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12. The tonnage of the United States in 1842, was 2069857, in 1846 it was 2500000 : what was the increase in 4 years ? 13. 14.

15. From 253760

3856031

54903670 Take 104523

462702

504089 16. 3576102-1750671.

28. 10000000-999999. 17. 4006723-5001.

29. 99999999_100000. 18. 3601900-1000000.

30. 83567000—438567. 19. 5317004-3565.

31. 40600056-7632. 20. 1000000—456321.

32. 56409250—1057245, 21. 2035024-27040.

33. 20030000—72534. 22. 45563075-460001.

34. 83175621–5256360. 23. 67030001-300452.

35. 70301604-250041. 24. 73256300—436020.

36. 60050376-6849005. 25. 56037431-735671.

37. 34200591-8888888. 26. 80200430-250.

38. 87035762—753017. 27. 96531768-873625.

39. 95246300—9438675. 40. From 6764+3764 take 6500+2430. 41. From 2890+8407 take 4251+3042. 42. From 7395+4036 take 8297+1750. 43. From 8404+7296 take 3201—1562. 44. From 6008+9270 take 5136—2352. 45. From 9234 +6850 take 9320—4783. 46. From 8564-2573 take 4431-1735. 47. From 7284–5362 take 60454-5729. 48. From 956144680 take 7352_6178. 49. From 8630—1763 take 2460+1743. 50. From 7561–2846 take 1734+2056.

51. From 9687-3401 take 3021 +1754. 52. A man having 55000 dollars, paid 7520 dollars for a house, 3260 dollars for furniture, 2375 dollars for a library, and invested the balance in bank stock: how much stock did he buy?

53. A gentleman worth 163250 dollars, bequeathed 15200 dollars apiece to his two sons, 16500 dollars to his daughter, and to his wife as much as to his three children, and the remainder to a hospital : how much did his wife receive, and how much the hos. pital?

54. A man bought three farms; for the first he paid 5260 dollars, for the second 3585, and for the third as much as for the first two. He afterwards sold them all for 15280 dollars: did he make or lose by the operation; and how much ?

55. What number is that, to which 3425 being added, the sum will be 175250 ?

56. A man being asked how much he was worth, replied, if you will give me 325263 dollars, I shall have two millions of dollars : how much was he worth?

57. A jockey gave 150 dollars for a horse, and meeting an acquaintance swapped with him, giving 37 dollars to boot; meeting another, he swapped and received 28 dollars to boot; he finally swapped again and gave 78 dollars to boot, and then sold his last horse for 140 dollars : how much did he lose by all his bargains ?

58. A speculator gained 3560 dollars, and afterwards lost 2500 dollars; at another time he gained 6283 dollars, and then lost 3450 dollars : how much more did he gain than lose ?

59. A man bought a house for MDCCCCXXXVII dollars, and sold it for DCXVIIII dollars less than he gave: how much did he sell it for ?

We perceive that the IIII in the lower number Operation. cannot be taken from II in the upper number; MDCCCCXXXVII dolls. we therefore borrow a V, which added to the II,

DCXVIIII dolls. makes IIIIIII; then IIII from IIIIIII, leaves Ans. MCCCXVIII dolls. III, which we set down. Now since we have borrowed the V in the upper number, there are no Vs left from which we can take the V in the lower number. We must therefore borrow an X; but X is equal to VV; and V from VV leaves V, which we set down. Having borrowed an X from the upper number, there are but XX left, and X from XX leaves X. C from CCCC leaves CCC. D from D leaves nothing. And nothing from M leaves M. Hence,

77. To subtract numbers expressed by the Roman Notation.

Write the less number under the greater ; then, beginning at the right hand, take the number in the lower line from that expressed by the same letters in the upper line, and set the remainder below. If the number in the lower line is larger than that expressed by the same letters in the upper line, borrow a letter next higher and add it to the number in the upper line; then subtract as before, observing to pay when you borrow as in subtraction of figures. (Art. 72.)

Oes. Other examples expressed by the Roman Notation, can be added by the teacher, if deemed expedient,

SECTION IV.

MULTIPLICATION.

Art. 79. Ex. 1. What will 3 melons cost, at 15 cents apiece ?

Solution.-If 1 melon costs 15 cents, 3 melons will cost 3 times 15 cents; and 3 times 15 cents are 45 cents. Ans. 45 cents.

2. What will 4 sleighs cost, at 21 dollars apiece ?

Solution.—Reasoning as before, if i sleigh costs 21 dollars, 4 sleighs will cost 4 times as much; and 4 times 21 dollars are 84 dollars. Ans. 84 dollars.

Obs. It is obvious that 3 times 15 cents is the same as 15 cents +15 cents +-15 cents, or 15 cents added to itself 3 times; and 4 times 21 dollars is the same as 21 dolls.+21 dolls.+21 dolls. +21 dolls., or 21 dollars added to itself 4 times.

80. This repeated addition of a number or quantity to itself, is called MULTIPLICATION.

The number to be repeated, or multiplied, is called the Multiplicand.

The number by which we multiply, is called the multiplier ; and shows how many times the multiplicand is to be repeated.

