42. From the preceding illustrations and principles we derive the following

GENERAL RULE FOR MULTIPLICATION.

I. Write the multiplier under the multiplicand, units under units, tens under tens, &c.

II. When the multiplier contains but ONE figure, begin with the units, and multiply each figure of the multiplicand by the multiplier, setting down the result and carrying as in addition. (Art. 23.)

III. If the multiplier contains MORE than one figure, multiply each figure of the multiplicand by each figure of the multiplier separately, and write the first figure of each partial product under the figure by which you are multiplying.

Finally, add the several partial products together, and the sum will be the whole product, or answer required.

43. PROOF.-Multiply the multiplier by the multiplicand, and if the second result is the same as the first, the work is right.

OBS. 1. It is immaterial as to the result which of the factors 18 taken for the multiplier. (Art. 38.) But it is more convenient and therefore customary to place the larger number for the multiplicand and the smaller for the multiplier. Thus, it is easier to multiply 254672381 by 7, than it is to multiply 7 by 254672381, but the product will be the same.

2. Multiplication may also be proved by division, and by casting out the nines; but neither of these methods can be explained here without anticipating principles belonging to division, with which the learner is supposed as yet to be unacquainted.

QUEST.-42. How do you write numbers for multiplication? When the multiplier contains but one figure, how do you proceed? When the multiplier contains more than one figure, how proceed? 41. Note. What is meant by partial products? What is to be done with the partial products? 43, How is multiplication proved?

EXAMPLES FOR PRACTICE.

1. Multip.y 63 by 4. 2. Multiply 78 by 5. 3. Multiply 81 by 7. 4. Multiply 97 by 6. 5. Multiply 120 by 7. 6. Multiply 231 by 5. 7. Multiply 446 by 8. 8. Multiply 307 by 9. 9. Multiply 560 by 7.

10. Multiply 46 by 10. 11. Multiply 52 by 11. 12. Multiply 68 by 12. 13. Multiply 84 by 13. 14. Multiply 78 by 15. 15. Multiply 95 by 23. 16. Multiply 129 by 35. 17. Multiply 293 by 42. 18. Multiply 461 by 55.

19. If 1 barrel of flour costs 9 dollars, how much will 38 barrels cost?

20. If 1 apple-tree bears 14 bushels of apples, how many bushels will 24 trees bear?

21. In 1 foot there are 12 inches: how many inches are there in 28 feet?

22. In 1 pound there are 20 shillings: how many shillings are there in 31 pounds?

23. What will 17 cows cost, at 23 dollars apiece?

24. What will 25 tons of hay cost, at 19 dollars per ton? 25. What will 37 sleighs cost, at 43 dollars apiece? 26. What will a drove of 150 sheep come to, at 13 shillings per head?

27. What cost 105 acres of land, at 15 dollars per acre? 28. How much will 135 yards of cloth come to, at 18 shillings per yard?

29. In 1 pound there are 16 ounces: how many ounces are there in 246 pounds?

30. A drover sold 283 oxen, at 38 dollars per head: how much did he receive for them?

31. If you walk 22 miles per day, how far will you walk in 305 days?

32. In one day there are 24 hours: how many hours are there in 365 days?

33. In 1 year there are 52 weeks: how many weeks are there in 175 years?

34. In 1 hour there are 60 minutes: how many minutes are there in 396 hours?

35. In 1 hogshead there are 63 gallons: how many gallons are there in 450 hogsheads?

36. What will 475 horses cost, at 73 dollars apiece? 37. In 1 square foot there are 144 square inches: how many square inches are there in 235 feet?

38. How far will a ship sail in 158 days, if she sails 165 miles per day?

44. It is a fundamental principle of notation, that each removal of a figure one place towards the left, increases its value ten times; (Art. 10;) consequently annexing a cipher to a number, increases its value ten times, or multiplies it by 10; annexing two ciphers, increases its value a hundred times, or multiplies it by 100; annexing three ciphers, increases it a thousand times, or multiplies it by 1000, &c.; for each cipher annexed, removes each figure in the number one place towards the left. Thus, 12 with a cipher annexed becomes 120, and is the same as 12×10; 12 with two ciphers annexed, becomes 1200, and is the same as 12X100; 12 with three ciphers annexed, becomes 12000, and is the same as 12X1000, &c. Hence,

45. To multiply by 10, 100, 1000, &c.

Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the number thus formed will be the product required.

Note.-To annex means to place after, or at the right hand.

QUEST.-44. What effect does it have to remove a figure one place towards the left hand? Two piaces? 45. How do you proceed when the multiplier is 10, 100, 1000, &c? Note. What is the meaning of the term annex?

40. What will 10 dresses cost, at 18 dollars apiece? Solution. If 1 dress costs 18 dollars, 10 dresses will cost 10 times 18 dollars.

number multiplies it by 10.

But annexing a cipher to a

We therefore annex a cipher

to the multiplicand, (18 dollars,) and it becomes 180 dolThe answer therefore is 180 dollars.

ars.

51. What will 20 wagons cost, at 67 dollars apiece?

Suggestion.-Since multiplying by

ciphers produces ciphers, we omit mul

Operation.

67

tiplying by the 0, and placing the sig

20

nificant figure 2 under the right hand Ans. 1340 dollars. figure of the multiplicand, multiply

by it in the usual way, and annex a cipher to the product. The answer is 1340 dollars. Hence,

46. When there are ciphers on the right hand of the multiplier.

Multiply the multiplicand by the significant figures of the multiplier, and to this product annex as many ciphers, as are found on the right hand of the multiplier.

QUEST. 46. When there are ciphers on the right of the multiplier, how do you proceed?

60. In one hour there are 60 minutes: how many minates are there in 125 hours?

61. What will 300 barrels of flour cost at 8 dollars barrel?

per

62. What cost 400 yds. of cloth, at 17 shills. per yd.? 63. If the expenses of 1 man are 135 dollars per month, how much will be the expenses of 200 men?

64. If 1500 men can build a fort in 235 days, how long will it take one man to build it?

47. When there are ciphers on the right of the multiplicand.

Multiply the significant figures of the multiplicand by the multiplier, and to the product annex as many ciphers, as are found on the right of the multiplicand.

65. What will 43 building lots cost, at 3500 dollars a lot?

Having placed the multiplier under the significant figures of the multiplicand, multiply by it as usual, and to the product thus produced, annex two ciphers, because there are two ciphers on the right of the multiplicand.

(68.)

Operation.

3500

43

105

140

Ans. 150500 dolls

(69.) 25000

21000

15

17

24

32

70. What is the product of 132000 multiplied by 25? 71. What is the product of 430000 multiplied by 34? 72. What is the product of 1520000 multiplied by 43? 73. What is the product of 2010000 multiplied by 52? 74. What is the product of 3004000 multiplied by 61?

QUEST.-47. When there are ciphers on the right of the multiplicand, how do you proceed?