33. French Money. 100 centimes = 1 franc (fr.)

34. German Money. 100 Pfenniges (pennies) = 1 Reichmark (mark). NOTE. The coin which represents the pound value is gold, and called a sovereign. The franc and the mark are both silver.

Square Root (Art. 675).

35. The following method of extracting square roots may be substituted for that given in the body of the book, if preferred. Let it be required to extract the square root of 1296.

=

Explanation. The formula Tens2 + 2 (tens x units) + units2 may be changed to the form, Tens2 + (2 x tens +units) x units. As the first part of the power, the square of the tens, is hundreds, the 12 hundreds of the given number must have in it the square of the tens of the root.

The greatest square contained in 12 (hundreds) is 9 (hundreds), the square root of which is 3 (tens). This we

write as the first term, or tens, of the root. Taking the square of 3 (tens): 9 (hundreds) out of 12 (hundreds), there remain 3 (hundreds), with which we unite the remaining part of the number, 96, making 396, which must contain the product of two times the tens plus the units multiplied by the units. If it contained only two times the tens multiplied by the units, we should find the number of units by dividing 396 by two times the tens. So we make two times the tens, or 6 tens, the trial divisor, and find that it is contained in 39 tens 6 times. Then 6 is probably the next term, or units, of the root. Adding 6 units to

the 6 tens (the trial divisor), we have now the true divisor, which, multiplied by 6, completes the square. So the given number is a perfect square, and 36 is its square root.

36. Rule.

To extract the square root of a number :

1. Beginning with the units' figure, point off the expression into periods of two figures each.

2. Find the greatest square in the number expressed by the left hand period, and write its square root as the first term of the root.

3. Subtract this square from the part of the number used, and with the remainder unite the next two terms of the given number for a dividend.

4. Double the part of the root already found for a trial divisor; and by it divide the dividend (rejecting the lowest term of the dividend) and write the quotient as the next term of the root. Also write it at the right of the trial divisor to express the true divisor.

5. Multiply the true divisor by this term, and subtract the product from the dividend.

6. If there are more terms of the root to be found, unite with the remainder the next two terms of the given number, take for a trial divisor double the part of the root now found, and proceed as before.

Cube Root (Art. 686).

37. The following method of extracting cube roots may be substituted for that given in the body of the book, if preferred. Let it be required to extract the cube root of 262144.

WRITTEN WORK.

Formula,

Explanation. The formula Tens3 + 3 (tens2 x units) + 3 (tens × 262′144 (64 units2) + units3 may be

Tenss+(3 x tens2+3 x tens x units + units2) x units.

(6 tens)3 =

216

changed to the form,

Tens+ (3 x tens2 + 3 x tens x units + units2) x

units.

As the first part of the power, the cube of

the tens, is thousands, we find the greatest cube contained in 262 (thousands), which is 216 (thousands), and write its cube root 6 (tens) as the first term, or tens, of the root.

Taking the cube of 6 (tens), 216 (thousands), out of 262 (thousands), there remain 46 (thousands), with which we unite the remaining part of the number, 144, making 46144, which must contain (3 x tens2 + 3 x tens x units + units2) x units.

If the number 46144 contained only 3 × tens2 × units, we should find the number of units by dividing 46144 by 3 times the square of the tens. So we make this number, 108 (hundreds), the trial divisor, and find that it is contained in 461 (hundreds) 4 times. Then 4 is probably the next term, or units, of the root.

Adding 3 times 6 tens × 4 units, and the square of 4 units to the trial divisor, we have the true divisor, which multiplied by 4 completes the cube. So the given number is a perfect cube, and 64 is its cube root.

38. Rule.

To extract the cube root of a number:

1. Beginning with the units' figure, point off the expression into periods of three figures each.

2. Find the greatest cube in the number expressed by the left-hand period, and write its cube root as the first term of the root.

3. Subtract the cube from the part of the number used, and with the remainder unite the next three terms of the given number for a dividend.

4. Take three times the square of the part of the root already found for a trial divisor, and by this divide the dividend (rejecting the lowest two terms of the dividend) and write the quotient as the next term of the root.

5. To the trial divisor (which is hundreds) add three times the first term of the root (tens) multiplied by the last term, also the square of the last term.

6. Multiply this sum by the last term of the root, and subtract the product from the dividend.

7. If there are more terms of the root to be found, unite with the remainder the next three terms of the given number, take for a trial divisor three times the square of the part of the root now found, and proceed as before.

