5' 40" west, then 39° 11′ 36′′ east : .n what longitude is she then, and how many days will it take her to sail to longitude 77° west, if she sails 3° 20' each day?

83. A privateer took a prize worth $25000, which was divided into 125 shares, of which the captain took 12 shares; 2 lieutenants, each 5 shares; 6 midshipmen, each 3 shares; and the remainder was divided equally among 85 seamen: how much did each receive?

84. If the longitude of Boston is 71° 4′, and a gentleman, in travelling from Boston to Chicago, finds that his watch is 1 hr. 5 m. 44 sec. too fast by the time of the latter place: what is the longitude of Chicago, provided his watch has kept accurate time?

85. What time would it be in Boston if it was 8 hr. 27 m. 30 sec., A. M., in Chicago?

86. What time would it be at Chicago if it was 12 M. at Boston?

87. Two places lie exactly east and west of each other, and by observation it is found that the sun comes to the meridian of the latter place 1 hour and 16 minutes after the former: how far apart are they in minutes and degrees of longitude?

88. In 12 bales of cloth, each bale containing 16 pieces, and each piece containing 20 ells English, how many yards?

89. A speculator gave $8968 for a certain number of barrels of flour, and sold a part of it for $2618, at $7 a barrel, and by so doing lost $2 on each barrel: for how much must he sell the remainder, to gain $1060 on the whole?

90. How many eagles can be made from 24 lb. 4 oz. 6 pwt. 18 gr. of gold, making no allowance for waste, if each eagle weighs 11 pwt. 9 gr.?

91. A man paid $3284.82 for some wheat. He sold 740 bushels at 2 dollars a bushel; the remainder stood him in $1.42 a bushel how many bushels did he purchase?

92. A man sold 105 A. 2 R. 20 P. of land for as many dollars as there were perches of land, payable in instalments, at the rate of 1 dollar an hour. If the contract was closed at 12 o'clock, M., April 1st, 1856, what length of time will be allowed the purchaser to pay the debt, reckoning 365 days 6 hours to the year?

93. The sum of 2 numbers is 98, and their difference is 46: what are the numbers?

94. A farmer paid $76 dollars more for a horse than for a cow; he paid $190 for both: what was the value of each?

95. How many days intervene between March 5th and August 21st, both days inclusive?

96. A merchant buys 870 barrels of flour, at $9.50 a barrel. He finds one-half of it injured, and is willing to lose onequarter on the value of that part: how much loss was that on each half barrel?

97. Three merchants, A, B, and C, are engaged together in business, and gain in one year $24612. This amount is to be equally divided among them, after paying A $675, and B $812, for extra services. How much did each receive?

98. Four merchants are in partnership. Their apparent profits during the year amount to $56895; but they have expended for clerk hire, $6750; for rent, $3500; for insurance, $156; and for incidental expenses, $364. The first is to have $250 for extra services; the second, $175 for travelling expenses; and the third, $95 for various articles furnished by him to the concern. What was the share of profit of each, after paying these expenses?

PROPERTIES OF NUMBERS.

Exact Divisors-Prime Numbers.

104. AN EXACT DIVISOR of a number, is any number, except 1 and the number itself, that will divide it without a remainder.

105. ONE NUMBER is divisible by another, when the remainder is 0.

106. AN ODD NUMBER is one not divisible by 2.

107. AN EVEN NUMBER is one divisible by 2.

108. A PRIME NUMBER is one which has no exact divisor. 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, &c.,

are prime numbers.

109. What numbers are exact divisors.

1. ANY FACTOR of a composite number is an exact divisor of the number.

2. THREE is an exact divisor of any number, the sum of whose digits is divisible by 3.

3. FOUR is an exact divisor of a number when it will exactly divide the number expressed by the two right-hand digits.

4. FIVE is an exact divisor of every number whose right-hand figure is 0 or 5.

5. Sıx is an exact divisor of any even number of which 3 is an exact divisor.

101. What is an exact divisor?-105. When is one number divisible by another?-106. What is an odd number?-107. What is an even number?-108. What is a prime number?

109.-1. Is a factor of a composite number an exact divisor?

2. What numbers will 3 exactly divide? 3. What numbers will 4 exactly divide ? 4. What numbers will 5 exactly divide ? 5. What numbers will 6 exactly divide? 6. What numbers will 9 exactly di vide? 7. What numbers will 10 exactly divide?

6. NINE is an exact divisor of any number, the sum of whose digits is divisible by 9.

7. TEN is an exact divisor of every number whose right-hand figure is 0.

110. To find the prime factors of a composite number.

1. What are the prime factors of 18? ANALYSIS.-Every factor of a composite number is an exact divisor. If any divisor is a composite number, resolve it again into factors, till each shall be prime: hence,

189 x 2

= 3 × 3 × 2

Every composite number is equal to the product of all its prime factors: hence, it is divisible by each of them.

Thus,

and

24 3 X 83 X 2 X 4 = 3 X 2 X2 X2;

60 5 X 125 X 2 X 6 = 5 x 2 x 3 x 2.

2. What are the prime factors of the composite number 105?

ANALYSIS.-Three being an exact divisor, and a prime number, we divide by it, giving the quotient 35; then, 5 and 7 are prime divisors of the quo

3)105
5)35
7

tient: hence, 3, 5, and 7 are the prime factors of 105.

Hence, to find the prime factors of any composite number, Rule.-Divide the given number by any PRIME number that is an exact divisor: then divide the quotient by any other exact prime divisor, and so on, till a quotient is found which is a prime number: the several divisors and the last quotient will be the prime factors.

Examples.

1. What are the prime factors of 9? 10? 12? 14? 16? 18? 24? 27? 28?

2. What are the prime factors of 30? 22? 32? 36? 3S? 40? 45? 49?

110. What is the rule for finding the prime factors of a composite number?