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discovery of America Ans. 190 years.

year 1302; how long was this before the by Columbus, which happened in 1492 ? 47. What number is that, to which if 6956 be added, the sum will be one million? Ans. 993044.

48. A man bought an estate for seventeen thousand five hundred and sixty-five dollars, and sold it for twenty-nine thousand three hundred and seventy-five dollars. Did he gain or lose, and how much? Ans. Gained $11810.

49. Bought a pair of oxen for 85 dollars, a horse for 126 dollars, three cows at 25 dollars apiece; and sold the whole for three hundred dollars; how much did I gain? Ans. $14. 50. Bonaparte was declared emperor in 1804; how many years since?

51. The union of the government of England and Scotland was in the year 1603; how long was it from this period to the time of the declaration of the independence of the United States? Ans. 173 years.

52. Jerusalem was taken and destroyed by Titus in the year 70; how long was it from this period to the time of the first Crusade, which was in the year 1096 ? Ans. 1026 years.

SECTION IV.

MULTIPLICATION.

MULTIPLICATION is the repetition of a number any proposed number of times. It consists of three parts, the Multiplicand, or number to be multiplied; the Multiplier, or number by which to multiply; and the result, which is called the Product. The Multiplicand and Multiplier are called factors.

RULE.

Place the larger number uppermost for the multiplicand, and the smaller number under it for a multiplier, arranging units under units, tens under tens, &c. Then multiply each figure of the multiplicand by each figure of the multiplier, beginning with the right-hand figure, and carrying for every ten as in addition. If the multiplier consists of more than one figure, the right-hand figure of each product must be placed directly under the figure of the multiplier that produces it, which will cause the successive products to recede each one place to the left. The sum of the several products will be the whole product required.

NOTE 1. When there are ciphers between the significant figures of the multiplier, pass over them in the operation, and multiply by the

significant figures only, remembering to set the first figure of the product directly under the figure of the multiplier that produces it. See Ex. 15. NOTE. 2.- - If there are ciphers at the right hand either of the multiplier or multiplicand, or of both, they may be neglected to the close of the operation, when they must be annexed to the product.

PROOF.

The correctness of the result in Multiplication may be conveniently ascertained in three ways; viz., by Division, by Multiplication, or by casting out the nines.

According to the first method,* divide the product by the multiplier; and, if the work is right, the quotient will be equal to the multiplicand.

According to the second method, take the multiplier for the multiplicand and the multiplicand for the multiplier, and -proceed according to the rule for multiplication; and, if the work be right, the product will be the same as by the former operation.

According to the third method, begin at the left hand of the multiplicand, and add together its successive figures towards the right, till the sum obtained equals or exceeds the number 9. If it equals it, drop the nine, and begin to add again at this point, and proceed till you obtain a sum equal to or greater than nine. If it exceeds nine, drop the nine as before, and carry the excess to the next figure, and then continue the addition as before. Proceed in this way till you have added all the figures in the multiplicand and rejected all the nines contained in it, and write the final excess at the right hand of the multiplicand. Proceed in the same manner with the multiplier, and write the final excess under that of the multiplicand. Multiply these excesses together and place the excess of nines in their product under the other excesses. Then proceed to find the excess of nines in the product obtained by the original operation, and, if the work be right, the excess thus found will be equal to the excess contained in the product of the above excesses of the multiplicand and multiplier. See Example 15.

NOTE. This method of proof, though perhaps sufficiently sure for common purposes, is not always a test of the correctness of an operation. Cases will sometimes occur in which the excesses above named will be equal, when the work is not right.

* As the pupil is presumed not to be acquainted with Division, he will pass over this method of proof for the present. It is placed here as a method important to be known, and because there seems to be no better place for it, though it presupposes an acquaintance with a rule yet to be learned.

TABLE OF PYTHAGORAS.

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36. Multiply 7008005 by 10008. 37. Multiply 4001100 by 40506. 38. Multiply 6716700 by 808070. 39. Multiply 987648 by 481007. 40. Multiply 18711000 by 470. 41. Multiply 10000 by 7000.

Ans. 70136114040. Ans. 162068556600. Ans. 5427563769000. Ans. 475065601536. Ans. 8794170000. Ans. 70000000.

42. Multiply 101010101 by 2020202. Ans. 204060808060402.

43. Multiply 70000 by 10000.
44. Multiply 800008 by 9009.
45. Multiply 900900 by 70070.
46. Multiply 4807658 by 706007.
47. Multiply 16789001 by 10080.

Ans. 700000000. Ans. 7207272072. Ans. 63126063000. Ans. 3394240201606.

Ans. 169233130080.

48. Multiply 304050607 by 3011101. Ans. 915527086788307. 49. Multiply 908007004 by 500123. Ans. 454115186861492. 50. Multiply 2003007001 by 6007023.

Ans. 12032109124168023. 51. Multiply 9000006 by 9000006. Ans. 81000108000036. 52. Multiply 1152921504606846976 by 1152921504606846976. Ans. 1329227995784915872903807060280344576.

53. What will 27 oxen cost at 35 dollars each ?

Ans. $945. 54. What will 365 acres of land cost at 73 dollars per acre?

Ans. $26645.

55. What will 97 tons of iron cost at 57 dollars a ton ?

Ans. $5529. 56. What will 397 yards of cloth cost at 7 dollars per yard? Ans. $2779.

57. What will 569 hogsheads of molasses cost at 37 dollars per hogshead? Ans. $21053. 58. If a man travel 37 miles in one day, how far will he travel in 365 days? Ans. 13505 miles. 59. If one quire of paper have 24 sheets, how many sheets are in a ream, which consists of 20 quires? Ans. 480 sheets. 60. If a vessel sails 169 miles in one day, how far will she sail in 144 days? Ans. 24336 miles.

61. What will 698 barrels of flour cost at 7 dollars a barrel? Ans. $4886.

62. What will 376 lbs. of sugar cost at 13 cents a pound?

Ans. 4888 cts.

63. What will 97 lbs. of tea cost at 93 cents a pound?

Ans. 9021 cts.

64. If a regiment of soldiers consists of 1128 men, how

many men are there in an army of 53 regiments?

Ans. 59784.

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