The number produced, or the answer to the question, is called the product. Thus, when we say, 8 times 12 are 96, 8 is the multiplier, 12 the multiplicand, and 96 the product.

81. The multiplier and multiplicand together are often called factors, because they make or produce the product.

Obs. 1. The term factor is derived from a Latin word which signifies an agent, a doer, or producer.

2. When the multiplicand denotes things of one denomination only, the operation is called Simple Multiplication.

QUEST.-80. What is multiplication? What is the number to be repeated called ? What the number by which we multiply? What does the multiplier show? What is the number produced called ? 81. What are the multiplicand and multiplier togethu called ? Why? Obs. What does the term factor signify?

MULTIPLICATION TABLE.

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4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 43 51 54 57 60
4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
6 12 18 24 30 35 42 48 54 60 66 72 78 84 90 96 102 108 114 120
7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140
8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160
9 18 27 36 45 51 63 72 81 9J 99 108 117 126 135 144 153 162 171180
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220
12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 2:28 240
13 26 39 52 65 78 91 104,117 13) 143 150 169 102 195 208 221 234 247 260
14 28 42 56 70 84 98 112 126 110 154 168 182 196 210 224 233 252 266 280
15 30 45 60 75 90 105 120 135 150 165 180 195 210 2.25 40 255 270 285 300
16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320
17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340
18 36 54 72 90 108 126 114 162 180 198 216 234 232 270 288 306 324 342 360
19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 351 380
20 40 60 80 100/120/140 160 180 200 220 240 260 280 300 320 340 360 380 400

Note: This Table was invented by Pythagoras, and is therefore sometimes called the Pythagorean Table.

The pupil will find assistance in learning the Multiplication Table by observing the following particulars.

1. The several results of multiplying by 10 are formed by simply adding a cipher to the figure that is to be multiplied. Thus, 10 times 2 are 20, 10 times 3 are 30, &c.

2. The results of multiplying by 5 terminate in 5 and 0, alternately. Thus, 5 times 1 are 5, 5 times 2 are 10, 5 times 3 are 15, &c.

3. The first nine results of multiplying by 11 are formed by repeating the figure to be multiplied. Thus, 11 times 2 are 22; 11 times 3 are 33, &c.

4. In the successive results of multiplying by 9, the right hand figure regalarly decreases by 1, and the left hand figure regularly increases by 1. Thus, 9 times 2 are 18; 9 times 3 are 27; 9 times 4 are 36, &c.

82. Multiplying by 1, is taking the multiplicand once : thus, 4 multiplied by l=4.

Multiplying by 2, is taking the multiplicand twice : thus, 2 times 4, or 4+4=8.

Multiplying by 3, is taking the multiplicand three times : thus, 3 times 4, or 4+4+4=12, &c. Hence,

Quest.--82. What is it to multiply by 1? By 2? By 3 ?

Multiplying by any whole number, is taking the multiplicand as many times, as there are units in the multiplier.

The application of this principle to fractional multipliers will be illustrated under fractions.

OBS. 1. From the definition of multiplication, it is manifest that the product is of the same kind or denomination as the multiplicand: for, repeating a number or quantity does not alter its nature. Thus, if we repeat dollars, they are still dollars; if we repeat yards, they are still yards, &c. Consequently, if the multiplicand is an abstract number, the product will be an abstract number; if money, the product will be money; if barrels, barrels, &c.

2. Every multiplier is to be considered an abstract number. In familiar language it is sometimes said, that the price multiplied by the weight will give the value of an article; and it is often asked how much 25 cents multiplied by 25 cents, &c., will produce. But these are abbreviated expressions, and are liable to convey an erroneous idea, or rather no idea at all. If taken literally, they are absurd; for multiplication is repeating a number or quantity a certain number of times. Now to say that the price is repeated as many times as the given quantity is heavy, or that 25 cents are repeated 25 cents times, is non

But we can multiply the price of 1 pound by a number equal to the number of pounds in the weight of the given article, and the product will be the value of the article. We can also multiply 25 cents by the number 25; that is, repeat 25 cents 25 times, and the product is 625 cents. Construed in this manner, the multiplier becomes an abstract number, and the expressions have a consistent meaning.

sense.

Ex. 3. What will 6 houses cost, at 2341 dollars apiece ?

Write the numbers on the slate as Operation. in the margin, and beginning at the 2341 Multiplicand right hand, proceed thus: 6 times 1

6 Multiplier. unit are 6 units; write the 6 under the Ans. 14046 Dollars. figure multiplied. 6 times 4 tens are 24 tens ; set the 4 or right hand figure under the figure multiplied, and carry the 2 or left hand figure to the next product figure, as in addition. (Art. 52.) 6 times 3 hundreds, are 18 hundreds, and 2 to carry make 20 hundreds; set the 0 under the figure multiplied, and carry the 2 to the next product as before. 6 times 2 thousands are 12 thousands, and 2 to carry make 14 thousands. Since there are no

QUEST.What is it to multiply by any whole number? Obs? Of what denomination is the product? How does this appear? What must every multiplier be considered ? Can you multiply by a given weight, a measure, or a sum of money ?

?

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