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48. Find the weight of a bar of iron having the following dimensions: length 3.6m, width 6cm, thickness 2cm; the specific gravity of the iron being 7.8.

49. How many lead balls each weighing 278 could be obtained by melting a mass of lead, cubic in form, the

edge measuring .356m, the specific gravity of the lead being 11.35?

50. Marble costs $30.95 a cubic meter, and the specific gravity of marble is 2.73. If a block of marble weighs 1260kg, what is its volume, and what is it worth?

51. Sea-water contains 28 parts, by weight, of salt in 1000. A liter of sea-water weighs 1.025kg. How many kilograms of salt could be obtained from 126.276842cbm of sea-water?

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 would it require to receive the wine from a vat containing 3.008cbm?

53. It takes about 204.81 of wheat to sow

Cask.

a hektar. How many cubic meters would it take to

sow a square kilometer?

54. A piece of road 1km long and 7m wide is to be macad

amized; the macadamizing is to be 33cm deep; it costs 43 cents a cubic meter to prepare the stones. What will enough for the road cost?

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

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. Ordi

nary 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 could 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. Give the specific gravity and volume of a body weighing 35kg in air and 30kg in water.

59. A ster of piled oak wood weighs 425; the specific gravity of the wood is .74. What is the volume occupied by the spaces between the logs? For how much must 100kg of separate sticks be sold in order to bring the same amount as when sold by the ster; a ster selling for $2.20?

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, in comparison with water, 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 .1mm thick. Calculate, 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; it weighs 2688. What is its thickness, reckoning the specific gravity of tin at 7.3?

64. The fine coal which collects about the shafts of the mines and in the coal-yards, 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 8kg of tar; the mixture is heated, and afterwards pressed in rectangular moulds of 14.75cm, 18.5cm, and 29cm; each one of these blocks weighs 10kg, they are sold at $3.00 a ton, and make excellent fuel for heating steam boilers. Give the specific gravity of this fuel; also, the sum which would be realized in thus utilizing 800,000 of coal dust, the cost of tar, mixing, etc., being $.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, the specific gravity of the iron being 7.8?

66. Fifty-three kilograms of starch are obtained from 100kg of wheat. A hektar of land produces 1363' of wheat; a hektoliter of wheat weighs 78kg. If the wheat harvested from a field measuring 2ha and 339m is taken to a starch factory, how much starch will be made from it?

67. A gardener wishes to provide glass for his hot-beds. The beds cover 2.65a; the panes will cover .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 12.540g. What is the capacity of the jar, and its weight? The specific gravity of the mercury

is 13.59.

69. A hektoliter of rape-seed weighs 63kg, 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 .97 of the weight of air, and a liter of air weighs 1.293. In a shop there are 65 jets, each one of which burns 123' an hour, and is used 5 hours in the winter evenings. Calculate the weight of the gas used in a month, 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 5' of the first with 8' 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. It is found in building that hewn stone of medium dur

ability ought not to support, as a permanent weight, more than .07 of the weight that it would require 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, the specific gravity of the stone being 2.1? 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?

CHAPTER X.

MULTIPLES AND MEASURES OF NUMBERS.

216. If one number can be divided by another number, without remainder, the divisor is called a factor, or measure, of the dividend, and the dividend a multiple of the divisor.

Thus, 35 can be divided by 5 without remainder; therefore 5 is called a factor or measure, of 35, and 35 a multiple of 5.

217. Numbers which can be divided, without remainder, only by themselves and 1, are called prime numbers.

The smaller ones are easily found by trial, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, etc.

218. Other numbers are each the product of a fixed set of prime factors, and are called composite numbers.

219. Numbers divisible by 2 are called even numbers. All other numbers are odd numbers. All primes (except 2) are odd.

220. Write a series of natural numbers in order; cancel the even ones; then place a dot over the multiples of 3: you produce this result:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, etc.

Each multiple of 6 has both the dot and the cancelling line, and the only numbers without the dot or line come just before or just after the multiples of 6. Therefore,

If a prime number be divided by 6 the remainder must be either 1 or 5.

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