52. An empty cask weighs 17.06kg; when filled with water it weighs 275.8kg. How many liters does it hold? How many casks of this size will it take for the wine from a vat containing 3.008cbm?

Hektoliter.

53. If it takes 2.048hl of wheat to sow a hektar, how many cubic meters will it take to sow a square kilometer?

54. A piece of road 1km long and 7m wide is to be macadamized to the depth of 33cm. What will the work cost at 43 cents a cubic meter?

55. A gasometer holds 28,000cbm of gas. How many jets will this gasometer feed for an evening, when each jet burns 1251 an hour, and is used 4 hours?

56. The city of Venice is situated in the midst of a great lake of salt water, communicating with the sea, and all the rain water is caught for the cisterns. Ordinary years the fall of rain in Venice is 82cm; the surface of the city, after the canals have been deducted, is 520ha. Reckoning the population at 115,530, how many liters a day of rain water can each inhabitant have?

57. Find the weight of a bar of iron 5.35m long, 4.56cm thick, and 3.54cm wide. Find, also, the width of an oak beam 4.30m long, 9.12cm thick, which has the same weight. The specific gravity of the oak to be reckoned at 1.026, that of the iron 7.788.

58. Find the specific gravity and volume of a body weighing 35kg in air and 30kg in water.

59. A ster of piled oak wood weighs 425kg; the specific gravity of the wood is 0.74. What is the volume occupied by the spaces between the logs? For how much must 100kg of separate sticks be sold to bring the same amount as when sold at $2.20 a ster?

60. Wrought iron sells for $7.00 per 100kg. A bar of iron 4.5cm wide, 3.3cm thick costs $5.08; what is its length, reckoning the specific gravity of the iron at 7.4 ?

61. Experiment shows that water weighs 770 times as much as air; and the specific gravity of mercury is 13.6. How many liters of air will it take to weigh as much as a liter of mercury?

62. A mass of lead weighing 753kg is made into sheets 0.1mm thick. Find, in square meters, the surface which can be covered by the sheets thus obtained. The specific gravity of the lead is 11.3.

63. A rectangular sheet of tin of uniform thickness is 85cm wide, 1.35m long, and weighs 268%. What is its thickness, if the specific gravity of tin is 7.3?

64. The fine coal which collects about the shafts of the mines and in the coalyards was for a long time wasted, because it could not be burned in stoves and grates. Now this dust is mixed with tar in the proportion of 92kg of dust and 8% of tar; the mixture is heated, and afterwards pressed in rectangular moulds 14.75cm by 18.5cm by 29cm; each one of these blocks weighs 10kg. They are sold at $3.00 a ton, and make excellent fuel for heating steam boilers. Find the specific gravity of this fuel; also the sum which would be realized in thus utilizing 800,000t of coal dust, the cost of tar, mixing, etc., being $0.50 a ton.

65. A bar of iron a millimeter square on the end will break under a tension of 30kg. Find the length at which a suspended bar of iron will break from its own weight, if the specific gravity of the iron is 7.8.

66. Fifty-three kilograms of starch are obtained from 100kg of wheat. A hektar of land produces 13631 of wheat; a hektoliter of wheat weighs 78kg. If the wheat harvested from a field measuring 2ha and 33m is taken to a starch factory, how much starch will be made from it?

67. A gardener wishes to provide glass for his hotbeds. The beds cover 2.65a; the panes will cover 0.75 of the whole surface, the rest being taken up by the frames and alleys. First find how many panes measuring 45cm by 37cm it will take to cover the beds; then find the price of the glass, at a cost of 95 cents a square meter.

68. A jar full of water weighs 1.325kg; filled with mercury it weighs 12.540kg. Find the capacity and the weight of the jar, if the specific gravity of the mercury is 13.59.

69. A hektoliter of rape seed weighs 63, and 321 of oil can be extracted from it. How many kilograms of the

seed will it take to make a hektoliter of oil?