39. To find the Capacity of a Cask or Barrel in Gallons.

Add to the head diameter of the difference between the head and bung diameters (or, if the staves are but little curved, 0.6 of the difference). This will give the mean diameter.

Multiply the square of the number of inches in the mean diameter by the number of inches in length, and this product by 0.0034.

40. Miscellaneous Examples.

47. If I lose 10% by selling goods at 18 cents per yard, for what should they have been sold to gain 20%?

48. If 30 men, working 11 hours a day, can do a piece of work in a certain time, how many more men must be employed, when it is half done, to finish it in the same number of days, working 10 hours a day?

49. Two armies are in opposite directions from a certain point, one being 300 miles east and the other 250 miles west of it, and marching towards each other, the first at the rate of 15 and the other of 18 miles in a day. In how many days will they meet, and where ?

50. If, by selling goods at 60 cents per lb., 20% is gained, what % would have been gained by selling them at 75 cents per lb.?

51. A broker purchases a lot of stocks at an average of 9% below par, and sells them at an average of 73% above par, and makes $300. What was the par value of the stocks?

52. How many bushels of corn at 50 cents a bushel must be mixed with 30 bushels of grain at 80 cents a bushel, that the mixture may be worth 75 cents a bushel?

NOTE. Take such a quantity of corn as shall make the gain in selling it at 75 cents a bushel equal the loss in selling the grain at 75 cents a bushel.

53. How much water must be mixed with a barrel of ink (31 gals.), which cost $34.10, that it may be sold at $1.10 a gallon and 25% be gained?

54. 20% of a lot of barley, originally 5000 bushels, was destroyed by fire, the cost having been $14 per bushel. What per cent will be gained on the lot by selling the remainder at $2 per bushel?

55. I sell of a lot of goods for $9, and thereby lose 25%. For what must I sell the remainder to make 83% on the whole?

56. I sold 4 ploughs at $ 24 each; on 2 of them I made 20%, and on 2 I lost 20%. What did I gain or lose on the whole?

57. Divide 52 into two such parts that of one part shall equal of the other.

58. How many cubic yards of earth must be removed for a cellar 10 feet deep and measuring inside the walls 27 feet long and 15 feet wide, the wall being 2 feet 6 inches thick?

59. If 10% is lost by selling boards at $7.20 per M., what % would be gained by selling them at 90 cents per C.?

60. A person takes a note on 2 months for $110 in payment for a watch. On getting the note discounted at a bank, he finds that he has lost 40% on the first cost of the watch. What was the cost?

61. What would be due May 1, 1878, on a note for $1000, dated March 26, 1875, at 8% interest, on which $200 were paid at the end of each year from the date of the note?

62. If I buy coal at $4.12 per ton on 6 months' credit, for what must I sell it immediately to gain 10%?

63. What will a pine log weigh whose length is 18 ft., measuring 3 ft. across the larger end, and 2 ft. across the smaller, pine being 0.6 as heavy as water, which weighs 62 lbs. to a cubic foot?

64. Required the number of square feet in the surface of a ditch surrounding a circular garden which is 25 yards across, the ditch being 2 ft. wide.

65. An aeronaut ascends at the rate of 41⁄2 miles an hour for 40 minutes, after which he maintains the same elevation; if his balloon is driven east 7 miles during the first hour from the time of his starting, and in an opposite direction at the rate of 10 miles an hour for the remaining time, how far from his starting-point in a straight line is he at the end of 5 hours?

66. 13% is lost by selling a lot of land for $783. What would it have brought if it had been sold at a loss of 83%?

67. What will be the per cent of gain on the cost of a Gas Co.'s stock, the par value of shares being $87.50, if it be bought at 15% below par, and sold at 19% above par?

68. What is the length of the edge of the largest cube that can be sawed from a globe 9 inches in diameter ?

69. Two boys tried their skill in running for pegs. Five pegs were set up in a line 6 feet apart. The starting-point was in the same line 6 feet from the first peg. How far must each boy run to fetch all the pegs one at a time to the starting-point?

70. John Barnes bought, June 8, 1875, 10 bales of cotton cloth, 14 pieces in a bale, 43 yds. in a piece, at 8 per yd., for which he gave his note on interest at 6%- On the 4th of Nov., 1877, he sold 1 bale at 30 a yd., and with the proceeds made part payment of his note. On the 3d of May, 1878, he sold 1 bale at 40%, and paid on his note the amount he received. On the 17th of Sept., 1878, he sold the remainder at 60, and settled the note. What did he gain by his speculation?