70. Common burning gas is 0.97 of the weight of air, and a liter of air weighs 1.2935. In a shop there are 65 jets, each one of which burns 1231 an hour, and is used 5 hours in the winter evenings. Find the weight of the gas used in a month of 26 days, and the expense of lighting the shop, when gas costs 6 cents a cubic meter.

71. A merchant buys one kind of wine at 30 cents a liter, another kind at 21 cents a liter; he mixes the two kinds by putting 51 of the first with 81 of the second. For how much a liter must he sell the mixture in order to gain $3.75 a hektoliter?

72. If it requires 360 tiles to drain an ar of land, what will it cost to drain 17.784ha, when the tiles cost $20 a thousand, and the expense of laying is the same as the cost of the tiles?

73. Hewn stone of medium durability ought not to support, as a permanent weight, more than 0.07 of the weight that is required to crush it. A certain kind of stone used for building will be crushed under a weight of 250kg a square centimeter. What is the greatest height to which a wall constructed of this material can be safely carried, if the specific gravity of the stone is 2.1?

са

74. Several different kinds of wine are mixed as follows: 2451 at 20 cents a liter, 5471 at 15 cents a liter, 3441 at 25 cents a liter. How much does the mixture cost a liter?

75. A farmer wishes to drain a field of 8.75ha. Each hektar requires 750m of ditches. The opening of these ditches costs 10 cents a running meter; the tiles are 30cm long and cost $15 a thousand. He pays 2 cents a meter for laying the tiles, and 4 cents a meter for filling the ditches. What is the cost of draining the field?

76. A silver five-franc piece weighs 25%, and is composed of 9 parts of pure silver and 1 part of pure copper. A silver two-franc piece weighs 10%, and is composed of 835 parts of pure silver and 165 parts of pure copper. A silver twenty-centime piece weighs 18, and has the same composition as the two-franc piece. Find the total weight of pure silver and of pure copper contained in 272 fivefranc pieces, 145 two-franc pieces, and 179 twenty-centime pieces.

77. The dimensions of the bottom of a rectangular box are 70cm by 50cm. If the box contains exactly an hektoliter of wheat when full, what is the height of the box?

78. If a stick of oak timber 54 centimeters wide and 65 centimeters thick costs $25 at $16 a cubic meter, what is the length of the stick?

79. A rectangular box whose bottom is a square 28cm on a side, and whose height is 19.2cm, is exactly filled with gold twenty-franc pieces, in piles touching each other. If a twenty-franc piece is 35mm in diameter, and 1.28mm thick, what is the value of the gold in the box?

80. If 111 of coal yields 1854cbm of gas, and one burner consumes 1401 of gas in an hour, how many hektoliters of coal are required to supply 2800 burners for 144 hours ? 81. How many liters of water in a cylindrical well 1.96m in diameter, if the water is 2.84m deep?

CHAPTER VI.

MEASURES AND MULTIPLES OF NUMBERS.

170. Factors of a Number. The factors of a number are the numbers whose product is that number.

171. Prime Numbers. A prime number is a number that has no integral factors, except itself and one.

Thus, 2, 3, 5, 7, 11, 13, 17, 19 are prime numbers.

172. Composite Numbers. A composite number is a number that is the product of two or more integral factors.

Thus, 10, 21, 143 are composite numbers, for 10 is 2 x 5; 21 is 3 × 7; 143 is 11 × 13.

NOTE. In speaking of the integral factors of a number we exclude the number itself and one.

173. Prime Factors. A prime factor is a factor that is a prime number.

174. A composite number can have but one set of prime factors.

Thus, 12 cannot be expressed as the product of any set of prime factors except 2 × 2 × 3. It is the product of 2 × 6, and of 3 × 4, but one of the factors of 2 × 6 and one of 3 x 4 is composite.

175. A number that can be divided by another without a remainder is said to be exactly divisible by that number ; and the divisor is called an exact divisor.

176. Even Numbers. An even number is a number that is exactly divisible by 2.

177. Odd Numbers. An odd number is a number that is not exactly divisible by 2.,